尝试划分4block但是效果不好，转为研究访存

Rank [  20]: Calc 0.000003 s

Rank [  35]: Calc 0.000003 s

Rank [  10]: Calc 0.000010 s

Rank [  17]: Calc 0.000005 s

Rank [  32]: Calc 0.000003 s，而且rank不固定，一般就是rank 10 和 rank 52；
但尽管有很多，比前者时间还是少很多
对于第二个Surf_Wave 调用 Interp_points，我发现前四个rank时间最长，比较固定，就是下面四个rank

Rank [  27]: Calc 0.331978 s

Rank [  35]: Calc 0.242219 s

Rank [  28]: Calc 0.242132 s

Rank [  36]: Calc 0.197024 s
因此下面surf_wave是核心

.gitignore

@@ -1,7 +1,6 @@
 __pycache__
 GW150914
 GW150914-origin
-GW150914-mini
 docs
 *.tmp
 

AMSS_NCKU_Input.py

@@ -16,14 +16,12 @@ import numpy
 File_directory   = "GW150914"                    ## output file directory
 Output_directory = "binary_output"               ## binary data file directory
                                                  ## The file directory name should not be too long
-MPI_processes    = 8                             ## number of mpi processes used in the simulation
+MPI_processes    = 64                             ## number of mpi processes used in the simulation
 
 GPU_Calculation  = "no"                          ## Use GPU or not 
                                                  ## (prefer "no" in the current version, because the GPU part may have bugs when integrated in this Python interface)
 CPU_Part         = 1.0
 GPU_Part         = 0.0
-Debug_NaN_Check          = 0                       ## enable NaN checks in compute_rhs_bssn: 0 (off) or 1 (on)
-
 
 #################################################
 

AMSS_NCKU_Input_Mini.py

@@ -1,233 +0,0 @@
-
-#################################################
-##
-## This file provides the input parameters required for numerical relativity.
-## XIAOQU
-## 2024/03/19 --- 2025/09/14
-## Modified for GW150914-mini test case
-##
-#################################################
-
-import numpy    
-
-#################################################
-
-## Setting MPI processes and the output file directory
-
-File_directory   = "GW150914-mini"               ## output file directory
-Output_directory = "binary_output"               ## binary data file directory
-                                                 ## The file directory name should not be too long
-MPI_processes    = 4                             ## number of mpi processes used in the simulation (Reduced for laptop)
-
-GPU_Calculation  = "no"                          ## Use GPU or not 
-                                                 ## (prefer "no" in the current version, because the GPU part may have bugs when integrated in this Python interface)
-CPU_Part         = 1.0
-GPU_Part         = 0.0
-
-#################################################
-
-
-#################################################
-
-## Setting the physical system and numerical method
-
-Symmetry                 = "equatorial-symmetry"   ## Symmetry of System: choose equatorial-symmetry、no-symmetry、octant-symmetry
-Equation_Class           = "BSSN"                  ## Evolution Equation: choose "BSSN", "BSSN-EScalar", "BSSN-EM", "Z4C" 
-                                                   ## If "BSSN-EScalar" is chosen, it is necessary to set other parameters below
-Initial_Data_Method      = "Ansorg-TwoPuncture"    ## initial data method: choose "Ansorg-TwoPuncture", "Lousto-Analytical", "Cao-Analytical", "KerrSchild-Analytical"
-Time_Evolution_Method    = "runge-kutta-45"        ## time evolution method: choose "runge-kutta-45"
-Finite_Diffenence_Method = "4th-order"             ## finite-difference method: choose "2nd-order", "4th-order", "6th-order", "8th-order"
-Debug_NaN_Check          = 0                       ## enable NaN checks in compute_rhs_bssn: 0 (off) or 1 (on)
-
-#################################################
-
-
-#################################################
-
-## Setting the time evolutionary information
-
-Start_Evolution_Time     = 0.0                    ## start evolution time t0
-Final_Evolution_Time     = 100.0                  ## final evolution time t1 (Reduced for quick test)
-Check_Time               = 10.0
-Dump_Time                = 10.0                   ## time inteval dT for dumping binary data
-D2_Dump_Time             = 10.0                   ## dump the ascii data for 2d surface after dT'
-Analysis_Time            = 1.0                    ## dump the puncture position and GW psi4 after dT"
-Evolution_Step_Number    = 10000000               ## stop the calculation after the maximal step number
-Courant_Factor           = 0.5                    ## Courant Factor
-Dissipation              = 0.15                   ## Kreiss-Oliger Dissipation Strength
-
-#################################################
-
-
-#################################################
-
-## Setting the grid structure
-
-basic_grid_set    = "Patch"                          ## grid structure: choose "Patch" or "Shell-Patch"
-grid_center_set   = "Cell"                           ## grid center: chose "Cell" or "Vertex"
-
-grid_level        = 7                                ## total number of AMR grid levels (Reduced from 9)
-static_grid_level = 4                                ## number of AMR static grid levels (Reduced from 5)
-moving_grid_level = grid_level - static_grid_level   ## number of AMR moving grid levels
-
-analysis_level    = 0
-refinement_level  = 3                                ## time refinement start from this grid level
-
-largest_box_xyz_max = [320.0, 320.0, 320.0]          ## scale of the largest box
-                                                     ## not ne cess ary to be cubic for "Patch" grid s tructure
-                                                     ## need to be a cubic box for "Shell-Patch" grid structure
-largest_box_xyz_min = - numpy.array(largest_box_xyz_max)  
-
-static_grid_number = 48                              ## grid points of each static AMR grid (in x direction) (Reduced from 96)
-                                                     ## (grid points in y and z directions are automatically adjusted)
-moving_grid_number = 24                              ## grid points of each moving AMR grid (Reduced from 48)
-shell_grid_number  = [32, 32, 100]                   ## grid points of Shell-Patch grid
-                                                     ## in (phi, theta, r) direction
-devide_factor      = 2.0                             ## resolution between different grid levels dh0/dh1, only support 2.0 now
-                                                     
-
-static_grid_type   = 'Linear'                        ## AMR static grid structure , only supports "Linear"
-moving_grid_type   = 'Linear'                        ## AMR moving grid structure , only supports "Linear"
-
-quarter_sphere_number = 48                           ## grid number of 1/4 s pher ical surface (Reduced from 96)
-                                                     ## (which is needed for evaluating the spherical surface integral)
-
-#################################################
-
-
-#################################################
-
-## Setting the puncture information
-
-puncture_number       = 2                                     
-
-position_BH           = numpy.zeros( (puncture_number, 3) )   
-parameter_BH          = numpy.zeros( (puncture_number, 3) )   
-dimensionless_spin_BH = numpy.zeros( (puncture_number, 3) )   
-momentum_BH           = numpy.zeros( (puncture_number, 3) )   
-
-puncture_data_set     = "Manually"                       ## Method to give Puncture’s positions and momentum
-                                                         ## choose "Manually" or "Automatically-BBH"
-                                                         ## Prefer to choose "Manually", because "Automatically-BBH" is developing now
-
-## initial orbital distance and ellipticity for BBHs system
-## ( needed for "Automatically-BBH" case , not affect the "Manually" case )
-Distance = 10.0
-e0       = 0.0
-
-## black hole parameter (M Q* a*)
-parameter_BH[0] = [ 36.0/(36.0+29.0),  0.0,  +0.31 ]   
-parameter_BH[1] = [ 29.0/(36.0+29.0),  0.0,  -0.46 ]  
-## dimensionless spin in each direction
-dimensionless_spin_BH[0] = [ 0.0,  0.0,  +0.31 ]   
-dimensionless_spin_BH[1] = [ 0.0,  0.0,  -0.46 ]  
-
-## use Brugmann's convention
-##  -----0-----> y
-##   -      +     
-
-#---------------------------------------------
-
-## If puncture_data_set is chosen to be "Manually", it is necessary to set the position and momentum of each puncture manually
-
-## initial position for each puncture
-position_BH[0]  = [  0.0,  10.0*29.0/(36.0+29.0), 0.0 ]  
-position_BH[1]  = [  0.0, -10.0*36.0/(36.0+29.0), 0.0 ] 
-
-## initial mumentum for each puncture
-## (needed for "Manually" case, does not affect the "Automatically-BBH" case)
-momentum_BH[0]  = [ -0.09530152296974252,  -0.00084541526517121,   0.0 ]
-momentum_BH[1]  = [ +0.09530152296974252,  +0.00084541526517121,   0.0 ]
-
-
-#################################################
-
-
-#################################################
-
-## Setting the gravitational wave information
-
-GW_L_max        = 4                      ## maximal L number in gravitational wave
-GW_M_max        = 4                      ## maximal M number in gravitational wave
-Detector_Number = 12                     ## number of dector
-Detector_Rmin   = 50.0                   ## nearest dector distance
-Detector_Rmax   = 160.0                  ## farest dector distance
-
-#################################################
-
-
-#################################################
-
-## Setting the apprent horizon
-
-AHF_Find       = "no"                    ## whether to find the apparent horizon: choose "yes" or "no"
-
-AHF_Find_Every = 24
-AHF_Dump_Time  = 20.0
-
-#################################################
-
-
-#################################################
-
-## Other parameters (testing)
-## Only influence the Equation_Class = "BSSN-EScalar" case
-
-FR_a2     = 3.0        ## f(R) = R + a2 * R^2    
-FR_l2     = 10000.0
-FR_phi0   = 0.00005
-FR_r0     = 120.0
-FR_sigma0 = 8.0
-FR_Choice = 2          ## Choice options: 1 2 3 4 5
-                       ## 1: phi(r) = phi0 * Exp(-(r-r0)**2/sigma0)   
-                       ##    V(r)   = 0
-                       ## 2: phi(r) =  phi0 * a2^2/(1+a2^2)  
-                       ##    V(r)   = Exp(-8*Sqrt(PI/3)*phi(r)) * (1-Exp(4*Sqrt(PI/3)*phi(r)))**2 / (32*PI*a2)
-                       ## 3: Schrodinger-Newton gived by system phi(r) 
-                       ##    V(r)   = Exp(-8*Sqrt(PI/3)*phi(r)) * (1-Exp(4*Sqrt(PI/3)*phi(r)))**2 / (32*PI*a2)
-                       ## 4: phi(r) = phi0 * 0.5 * ( tanh((r+r0)/sigma0) - tanh((r-r0)/sigma0) )  
-                       ##    V(r)   = 0
-                       ##    f(R)   = R + a2*R^2  with a2 = +oo
-                       ## 5: phi(r) = phi0 * Exp(-(r-r0)**2/sigma)   
-                       ##    V(r)   = 0
-
-#################################################
-
-
-#################################################
-
-## Other parameters (testing)
-## (please do not change if not necessary)
-
-boundary_choice = "BAM-choice"     ## Sommerfeld boundary condition : choose "BAM-choice" or "Shibata-choice" 
-                                   ## prefer "BAM-choice"
-
-gauge_choice  = 0                  ## gauge choice
-                                   ## 0: B^i gauge
-                                   ## 1: David's puncture gauge
-                                   ## 2: MB B^i gauge               
-                                   ## 3: RIT B^i gauge
-                                   ## 4: MB beta gauge 
-                                   ## 5: RIT beta gauge 
-                                   ## 6: MGB1 B^i gauge
-                                   ## 7: MGB2 B^i gauge
-                                   ## prefer 0 or 1
-                                   
-tetrad_type  = 2                   ## tetradtype 
-                                   ##  v:r; u: phi; w: theta
-                                   ##      v^a = (x,y,z)
-                                   ## 0: orthonormal order: v,u,w
-                                   ##    v^a = (x,y,z)   
-                                   ##    m = (phi - i theta)/sqrt(2) 
-                                   ##    following Frans, Eq.(8) of  PRD 75, 124018(2007)
-                                   ## 1: orthonormal order: w,u,v
-                                   ##    m = (theta + i phi)/sqrt(2) 
-                                   ##    following Sperhake, Eq.(3.2) of  PRD 85, 124062(2012)    
-                                   ## 2: orthonormal order: v,u,w
-                                   ##    v_a = (x,y,z)
-                                   ##    m = (phi - i theta)/sqrt(2) 
-                                   ##    following Frans, Eq.(8) of  PRD 75, 124018(2007)
-                                   ## this version recommend set to 2
-                                   ## prefer 2
-                                   
-#################################################

AMSS_NCKU_MiniProgram.py

@@ -1,224 +0,0 @@
-##################################################################
-##
-## AMSS-NCKU Numerical Relativity Mini Test Program
-## Author: Assistant (based on Xiaoqu's code)
-## 2026/01/20
-##
-## This script runs a scaled-down version of the GW150914 test case
-## suitable for laptop testing.
-##
-##################################################################
-
-import os
-import shutil
-import sys
-import time
-
-# --- Context Manager for Input File Swapping ---
-class InputFileSwapper:
-    def __init__(self, mini_file="AMSS_NCKU_Input_Mini.py", target_file="AMSS_NCKU_Input.py"):
-        self.mini_file = mini_file
-        self.target_file = target_file
-        self.backup_file = target_file + ".bak"
-        self.swapped = False
-
-    def __enter__(self):
-        print(f"[MiniProgram] Swapping {self.target_file} with {self.mini_file}...")
-        if os.path.exists(self.target_file):
-            shutil.move(self.target_file, self.backup_file)
-        shutil.copy(self.mini_file, self.target_file)
-        self.swapped = True
-        return self
-
-    def __exit__(self, exc_type, exc_value, traceback):
-        if self.swapped:
-            print(f"[MiniProgram] Restoring original {self.target_file}...")
-            os.remove(self.target_file)
-            if os.path.exists(self.backup_file):
-                shutil.move(self.backup_file, self.target_file)
-
-def main():
-    # Use the swapper to ensure all imported modules see the mini configuration
-    with InputFileSwapper():
-        
-        # Import modules AFTER swapping input file
-        try:
-            import AMSS_NCKU_Input as input_data
-            import print_information
-            import setup
-            import numerical_grid
-            import generate_macrodef
-            import makefile_and_run
-            import generate_TwoPuncture_input
-            import renew_puncture_parameter
-            import plot_xiaoqu
-            import plot_GW_strain_amplitude_xiaoqu
-        except ImportError as e:
-            print(f"Error importing modules: {e}")
-            return
-
-        print_information.print_program_introduction()
-        
-        print("\n" + "#"*60)
-        print(" RUNNING MINI TEST CASE: GW150914-mini")
-        print("#"*60 + "\n")
-
-        # --- Directory Setup ---
-        File_directory = os.path.join(input_data.File_directory)
-        
-        if os.path.exists(File_directory):
-            print(f" Output directory '{File_directory}' exists. Removing for mini test...")
-            shutil.rmtree(File_directory, ignore_errors=True)
-        
-        os.mkdir(File_directory)
-        shutil.copy("AMSS_NCKU_Input.py", File_directory) # Copies the current (mini) input
-
-        output_directory = os.path.join(File_directory, "AMSS_NCKU_output")
-        os.mkdir(output_directory)
-
-        binary_results_directory = os.path.join(output_directory, input_data.Output_directory)
-        os.mkdir(binary_results_directory)
-
-        figure_directory = os.path.join(File_directory, "figure")
-        os.mkdir(figure_directory)
-
-        print(" Output directories generated.\n")
-
-        # --- Setup and Input Generation ---
-        setup.print_input_data(File_directory)
-        setup.generate_AMSSNCKU_input()
-        setup.print_puncture_information()
-
-        print("\n Generating AMSS-NCKU input parfile...")
-        numerical_grid.append_AMSSNCKU_cgh_input()
-        
-        print("\n Plotting initial grid...")
-        numerical_grid.plot_initial_grid()
-
-        print("\n Generating macro files...")
-        generate_macrodef.generate_macrodef_h()
-        generate_macrodef.generate_macrodef_fh()
-        
-        # --- Compilation Preparation ---
-        print("\n Preparing to compile and run...")
-        
-        AMSS_NCKU_source_path = "AMSS_NCKU_source"
-        AMSS_NCKU_source_copy = os.path.join(File_directory, "AMSS_NCKU_source_copy")
-
-        if not os.path.exists(AMSS_NCKU_source_path):
-             print(" Error: AMSS_NCKU_source not found! Please run in the project root.")
-             return
-
-        shutil.copytree(AMSS_NCKU_source_path, AMSS_NCKU_source_copy)
-        
-        macrodef_h_path  = os.path.join(File_directory, "macrodef.h") 
-        macrodef_fh_path = os.path.join(File_directory, "macrodef.fh") 
-
-        shutil.copy2(macrodef_h_path,  AMSS_NCKU_source_copy)
-        shutil.copy2(macrodef_fh_path, AMSS_NCKU_source_copy)
-
-        # --- Compilation ---
-        cwd = os.getcwd()
-        os.chdir(AMSS_NCKU_source_copy)
-        
-        print(" Compiling ABE...")
-        makefile_and_run.makefile_ABE()
-        
-        if (input_data.Initial_Data_Method == "Ansorg-TwoPuncture" ): 
-            print(" Compiling TwoPunctureABE...")
-            makefile_and_run.makefile_TwoPunctureABE()
-            
-        os.chdir(cwd)
-
-        # --- Copy Executables ---
-        if (input_data.GPU_Calculation == "no"):
-            ABE_file = os.path.join(AMSS_NCKU_source_copy, "ABE")
-        else:
-            ABE_file = os.path.join(AMSS_NCKU_source_copy, "ABEGPU")
-            
-        if not os.path.exists(ABE_file):
-            print(" Error: ABE executable compilation failed.")
-            return
-            
-        shutil.copy2(ABE_file, output_directory)
-
-        TwoPuncture_file = os.path.join(AMSS_NCKU_source_copy, "TwoPunctureABE")
-        if (input_data.Initial_Data_Method == "Ansorg-TwoPuncture" ):
-            if not os.path.exists(TwoPuncture_file):
-                print(" Error: TwoPunctureABE compilation failed.")
-                return
-            shutil.copy2(TwoPuncture_file, output_directory)
-
-        # --- Execution ---
-        start_time = time.time()
-        
-        if (input_data.Initial_Data_Method == "Ansorg-TwoPuncture" ):
-             print("\n Generating TwoPuncture input...")
-             generate_TwoPuncture_input.generate_AMSSNCKU_TwoPuncture_input()
-             
-             AMSS_NCKU_TwoPuncture_inputfile = 'AMSS-NCKU-TwoPuncture.input'
-             AMSS_NCKU_TwoPuncture_inputfile_path = os.path.join( File_directory, AMSS_NCKU_TwoPuncture_inputfile )
-             shutil.copy2( AMSS_NCKU_TwoPuncture_inputfile_path, os.path.join(output_directory, 'TwoPunctureinput.par') )
-             
-             print(" Running TwoPunctureABE...")
-             os.chdir(output_directory)
-             makefile_and_run.run_TwoPunctureABE()
-             os.chdir(cwd)
-
-        # Update Puncture Parameter
-        renew_puncture_parameter.append_AMSSNCKU_BSSN_input(File_directory, output_directory)
-        
-        AMSS_NCKU_inputfile = 'AMSS-NCKU.input'
-        AMSS_NCKU_inputfile_path = os.path.join(File_directory, AMSS_NCKU_inputfile)
-        shutil.copy2( AMSS_NCKU_inputfile_path, os.path.join(output_directory, 'input.par') )
-        
-        print("\n Input files ready. Launching ABE...")
-        
-        os.chdir(output_directory)
-        makefile_and_run.run_ABE()
-        os.chdir(cwd)
-        
-        end_time = time.time()
-        elapsed_time = end_time - start_time
-        
-        # --- Post-processing ---
-        print("\n Copying output files for inspection...")
-        AMSS_NCKU_error_file_path = os.path.join(binary_results_directory, "setting.par")
-        if os.path.exists(AMSS_NCKU_error_file_path):
-            shutil.copy( AMSS_NCKU_error_file_path, os.path.join(output_directory, "AMSSNCKU_setting_parameter") )
-
-        AMSS_NCKU_error_file_path = os.path.join(binary_results_directory, "Error.log")
-        if os.path.exists(AMSS_NCKU_error_file_path):
-            shutil.copy( AMSS_NCKU_error_file_path, os.path.join(output_directory, "Error.log") )
-
-        for fname in ["bssn_BH.dat", "bssn_ADMQs.dat", "bssn_psi4.dat", "bssn_constraint.dat"]:
-            fpath = os.path.join(binary_results_directory, fname)
-            if os.path.exists(fpath):
-                shutil.copy(fpath, os.path.join(output_directory, fname))
-
-        # --- Plotting ---
-        print("\n Plotting results...")
-        try:
-            plot_xiaoqu.generate_puncture_orbit_plot(   binary_results_directory, figure_directory )
-            plot_xiaoqu.generate_puncture_orbit_plot3D( binary_results_directory, figure_directory )
-            plot_xiaoqu.generate_puncture_distence_plot( binary_results_directory, figure_directory )
-            
-            for i in range(input_data.Detector_Number):
-                plot_xiaoqu.generate_gravitational_wave_psi4_plot( binary_results_directory, figure_directory, i )
-                plot_GW_strain_amplitude_xiaoqu.generate_gravitational_wave_amplitude_plot( binary_results_directory, figure_directory, i )
-
-            for i in range(input_data.Detector_Number):
-                plot_xiaoqu.generate_ADMmass_plot( binary_results_directory, figure_directory, i )
-
-            for i in range(input_data.grid_level):
-                plot_xiaoqu.generate_constraint_check_plot( binary_results_directory, figure_directory, i )
-                
-            plot_xiaoqu.generate_binary_data_plot( binary_results_directory, figure_directory )
-        except Exception as e:
-            print(f"Warning: Plotting failed: {e}")
-
-        print(f"\n Program Cost = {elapsed_time:.2f} Seconds \n")
-        print(" AMSS-NCKU-Python simulation finished (Mini Test).\n")
-
-if __name__ == "__main__":
-    main()

AMSS_NCKU_Program.py

@@ -8,6 +8,14 @@
 ##
 ##################################################################
 
+## Guard against re-execution by multiprocessing child processes.
+## Without this, using 'spawn' or 'forkserver' context would cause every
+## worker to re-run the entire script, spawning exponentially more
+## workers (fork bomb).
+if __name__ != '__main__':
+    import sys as _sys
+    _sys.exit(0)
+
 
 ##################################################################
 
@@ -58,7 +66,8 @@ if os.path.exists(File_directory):
     ## Prompt whether to overwrite the existing directory
     while True:
         try:
-            inputvalue = input()
+            ## inputvalue = input()
+            inputvalue = "continue"
             ## If the user agrees to overwrite, proceed and remove the existing directory
             if ( inputvalue == "continue" ):
                 print( " Continue the calculation !!! " )
@@ -424,26 +433,31 @@ print(
 
 import plot_xiaoqu
 import plot_GW_strain_amplitude_xiaoqu
+from parallel_plot_helper import run_plot_tasks_parallel
+
+plot_tasks = []
 
 ## Plot black hole trajectory
-plot_xiaoqu.generate_puncture_orbit_plot(   binary_results_directory, figure_directory )
-plot_xiaoqu.generate_puncture_orbit_plot3D( binary_results_directory, figure_directory )
+plot_tasks.append( ( plot_xiaoqu.generate_puncture_orbit_plot,   (binary_results_directory, figure_directory) ) )
+plot_tasks.append( ( plot_xiaoqu.generate_puncture_orbit_plot3D, (binary_results_directory, figure_directory) ) )
 
 ## Plot black hole separation vs. time
-plot_xiaoqu.generate_puncture_distence_plot( binary_results_directory, figure_directory )
+plot_tasks.append( ( plot_xiaoqu.generate_puncture_distence_plot, (binary_results_directory, figure_directory) ) )
 
 ## Plot gravitational waveforms (psi4 and strain amplitude)
 for i in range(input_data.Detector_Number):
-    plot_xiaoqu.generate_gravitational_wave_psi4_plot( binary_results_directory, figure_directory, i )
-    plot_GW_strain_amplitude_xiaoqu.generate_gravitational_wave_amplitude_plot( binary_results_directory, figure_directory, i )
+    plot_tasks.append( ( plot_xiaoqu.generate_gravitational_wave_psi4_plot, (binary_results_directory, figure_directory, i) ) )
+    plot_tasks.append( ( plot_GW_strain_amplitude_xiaoqu.generate_gravitational_wave_amplitude_plot, (binary_results_directory, figure_directory, i) ) )
 
 ## Plot ADM mass evolution
 for i in range(input_data.Detector_Number):
-    plot_xiaoqu.generate_ADMmass_plot( binary_results_directory, figure_directory, i )
+    plot_tasks.append( ( plot_xiaoqu.generate_ADMmass_plot, (binary_results_directory, figure_directory, i) ) )
 
 ## Plot Hamiltonian constraint violation over time
 for i in range(input_data.grid_level):
-    plot_xiaoqu.generate_constraint_check_plot( binary_results_directory, figure_directory, i )
+    plot_tasks.append( ( plot_xiaoqu.generate_constraint_check_plot, (binary_results_directory, figure_directory, i) ) )
+
+run_plot_tasks_parallel(plot_tasks)
 
 ## Plot stored binary data
 plot_xiaoqu.generate_binary_data_plot( binary_results_directory, figure_directory )

AMSS_NCKU_source/MPatch.C

@@ -341,8 +341,9 @@ void Patch::Interp_Points(MyList<var> *VarList,
                           double *Shellf, int Symmetry)
 {
   // NOTE: we do not Synchnize variables here, make sure of that before calling this routine
-  int myrank;
+  int myrank, nprocs;
   MPI_Comm_rank(MPI_COMM_WORLD, &myrank);
+  MPI_Comm_size(MPI_COMM_WORLD, &nprocs);
 
   int ordn = 2 * ghost_width;
   MyList<var> *varl;
@@ -354,24 +355,18 @@ void Patch::Interp_Points(MyList<var> *VarList,
     varl = varl->next;
   }
 
-  double *shellf;
-  shellf = new double[NN * num_var];
-  memset(shellf, 0, sizeof(double) * NN * num_var);
+  memset(Shellf, 0, sizeof(double) * NN * num_var);
 
-  // we use weight to monitor code, later some day we can move it for optimization
-  int *weight;
-  weight = new int[NN];
-  memset(weight, 0, sizeof(int) * NN);
-
-  double *DH, *llb, *uub;
-  DH = new double[dim];
+  // owner_rank[j] records which MPI rank owns point j
+  // All ranks traverse the same block list so they all agree on ownership
+  int *owner_rank;
+  owner_rank = new int[NN];
+  for (int j = 0; j < NN; j++)
+    owner_rank[j] = -1;
 
+  double DH[dim], llb[dim], uub[dim];
   for (int i = 0; i < dim; i++)
-  {
     DH[i] = getdX(i);
-  }
-  llb = new double[dim];
-  uub = new double[dim];
 
   for (int j = 0; j < NN; j++) // run along points
   {
@@ -403,12 +398,6 @@ void Patch::Interp_Points(MyList<var> *VarList,
       bool flag = true;
       for (int i = 0; i < dim; i++)
       {
-// NOTE: our dividing structure is (exclude ghost)
-// -1 0
-//       1  2
-// so (0,1) does not belong to any part for vertex structure
-// here we put (0,0.5) to left part and (0.5,1) to right part
-// BUT for cell structure the bbox is (-1.5,0.5) and (0.5,2.5), there is no missing region at all
 #ifdef Vertex
 #ifdef Cell
 #error Both Cell and Vertex are defined
@@ -433,6 +422,7 @@ void Patch::Interp_Points(MyList<var> *VarList,
       if (flag)
       {
         notfind = false;
+        owner_rank[j] = BP->rank;
         if (myrank == BP->rank)
         {
           //---> interpolation
@@ -440,14 +430,11 @@ void Patch::Interp_Points(MyList<var> *VarList,
           int k = 0;
           while (varl) // run along variables
           {
-            //              shellf[j*num_var+k] = Parallel::global_interp(dim,BP->shape,BP->X,BP->fgfs[varl->data->sgfn],
-            //	  		                                    pox,ordn,varl->data->SoA,Symmetry);
-            f_global_interp(BP->shape, BP->X[0], BP->X[1], BP->X[2], BP->fgfs[varl->data->sgfn], shellf[j * num_var + k],
+            f_global_interp(BP->shape, BP->X[0], BP->X[1], BP->X[2], BP->fgfs[varl->data->sgfn], Shellf[j * num_var + k],
                             pox[0], pox[1], pox[2], ordn, varl->data->SoA, Symmetry);
             varl = varl->next;
             k++;
           }
-          weight[j] = 1;
         }
       }
       if (Bp == ble)
@@ -455,62 +442,128 @@ void Patch::Interp_Points(MyList<var> *VarList,
       Bp = Bp->next;
     }
   }
-
-  MPI_Allreduce(shellf, Shellf, NN * num_var, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
-  int *Weight;
-  Weight = new int[NN];
-  MPI_Allreduce(weight, Weight, NN, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
-
-  //  misc::tillherecheck("print me");
-
-  for (int i = 0; i < NN; i++)
+  // Replace MPI_Allreduce with per-owner MPI_Bcast:
+  // Group consecutive points by owner rank and broadcast each group.
+  // Since each point's data is non-zero only on the owner rank,
+  // Bcast from owner is equivalent to Allreduce(MPI_SUM) but much cheaper.
   {
-    if (Weight[i] > 1)
+    int j = 0;
+    while (j < NN)
+    {
+      int cur_owner = owner_rank[j];
+      if (cur_owner < 0)
       {
         if (myrank == 0)
-        cout << "WARNING: Patch::Interp_Points meets multiple weight" << endl;
-      for (int j = 0; j < num_var; j++)
-        Shellf[j + i * num_var] = Shellf[j + i * num_var] / Weight[i];
-    }
-    else if (Weight[i] == 0 && myrank == 0)
         {
           cout << "ERROR: Patch::Interp_Points fails to find point (";
-      for (int j = 0; j < dim; j++)
+          for (int d = 0; d < dim; d++)
           {
-        cout << XX[j][i];
-        if (j < dim - 1)
+            cout << XX[d][j];
+            if (d < dim - 1)
               cout << ",";
             else
               cout << ")";
           }
           cout << " on Patch (";
-      for (int j = 0; j < dim; j++)
+          for (int d = 0; d < dim; d++)
           {
-        cout << bbox[j] << "+" << lli[j] * getdX(j);
-        if (j < dim - 1)
+            cout << bbox[d] << "+" << lli[d] * DH[d];
+            if (d < dim - 1)
               cout << ",";
             else
               cout << ")--";
           }
           cout << "(";
-      for (int j = 0; j < dim; j++)
+          for (int d = 0; d < dim; d++)
           {
-        cout << bbox[dim + j] << "-" << uui[j] * getdX(j);
-        if (j < dim - 1)
+            cout << bbox[dim + d] << "-" << uui[d] * DH[d];
+            if (d < dim - 1)
               cout << ",";
             else
               cout << ")" << endl;
           }
-#if 0
-       checkBlock();
-#else
-      cout << "splited domains:" << endl;
+          MPI_Abort(MPI_COMM_WORLD, 1);
+        }
+        j++;
+        continue;
+      }
+      // Find contiguous run of points with the same owner
+      int jstart = j;
+      while (j < NN && owner_rank[j] == cur_owner)
+        j++;
+      int count = (j - jstart) * num_var;
+      MPI_Bcast(Shellf + jstart * num_var, count, MPI_DOUBLE, cur_owner, MPI_COMM_WORLD);
+    }
+  }
+
+  delete[] owner_rank;
+}
+void Patch::Interp_Points(MyList<var> *VarList,
+                          int NN, double **XX,
+                          double *Shellf, int Symmetry,
+                          int Nmin_consumer, int Nmax_consumer)
+{
+  // Targeted point-to-point overload: each owner sends each point only to
+  // the one rank that needs it for integration (consumer), reducing
+  // communication volume by ~nprocs times compared to the Bcast version.
+  /*
+  double t_calc_end, t_calc_total = 0;
+  double t_calc_start = MPI_Wtime();*/
+  int myrank, nprocs;
+  MPI_Comm_rank(MPI_COMM_WORLD, &myrank);
+  MPI_Comm_size(MPI_COMM_WORLD, &nprocs);
+
+  int ordn = 2 * ghost_width;
+  MyList<var> *varl;
+  int num_var = 0;
+  varl = VarList;
+  while (varl)
   {
+    num_var++;
+    varl = varl->next;
+  }
+
+  memset(Shellf, 0, sizeof(double) * NN * num_var);
+
+  // owner_rank[j] records which MPI rank owns point j
+  int *owner_rank;
+  owner_rank = new int[NN];
+  for (int j = 0; j < NN; j++)
+    owner_rank[j] = -1;
+
+  double DH[dim], llb[dim], uub[dim];
+  for (int i = 0; i < dim; i++)
+    DH[i] = getdX(i);
+
+  // --- Interpolation phase (identical to original) ---
+  for (int j = 0; j < NN; j++)
+  {
+    double pox[dim];
+    for (int i = 0; i < dim; i++)
+    {
+      pox[i] = XX[i][j];
+      if (myrank == 0 && (XX[i][j] < bbox[i] + lli[i] * DH[i] || XX[i][j] > bbox[dim + i] - uui[i] * DH[i]))
+      {
+        cout << "Patch::Interp_Points: point (";
+        for (int k = 0; k < dim; k++)
+        {
+          cout << XX[k][j];
+          if (k < dim - 1)
+            cout << ",";
+          else
+            cout << ") is out of current Patch." << endl;
+        }
+        MPI_Abort(MPI_COMM_WORLD, 1);
+      }
+    }
+
     MyList<Block> *Bp = blb;
-        while (Bp)
+    bool notfind = true;
+    while (notfind && Bp)
     {
       Block *BP = Bp->data;
 
+      bool flag = true;
       for (int i = 0; i < dim; i++)
       {
 #ifdef Vertex
@@ -527,32 +580,252 @@ void Patch::Interp_Points(MyList<var> *VarList,
 #error Not define Vertex nor Cell
 #endif
 #endif
-          }
-          cout << "(";
-          for (int j = 0; j < dim; j++)
+        if (XX[i][j] - llb[i] < -DH[i] / 2 || XX[i][j] - uub[i] > DH[i] / 2)
         {
-            cout << llb[j] << ":" << uub[j];
-            if (j < dim - 1)
-              cout << ",";
-            else
-              cout << ")" << endl;
+          flag = false;
+          break;
+        }
+      }
+
+      if (flag)
+      {
+        notfind = false;
+        owner_rank[j] = BP->rank;
+        if (myrank == BP->rank)
+        {
+          varl = VarList;
+          int k = 0;
+          while (varl)
+          {
+            f_global_interp(BP->shape, BP->X[0], BP->X[1], BP->X[2], BP->fgfs[varl->data->sgfn], Shellf[j * num_var + k],
+                            pox[0], pox[1], pox[2], ordn, varl->data->SoA, Symmetry);
+            varl = varl->next;
+            k++;
+          }
+        }
       }
       if (Bp == ble)
         break;
       Bp = Bp->next;
     }
   }
-#endif
+  /*
+      t_calc_end = MPI_Wtime();
+      t_calc_total = t_calc_end - t_calc_start;*/
+  // --- Error check for unfound points ---
+  for (int j = 0; j < NN; j++)
+  {
+    if (owner_rank[j] < 0 && myrank == 0)
+    {
+      cout<<owner_rank[j-1]<<endl;
+      cout << "ERROR: Patch::Interp_Points fails to find point (";
+      for (int d = 0; d < dim; d++)
+      {
+        cout << XX[d][j];
+        if (d < dim - 1)
+          cout << ",";
+        else
+          cout << ")";
+      }
+      cout << " on Patch (";
+      for (int d = 0; d < dim; d++)
+      {
+        cout << bbox[d] << "+" << lli[d] * DH[d];
+        if (d < dim - 1)
+          cout << ",";
+        else
+          cout << ")--";
+      }
+      cout << "(";
+      for (int d = 0; d < dim; d++)
+      {
+        cout << bbox[dim + d] << "-" << uui[d] * DH[d];
+        if (d < dim - 1)
+          cout << ",";
+        else
+          cout << ")" << endl;
+      }
       MPI_Abort(MPI_COMM_WORLD, 1);
     }
   }
 
-  delete[] shellf;
-  delete[] weight;
-  delete[] Weight;
-  delete[] DH;
-  delete[] llb;
-  delete[] uub;
+  // --- Targeted point-to-point communication phase ---
+  // Compute consumer_rank[j] using the same deterministic formula as surface_integral
+  int *consumer_rank = new int[NN];
+  {
+    int mp = NN / nprocs;
+    int Lp = NN - nprocs * mp;
+    for (int j = 0; j < NN; j++)
+    {
+      if (j < Lp * (mp + 1))
+        consumer_rank[j] = j / (mp + 1);
+      else
+        consumer_rank[j] = Lp + (j - Lp * (mp + 1)) / mp;
+    }
+  }
+
+  // Count sends and recvs per rank
+  int *send_count = new int[nprocs];
+  int *recv_count = new int[nprocs];
+  memset(send_count, 0, sizeof(int) * nprocs);
+  memset(recv_count, 0, sizeof(int) * nprocs);
+
+  for (int j = 0; j < NN; j++)
+  {
+    int own = owner_rank[j];
+    int con = consumer_rank[j];
+    if (own == con)
+      continue; // local — no communication needed
+    if (own == myrank)
+      send_count[con]++;
+    if (con == myrank)
+      recv_count[own]++;
+  }
+
+  // Build send buffers: for each destination rank, pack (index, data) pairs
+  // Each entry: 1 int (point index j) + num_var doubles
+  int total_send = 0, total_recv = 0;
+  int *send_offset = new int[nprocs];
+  int *recv_offset = new int[nprocs];
+  for (int r = 0; r < nprocs; r++)
+  {
+    send_offset[r] = total_send;
+    total_send += send_count[r];
+    recv_offset[r] = total_recv;
+    total_recv += recv_count[r];
+  }
+
+  // Pack send buffers: each message contains (j, data[0..num_var-1]) per point
+  int stride = 1 + num_var; // 1 double for index + num_var doubles for data
+  double *sendbuf = new double[total_send * stride];
+  double *recvbuf = new double[total_recv * stride];
+
+  // Temporary counters for packing
+  int *pack_pos = new int[nprocs];
+  memset(pack_pos, 0, sizeof(int) * nprocs);
+
+  for (int j = 0; j < NN; j++)
+  {
+    int own = owner_rank[j];
+    int con = consumer_rank[j];
+    if (own != myrank || con == myrank)
+      continue;
+    int pos = (send_offset[con] + pack_pos[con]) * stride;
+    sendbuf[pos] = (double)j; // point index
+    for (int v = 0; v < num_var; v++)
+      sendbuf[pos + 1 + v] = Shellf[j * num_var + v];
+    pack_pos[con]++;
+  }
+
+  // Post non-blocking recvs and sends
+  int n_req = 0;
+  for (int r = 0; r < nprocs; r++)
+  {
+    if (recv_count[r] > 0) n_req++;
+    if (send_count[r] > 0) n_req++;
+  }
+
+  MPI_Request *reqs = new MPI_Request[n_req];
+  int req_idx = 0;
+
+  for (int r = 0; r < nprocs; r++)
+  {
+    if (recv_count[r] > 0)
+    {
+      MPI_Irecv(recvbuf + recv_offset[r] * stride,
+                recv_count[r] * stride, MPI_DOUBLE,
+                r, 0, MPI_COMM_WORLD, &reqs[req_idx++]);
+    }
+  }
+  for (int r = 0; r < nprocs; r++)
+  {
+    if (send_count[r] > 0)
+    {
+      MPI_Isend(sendbuf + send_offset[r] * stride,
+                send_count[r] * stride, MPI_DOUBLE,
+                r, 0, MPI_COMM_WORLD, &reqs[req_idx++]);
+    }
+  }
+
+  if (n_req > 0)
+    MPI_Waitall(n_req, reqs, MPI_STATUSES_IGNORE);
+
+  // Unpack recv buffers into Shellf
+  for (int i = 0; i < total_recv; i++)
+  {
+    int pos = i * stride;
+    int j = (int)recvbuf[pos];
+    for (int v = 0; v < num_var; v++)
+      Shellf[j * num_var + v] = recvbuf[pos + 1 + v];
+  }
+
+  delete[] reqs;
+  delete[] sendbuf;
+  delete[] recvbuf;
+  delete[] pack_pos;
+  delete[] send_offset;
+  delete[] recv_offset;
+  delete[] send_count;
+  delete[] recv_count;
+  delete[] consumer_rank;
+  delete[] owner_rank;
+  /*
+  // 4. 汇总并输出真正干活最慢的 Top 4
+  struct RankStats {
+    int rank;
+    double calc_time; // 净计算时间
+  };
+
+  // 创建当前进程的统计数据
+  RankStats local_stat;
+  local_stat.rank = myrank;
+  local_stat.calc_time = t_calc_total;
+
+  // 为所有进程的统计数据分配内存
+  RankStats *all_stats = nullptr;
+  if (myrank == 0) {
+    all_stats = new RankStats[nprocs];
+  }
+
+  // 使用MPI_Gather收集所有进程的数据到rank 0
+  MPI_Gather(&local_stat, sizeof(RankStats), MPI_BYTE,
+             all_stats, sizeof(RankStats), MPI_BYTE,
+             0, MPI_COMM_WORLD);
+
+  // 准备输出前4个rank的信息（所有rank都参与，确保广播后一致）
+  int top10_ranks[10] = { -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 };
+  double top10_times[10] = { 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 };
+  int num_top10 = 0;
+
+  if (myrank == 0) {
+    // 按 calc_time（净计算时间）排序
+    std::sort(all_stats, all_stats + nprocs, [](const RankStats& a, const RankStats& b) {
+        return a.calc_time > b.calc_time;
+    });
+
+    // 取前4个
+    num_top10 = (nprocs < 10) ? nprocs : 10;
+    for (int i = 0; i < num_top10; i++) {
+      top10_ranks[i] = all_stats[i].rank;
+      top10_times[i] = all_stats[i].calc_time;
+    }
+
+    printf("\n--- Top %d Ranks by ACTIVE COMPUTATION (CPU Time) ---\n", num_top10);
+    for (int i = 0; i < num_top10; i++) {
+      printf("Rank [%4d]: Calc %.6f s\n", top10_ranks[i], top10_times[i]);
+    }
+
+    // 清理分配的内存
+    delete[] all_stats;
+  }
+
+  // 广播前4个rank的信息给所有进程
+  MPI_Bcast(&num_top10, 1, MPI_INT, 0, MPI_COMM_WORLD);
+  if (num_top10 > 0) {
+    MPI_Bcast(top10_ranks, 10, MPI_INT, 0, MPI_COMM_WORLD);
+    MPI_Bcast(top10_times, 10, MPI_DOUBLE, 0, MPI_COMM_WORLD);
+  }
+*/
 }
 void Patch::Interp_Points(MyList<var> *VarList,
                           int NN, double **XX,
@@ -573,24 +846,22 @@ void Patch::Interp_Points(MyList<var> *VarList,
     varl = varl->next;
   }
 
-  double *shellf;
-  shellf = new double[NN * num_var];
-  memset(shellf, 0, sizeof(double) * NN * num_var);
+  memset(Shellf, 0, sizeof(double) * NN * num_var);
 
-  // we use weight to monitor code, later some day we can move it for optimization
-  int *weight;
-  weight = new int[NN];
-  memset(weight, 0, sizeof(int) * NN);
+  // owner_rank[j] stores the global rank that owns point j
+  int *owner_rank;
+  owner_rank = new int[NN];
+  for (int j = 0; j < NN; j++)
+    owner_rank[j] = -1;
 
-  double *DH, *llb, *uub;
-  DH = new double[dim];
+  // Build global-to-local rank translation for Comm_here
+  MPI_Group world_group, local_group;
+  MPI_Comm_group(MPI_COMM_WORLD, &world_group);
+  MPI_Comm_group(Comm_here, &local_group);
 
+  double DH[dim], llb[dim], uub[dim];
   for (int i = 0; i < dim; i++)
-  {
     DH[i] = getdX(i);
-  }
-  llb = new double[dim];
-  uub = new double[dim];
 
   for (int j = 0; j < NN; j++) // run along points
   {
@@ -622,12 +893,6 @@ void Patch::Interp_Points(MyList<var> *VarList,
       bool flag = true;
       for (int i = 0; i < dim; i++)
       {
-// NOTE: our dividing structure is (exclude ghost)
-// -1 0
-//       1  2
-// so (0,1) does not belong to any part for vertex structure
-// here we put (0,0.5) to left part and (0.5,1) to right part
-// BUT for cell structure the bbox is (-1.5,0.5) and (0.5,2.5), there is no missing region at all
 #ifdef Vertex
 #ifdef Cell
 #error Both Cell and Vertex are defined
@@ -652,6 +917,7 @@ void Patch::Interp_Points(MyList<var> *VarList,
       if (flag)
       {
         notfind = false;
+        owner_rank[j] = BP->rank;
         if (myrank == BP->rank)
         {
           //---> interpolation
@@ -659,14 +925,11 @@ void Patch::Interp_Points(MyList<var> *VarList,
           int k = 0;
           while (varl) // run along variables
           {
-            //              shellf[j*num_var+k] = Parallel::global_interp(dim,BP->shape,BP->X,BP->fgfs[varl->data->sgfn],
-            //	  		                                    pox,ordn,varl->data->SoA,Symmetry);
-            f_global_interp(BP->shape, BP->X[0], BP->X[1], BP->X[2], BP->fgfs[varl->data->sgfn], shellf[j * num_var + k],
+            f_global_interp(BP->shape, BP->X[0], BP->X[1], BP->X[2], BP->fgfs[varl->data->sgfn], Shellf[j * num_var + k],
                             pox[0], pox[1], pox[2], ordn, varl->data->SoA, Symmetry);
             varl = varl->next;
             k++;
           }
-          weight[j] = 1;
         }
       }
       if (Bp == ble)
@@ -675,97 +938,35 @@ void Patch::Interp_Points(MyList<var> *VarList,
     }
   }
 
-  MPI_Allreduce(shellf, Shellf, NN * num_var, MPI_DOUBLE, MPI_SUM, Comm_here);
-  int *Weight;
-  Weight = new int[NN];
-  MPI_Allreduce(weight, Weight, NN, MPI_INT, MPI_SUM, Comm_here);
+  // Collect unique global owner ranks and translate to local ranks in Comm_here
+  // Then broadcast each owner's points via MPI_Bcast on Comm_here
+  {
+    int j = 0;
+    while (j < NN)
+    {
+      int cur_owner_global = owner_rank[j];
+      if (cur_owner_global < 0)
+      {
+        // Point not found — skip (error check disabled for sub-communicator levels)
+        j++;
+        continue;
+      }
+      // Translate global rank to local rank in Comm_here
+      int cur_owner_local;
+      MPI_Group_translate_ranks(world_group, 1, &cur_owner_global, local_group, &cur_owner_local);
 
-  //  misc::tillherecheck("print me");
-  //  if(lmyrank == 0) cout<<"myrank = "<<myrank<<"print me"<<endl;
-
-  for (int i = 0; i < NN; i++)
-  {
-    if (Weight[i] > 1)
-    {
-      if (lmyrank == 0)
-        cout << "WARNING: Patch::Interp_Points meets multiple weight" << endl;
-      for (int j = 0; j < num_var; j++)
-        Shellf[j + i * num_var] = Shellf[j + i * num_var] / Weight[i];
+      // Find contiguous run of points with the same owner
+      int jstart = j;
+      while (j < NN && owner_rank[j] == cur_owner_global)
+        j++;
+      int count = (j - jstart) * num_var;
+      MPI_Bcast(Shellf + jstart * num_var, count, MPI_DOUBLE, cur_owner_local, Comm_here);
     }
-#if 0 // for not involved levels, this may fail     
-     else if(Weight[i] == 0 && lmyrank == 0)
-     {
-       cout<<"ERROR: Patch::Interp_Points fails to find point (";
-       for(int j=0;j<dim;j++)
-       {
-	  cout<<XX[j][i];
-	  if(j<dim-1) cout<<",";
-	  else        cout<<")";
-       }
-       cout<<" on Patch (";
-       for(int j=0;j<dim;j++)
-       {
-	  cout<<bbox[j]<<"+"<<lli[j]*getdX(j);
-	  if(j<dim-1) cout<<",";
-	  else        cout<<")--";
-       }
-       cout<<"(";
-       for(int j=0;j<dim;j++)
-       {
-	  cout<<bbox[dim+j]<<"-"<<uui[j]*getdX(j);
-	  if(j<dim-1) cout<<",";
-	  else        cout<<")"<<endl;
-       }
-#if 0
-       checkBlock();
-#else
-  cout<<"splited domains:"<<endl;
-  {
-     MyList<Block> *Bp=blb;
-     while(Bp)
-     {
-	Block *BP=Bp->data;
-
-	for(int i=0;i<dim;i++)
-	{
-#ifdef Vertex
-#ifdef Cell
-#error Both Cell and Vertex are defined
-#endif    
-          llb[i] = (feq(BP->bbox[i]    ,bbox[i]    ,DH[i]/2)) ? BP->bbox[i]+lli[i]*DH[i]     : BP->bbox[i]    +(ghost_width-0.5)*DH[i];
-          uub[i] = (feq(BP->bbox[dim+i],bbox[dim+i],DH[i]/2)) ? BP->bbox[dim+i]-uui[i]*DH[i] : BP->bbox[dim+i]-(ghost_width-0.5)*DH[i];
-#else
-#ifdef Cell
-          llb[i] = (feq(BP->bbox[i]    ,bbox[i]    ,DH[i]/2)) ? BP->bbox[i]+lli[i]*DH[i]     : BP->bbox[i]    +ghost_width*DH[i];
-          uub[i] = (feq(BP->bbox[dim+i],bbox[dim+i],DH[i]/2)) ? BP->bbox[dim+i]-uui[i]*DH[i] : BP->bbox[dim+i]-ghost_width*DH[i];
-#else
-#error Not define Vertex nor Cell
-#endif
-#endif 
-	}       
-       cout<<"(";
-       for(int j=0;j<dim;j++)
-       {
-	  cout<<llb[j]<<":"<<uub[j];
-	  if(j<dim-1) cout<<",";
-	  else        cout<<")"<<endl;
-       }
-	if(Bp == ble) break;
-	Bp=Bp->next;
-     }
-  }
-#endif       
-       MPI_Abort(MPI_COMM_WORLD,1);
-     }
-#endif
   }
 
-  delete[] shellf;
-  delete[] weight;
-  delete[] Weight;
-  delete[] DH;
-  delete[] llb;
-  delete[] uub;
+  MPI_Group_free(&world_group);
+  MPI_Group_free(&local_group);
+  delete[] owner_rank;
 }
 void Patch::checkBlock()
 {

AMSS_NCKU_source/MPatch.h

@@ -39,6 +39,10 @@ public:
 
    bool Find_Point(double *XX);
 
+   void Interp_Points(MyList<var> *VarList,
+                      int NN, double **XX,
+                      double *Shellf, int Symmetry,
+                      int Nmin_consumer, int Nmax_consumer);
    void Interp_Points(MyList<var> *VarList,
                       int NN, double **XX,
                       double *Shellf, int Symmetry, MPI_Comm Comm_here);

AMSS_NCKU_source/NullShellPatch.h

@@ -24,6 +24,7 @@ using namespace std;
 #endif
 
 #include <mpi.h>
+#include <memory.h>
 #include "MyList.h"
 #include "Block.h"
 #include "Parallel.h"

AMSS_NCKU_source/Parallel.h

@@ -11,7 +11,7 @@
 #include <cmath>
 #include <new>
 using namespace std;
-
+#include <memory.h>
 #include "Parallel_bam.h"
 #include "var.h"
 #include "MPatch.h"
@@ -32,6 +32,25 @@ namespace Parallel
   int partition2(int *nxy, int split_size, int *min_width, int cpusize, int *shape); // special for 2 diemnsions
   int partition3(int *nxyz, int split_size, int *min_width, int cpusize, int *shape);
   MyList<Block> *distribute(MyList<Patch> *PatchLIST, int cpusize, int ingfsi, int fngfs, bool periodic, int nodes = 0); // produce corresponding Blocks
+
+  MyList<Block> *distribute_hard(MyList<Patch> *PatchLIST, int cpusize, int ingfsi, int fngfs, bool periodic, int nodes = 0); // produce corresponding Blocks
+  Block* splitHotspotBlock(MyList<Block>* &BlL, int _dim, 
+                                 int ib0_orig, int ib3_orig, 
+                                 int jb1_orig, int jb4_orig, 
+                                 int kb2_orig, int kb5_orig, 
+                                 Patch* PP, int r_1, int r_2, 
+                                 int ingfsi, int fngfsi, bool periodic,
+                                 Block* &split_first_block, Block* &split_last_block);
+
+Block* splitHotspotBlock(MyList<Block>* &BlL, int _dim, 
+                                 int ib0_orig, int ib3_orig, 
+                                 int jb1_orig, int jb4_orig, 
+                                 int kb2_orig, int kb5_orig, 
+                                 Patch* PP, int r_1, int r_2, int r_3, int r_4,
+                                 int ingfsi, int fngfsi, bool periodic,
+                                 Block* &split_first_block, Block* &split_last_block);
+  Block* createMappedBlock(MyList<Block>* &BlL, int _dim, int* shape, double* bbox,
+                        int block_id, int ingfsi, int fngfsi, int lev); 
   void KillBlocks(MyList<Patch> *PatchLIST);
 
   void setfunction(MyList<Block> *BlL, var *vn, double func(double x, double y, double z));
@@ -81,6 +100,43 @@ namespace Parallel
                    int Symmetry);
   void Sync(Patch *Pat, MyList<var> *VarList, int Symmetry);
   void Sync(MyList<Patch> *PatL, MyList<var> *VarList, int Symmetry);
+  void Sync_merged(MyList<Patch> *PatL, MyList<var> *VarList, int Symmetry);
+
+  struct SyncCache {
+    bool valid;
+    int cpusize;
+    MyList<gridseg> **combined_src;
+    MyList<gridseg> **combined_dst;
+    int *send_lengths;
+    int *recv_lengths;
+    double **send_bufs;
+    double **recv_bufs;
+    int *send_buf_caps;
+    int *recv_buf_caps;
+    MPI_Request *reqs;
+    MPI_Status *stats;
+    int max_reqs;
+    bool lengths_valid;
+    SyncCache();
+    void invalidate();
+    void destroy();
+  };
+
+  void Sync_cached(MyList<Patch> *PatL, MyList<var> *VarList, int Symmetry, SyncCache &cache);
+  void transfer_cached(MyList<gridseg> **src, MyList<gridseg> **dst,
+                       MyList<var> *VarList1, MyList<var> *VarList2,
+                       int Symmetry, SyncCache &cache);
+
+  struct AsyncSyncState {
+    int req_no;
+    bool active;
+    AsyncSyncState() : req_no(0), active(false) {}
+  };
+
+  void Sync_start(MyList<Patch> *PatL, MyList<var> *VarList, int Symmetry,
+                  SyncCache &cache, AsyncSyncState &state);
+  void Sync_finish(SyncCache &cache, AsyncSyncState &state,
+                   MyList<var> *VarList, int Symmetry);
   void OutBdLow2Hi(Patch *Patc, Patch *Patf,
                    MyList<var> *VarList1 /* source */, MyList<var> *VarList2 /* target */,
                    int Symmetry);
@@ -93,6 +149,15 @@ namespace Parallel
   void OutBdLow2Himix(MyList<Patch> *PatcL, MyList<Patch> *PatfL,
                       MyList<var> *VarList1 /* source */, MyList<var> *VarList2 /* target */,
                       int Symmetry);
+  void Restrict_cached(MyList<Patch> *PatcL, MyList<Patch> *PatfL,
+                       MyList<var> *VarList1, MyList<var> *VarList2,
+                       int Symmetry, SyncCache &cache);
+  void OutBdLow2Hi_cached(MyList<Patch> *PatcL, MyList<Patch> *PatfL,
+                          MyList<var> *VarList1, MyList<var> *VarList2,
+                          int Symmetry, SyncCache &cache);
+  void OutBdLow2Himix_cached(MyList<Patch> *PatcL, MyList<Patch> *PatfL,
+                             MyList<var> *VarList1, MyList<var> *VarList2,
+                             int Symmetry, SyncCache &cache);
   void Prolong(Patch *Patc, Patch *Patf,
                MyList<var> *VarList1 /* source */, MyList<var> *VarList2 /* target */,
                int Symmetry);
@@ -162,6 +227,18 @@ namespace Parallel
 #if (PSTR == 1 || PSTR == 2 || PSTR == 3)
   MyList<Block> *distribute(MyList<Patch> *PatchLIST, int cpusize, int ingfsi, int fngfsi,
                             bool periodic, int start_rank, int end_rank, int nodes = 0);
+  
+  // Redistribute blocks with time statistics for load balancing
+  MyList<Block> *distribute(MyList<Patch> *PatchLIST, MyList<Block> *OldBlockL,
+                            int cpusize, int ingfsi, int fngfsi,
+                            bool periodic, int start_rank, int end_rank, int nodes = 0);
 #endif
-}
-#endif /*PARALLEL_H */
+
+  // Dynamic load balancing: split blocks for heavy ranks
+    void split_heavy_blocks(MyList<Patch> *PatL, int *heavy_ranks, int num_heavy,
+                            int split_factor, int cpusize, int ingfsi, int fngfsi);
+
+    // Check if load balancing is needed based on interpolation times
+    bool check_load_balance_need(double *rank_times, int nprocs, int &num_heavy, int *heavy_ranks);
+  }
+  #endif /*PARALLEL_H */

AMSS_NCKU_source/TwoPunctures.h

@@ -1,7 +1,8 @@
-
 #ifndef TWO_PUNCTURES_H
 #define TWO_PUNCTURES_H
 
+#include <omp.h>
+
 #define StencilSize 19
 #define N_PlaneRelax 1
 #define NRELAX 200
@@ -42,6 +43,18 @@ private:
 
        int ntotal;
 
+       // ===== Precomputed spectral derivative matrices =====
+       double *D1_A, *D2_A;
+       double *D1_B, *D2_B;
+       double *DF1_phi, *DF2_phi;
+
+       // ===== Pre-allocated workspace for LineRelax (per-thread) =====
+       int max_threads;
+       double **ws_diag_be, **ws_e_be, **ws_f_be, **ws_b_be, **ws_x_be;
+       double **ws_l_be, **ws_u_be, **ws_d_be, **ws_y_be;
+       double **ws_diag_al, **ws_e_al, **ws_f_al, **ws_b_al, **ws_x_al;
+       double **ws_l_al, **ws_u_al, **ws_d_al, **ws_y_al;
+
        struct parameters
        {
               int nvar, n1, n2, n3;
@@ -58,6 +71,28 @@ public:
                     int Newtonmaxit);
        ~TwoPunctures();
 
+       // 02/07: New/modified methods
+       void allocate_workspace();
+       void free_workspace();
+       void precompute_derivative_matrices();
+       void build_cheb_deriv_matrices(int n, double *D1, double *D2);
+       void build_fourier_deriv_matrices(int N, double *DF1, double *DF2);
+       void Derivatives_AB3_MatMul(int nvar, int n1, int n2, int n3, derivs v);
+       void ThomasAlgorithm_ws(int N, double *b, double *a, double *c, double *x, double *q,
+                                double *l, double *u_ws, double *d, double *y);
+       void LineRelax_be_omp(double *dv,
+                         int const i, int const k, int const nvar,
+                         int const n1, int const n2, int const n3,
+                         double const *rhs, int const *ncols, int **cols,
+                         double **JFD, int tid);
+       void LineRelax_al_omp(double *dv,
+                         int const j, int const k, int const nvar,
+                         int const n1, int const n2, int const n3,
+                         double const *rhs, int const *ncols,
+                         int **cols, double **JFD, int tid);
+       void relax_omp(double *dv, int const nvar, int const n1, int const n2, int const n3,
+                  double const *rhs, int const *ncols, int **cols, double **JFD);
+
        void Solve();
        void set_initial_guess(derivs v);
        int index(int i, int j, int k, int l, int a, int b, int c, int d);
@@ -116,23 +151,11 @@ public:
        double BY_KKofxyz(double x, double y, double z);
        void SetMatrix_JFD(int nvar, int n1, int n2, int n3, derivs u, int *ncols, int **cols, double **Matrix);
        void J_times_dv(int nvar, int n1, int n2, int n3, derivs dv, double *Jdv, derivs u);
-       void relax(double *dv, int const nvar, int const n1, int const n2, int const n3,
-                  double const *rhs, int const *ncols, int **cols, double **JFD);
-       void LineRelax_be(double *dv,
-                         int const i, int const k, int const nvar,
-                         int const n1, int const n2, int const n3,
-                         double const *rhs, int const *ncols, int **cols,
-                         double **JFD);
        void JFD_times_dv(int i, int j, int k, int nvar, int n1, int n2,
                          int n3, derivs dv, derivs u, double *values);
        void LinEquations(double A, double B, double X, double R,
                          double x, double r, double phi,
                          double y, double z, derivs dU, derivs U, double *values);
-       void LineRelax_al(double *dv,
-                         int const j, int const k, int const nvar,
-                         int const n1, int const n2, int const n3,
-                         double const *rhs, int const *ncols,
-                         int **cols, double **JFD);
        void ThomasAlgorithm(int N, double *b, double *a, double *c, double *x, double *q);
        void Save(char *fname);
        // provided by Vasileios Paschalidis (vpaschal@illinois.edu)

AMSS_NCKU_source/bssn_class.C

@@ -730,6 +730,12 @@ void bssn_class::Initialize()
     PhysTime = StartTime;
     Setup_Black_Hole_position();
   }
+
+  // Initialize sync caches (per-level, for predictor and corrector)
+  sync_cache_pre = new Parallel::SyncCache[GH->levels];
+  sync_cache_cor = new Parallel::SyncCache[GH->levels];
+  sync_cache_rp_coarse = new Parallel::SyncCache[GH->levels];
+  sync_cache_rp_fine = new Parallel::SyncCache[GH->levels];
 }
 
 //================================================================================================
@@ -981,6 +987,32 @@ bssn_class::~bssn_class()
   delete Azzz;
 #endif
 
+  // Destroy sync caches before GH
+  if (sync_cache_pre)
+  {
+    for (int i = 0; i < GH->levels; i++)
+      sync_cache_pre[i].destroy();
+    delete[] sync_cache_pre;
+  }
+  if (sync_cache_cor)
+  {
+    for (int i = 0; i < GH->levels; i++)
+      sync_cache_cor[i].destroy();
+    delete[] sync_cache_cor;
+  }
+  if (sync_cache_rp_coarse)
+  {
+    for (int i = 0; i < GH->levels; i++)
+      sync_cache_rp_coarse[i].destroy();
+    delete[] sync_cache_rp_coarse;
+  }
+  if (sync_cache_rp_fine)
+  {
+    for (int i = 0; i < GH->levels; i++)
+      sync_cache_rp_fine[i].destroy();
+    delete[] sync_cache_rp_fine;
+  }
+
   delete GH;
 #ifdef WithShell
   delete SH;
@@ -2181,6 +2213,7 @@ void bssn_class::Evolve(int Steps)
     GH->Regrid(Symmetry, BH_num, Porgbr, Porg0,
                SynchList_cor, OldStateList, StateList, SynchList_pre,
                fgt(PhysTime - dT_mon, StartTime, dT_mon / 2), ErrorMonitor);
+    for (int il = 0; il < GH->levels; il++) { sync_cache_pre[il].invalidate(); sync_cache_cor[il].invalidate(); sync_cache_rp_coarse[il].invalidate(); sync_cache_rp_fine[il].invalidate(); }
 #endif
 
 #if (REGLEV == 0 && (PSTR == 1 || PSTR == 2))
@@ -2396,6 +2429,7 @@ void bssn_class::RecursiveStep(int lev)
   GH->Regrid_Onelevel(lev, Symmetry, BH_num, Porgbr, Porg0,
                       SynchList_cor, OldStateList, StateList, SynchList_pre,
                       fgt(PhysTime - dT_lev, StartTime, dT_lev / 2), ErrorMonitor);
+  for (int il = 0; il < GH->levels; il++) { sync_cache_pre[il].invalidate(); sync_cache_cor[il].invalidate(); sync_cache_rp_coarse[il].invalidate(); sync_cache_rp_fine[il].invalidate(); }
 #endif
 }
 
@@ -2574,6 +2608,7 @@ void bssn_class::ParallelStep()
   GH->Regrid_Onelevel(GH->mylev, Symmetry, BH_num, Porgbr, Porg0,
                       SynchList_cor, OldStateList, StateList, SynchList_pre,
                       fgt(PhysTime - dT_lev, StartTime, dT_lev / 2), ErrorMonitor);
+  for (int il = 0; il < GH->levels; il++) { sync_cache_pre[il].invalidate(); sync_cache_cor[il].invalidate(); sync_cache_rp_coarse[il].invalidate(); sync_cache_rp_fine[il].invalidate(); }
 #endif
 }
 
@@ -2740,6 +2775,7 @@ void bssn_class::ParallelStep()
         GH->Regrid_Onelevel(lev + 1, Symmetry, BH_num, Porgbr, Porg0,
                             SynchList_cor, OldStateList, StateList, SynchList_pre,
                             fgt(PhysTime - dT_levp1, StartTime, dT_levp1 / 2), ErrorMonitor);
+        for (int il = 0; il < GH->levels; il++) { sync_cache_pre[il].invalidate(); sync_cache_cor[il].invalidate(); sync_cache_rp_coarse[il].invalidate(); sync_cache_rp_fine[il].invalidate(); }
 
         //               a_stream.clear();
         //               a_stream.str("");
@@ -2754,6 +2790,7 @@ void bssn_class::ParallelStep()
       GH->Regrid_Onelevel(lev, Symmetry, BH_num, Porgbr, Porg0,
                           SynchList_cor, OldStateList, StateList, SynchList_pre,
                           fgt(PhysTime - dT_lev, StartTime, dT_lev / 2), ErrorMonitor);
+      for (int il = 0; il < GH->levels; il++) { sync_cache_pre[il].invalidate(); sync_cache_cor[il].invalidate(); sync_cache_rp_coarse[il].invalidate(); sync_cache_rp_fine[il].invalidate(); }
 
       //               a_stream.clear();
       //               a_stream.str("");
@@ -2772,6 +2809,7 @@ void bssn_class::ParallelStep()
           GH->Regrid_Onelevel(lev - 1, Symmetry, BH_num, Porgbr, Porg0,
                               SynchList_cor, OldStateList, StateList, SynchList_pre,
                               fgt(PhysTime - dT_lev, StartTime, dT_levm1 / 2), ErrorMonitor);
+          for (int il = 0; il < GH->levels; il++) { sync_cache_pre[il].invalidate(); sync_cache_cor[il].invalidate(); sync_cache_rp_coarse[il].invalidate(); sync_cache_rp_fine[il].invalidate(); }
 
           //               a_stream.clear();
           //               a_stream.str("");
@@ -2787,6 +2825,7 @@ void bssn_class::ParallelStep()
           GH->Regrid_Onelevel(lev - 1, Symmetry, BH_num, Porgbr, Porg0,
                               SynchList_cor, OldStateList, StateList, SynchList_pre,
                               fgt(PhysTime - dT_lev, StartTime, dT_levm1 / 2), ErrorMonitor);
+          for (int il = 0; il < GH->levels; il++) { sync_cache_pre[il].invalidate(); sync_cache_cor[il].invalidate(); sync_cache_rp_coarse[il].invalidate(); sync_cache_rp_fine[il].invalidate(); }
 
           //               a_stream.clear();
           //               a_stream.str("");
@@ -3158,21 +3197,7 @@ void bssn_class::Step(int lev, int YN)
     }
     Pp = Pp->next;
   }
-  // check error information
-  {
-    int erh = ERROR;
-    MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
-  }
-  if (ERROR)
-  {
-    Parallel::Dump_Data(GH->PatL[lev], StateList, 0, PhysTime, dT_lev);
-    if (myrank == 0)
-    {
-      if (ErrorMonitor->outfile)
-        ErrorMonitor->outfile << "find NaN in state variables at t = " << PhysTime << ", lev = " << lev << endl;
-      MPI_Abort(MPI_COMM_WORLD, 1);
-    }
-  }
+  // NOTE: error check deferred to after Shell Patch computation to reduce MPI_Allreduce calls
 
 #ifdef WithShell
   // evolve Shell Patches
@@ -3316,25 +3341,16 @@ void bssn_class::Step(int lev, int YN)
 #endif
   }
 
-  // check error information
+  // Non-blocking error reduction overlapped with Sync to hide Allreduce latency
+  MPI_Request err_req;
   {
     int erh = ERROR;
-    MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
-  }
-
-  if (ERROR)
-  {
-    SH->Dump_Data(StateList, 0, PhysTime, dT_lev);
-    if (myrank == 0)
-    {
-      if (ErrorMonitor->outfile)
-        ErrorMonitor->outfile << "find NaN in state variables on Shell Patches at t = " << PhysTime << endl;
-      MPI_Abort(MPI_COMM_WORLD, 1);
-    }
+    MPI_Iallreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD, &err_req);
   }
 #endif
 
-  Parallel::Sync(GH->PatL[lev], SynchList_pre, Symmetry);
+  Parallel::AsyncSyncState async_pre;
+  Parallel::Sync_start(GH->PatL[lev], SynchList_pre, Symmetry, sync_cache_pre[lev], async_pre);
 
 #ifdef WithShell
   if (lev == 0)
@@ -3353,6 +3369,23 @@ void bssn_class::Step(int lev, int YN)
     }
   }
 #endif
+  Parallel::Sync_finish(sync_cache_pre[lev], async_pre, SynchList_pre, Symmetry);
+
+#ifdef WithShell
+  // Complete non-blocking error reduction and check
+  MPI_Wait(&err_req, MPI_STATUS_IGNORE);
+  if (ERROR)
+  {
+    Parallel::Dump_Data(GH->PatL[lev], StateList, 0, PhysTime, dT_lev);
+    SH->Dump_Data(StateList, 0, PhysTime, dT_lev);
+    if (myrank == 0)
+    {
+      if (ErrorMonitor->outfile)
+        ErrorMonitor->outfile << "find NaN in state variables at t = " << PhysTime << ", lev = " << lev << endl;
+      MPI_Abort(MPI_COMM_WORLD, 1);
+    }
+  }
+#endif
 
 #if (MAPBH == 0)
   // for black hole position
@@ -3528,24 +3561,7 @@ void bssn_class::Step(int lev, int YN)
       Pp = Pp->next;
     }
 
-    // check error information
-    {
-      int erh = ERROR;
-      MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
-    }
-
-    if (ERROR)
-    {
-      Parallel::Dump_Data(GH->PatL[lev], SynchList_pre, 0, PhysTime, dT_lev);
-      if (myrank == 0)
-      {
-        if (ErrorMonitor->outfile)
-          ErrorMonitor->outfile << "find NaN in RK4 substep#" << iter_count 
-                                << " variables at t = " << PhysTime 
-                                << ", lev = " << lev << endl;
-        MPI_Abort(MPI_COMM_WORLD, 1);
-      }
-    }
+    // NOTE: error check deferred to after Shell Patch computation to reduce MPI_Allreduce calls
 
 #ifdef WithShell
     // evolve Shell Patches
@@ -3685,26 +3701,16 @@ void bssn_class::Step(int lev, int YN)
         sPp = sPp->next;
       }
     }
-    // check error information
+    // Non-blocking error reduction overlapped with Sync to hide Allreduce latency
+    MPI_Request err_req_cor;
     {
       int erh = ERROR;
-      MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
-    }
-    if (ERROR)
-    {
-      SH->Dump_Data(SynchList_pre, 0, PhysTime, dT_lev);
-      if (myrank == 0)
-      {
-        if (ErrorMonitor->outfile)
-          ErrorMonitor->outfile << "find NaN on Shell Patches in RK4 substep#" 
-                                << iter_count << " variables at t = " 
-                                << PhysTime << endl;
-        MPI_Abort(MPI_COMM_WORLD, 1);
-      }
+      MPI_Iallreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD, &err_req_cor);
     }
 #endif
 
-    Parallel::Sync(GH->PatL[lev], SynchList_cor, Symmetry);
+    Parallel::AsyncSyncState async_cor;
+    Parallel::Sync_start(GH->PatL[lev], SynchList_cor, Symmetry, sync_cache_cor[lev], async_cor);
 
 #ifdef WithShell
     if (lev == 0)
@@ -3723,6 +3729,25 @@ void bssn_class::Step(int lev, int YN)
       }
     }
 #endif
+    Parallel::Sync_finish(sync_cache_cor[lev], async_cor, SynchList_cor, Symmetry);
+
+#ifdef WithShell
+    // Complete non-blocking error reduction and check
+    MPI_Wait(&err_req_cor, MPI_STATUS_IGNORE);
+    if (ERROR)
+    {
+      Parallel::Dump_Data(GH->PatL[lev], SynchList_pre, 0, PhysTime, dT_lev);
+      SH->Dump_Data(SynchList_pre, 0, PhysTime, dT_lev);
+      if (myrank == 0)
+      {
+        if (ErrorMonitor->outfile)
+          ErrorMonitor->outfile << "find NaN in RK4 substep#" << iter_count
+                                << " variables at t = " << PhysTime
+                                << ", lev = " << lev << endl;
+        MPI_Abort(MPI_COMM_WORLD, 1);
+      }
+    }
+#endif
 
 #if (MAPBH == 0)
     // for black hole position
@@ -4034,22 +4059,7 @@ void bssn_class::Step(int lev, int YN)
     }
     Pp = Pp->next;
   }
-  // check error information
-  {
-    int erh = ERROR;
-    MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
-  }
-  if (ERROR)
-  {
-    Parallel::Dump_Data(GH->PatL[lev], StateList, 0, PhysTime, dT_lev);
-    if (myrank == 0)
-    {
-      if (ErrorMonitor->outfile)
-        ErrorMonitor->outfile << "find NaN in state variables at t = " << PhysTime 
-                              << ", lev = " << lev << endl;
-      MPI_Abort(MPI_COMM_WORLD, 1);
-    }
-  }
+  // NOTE: error check deferred to after Shell Patch computation to reduce MPI_Allreduce calls
 
 #ifdef WithShell
   // evolve Shell Patches
@@ -4190,25 +4200,16 @@ void bssn_class::Step(int lev, int YN)
   }
 #endif
   }
-  // check error information
+  // Non-blocking error reduction overlapped with Sync to hide Allreduce latency
+  MPI_Request err_req;
   {
     int erh = ERROR;
-    MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
-  }
-  if (ERROR)
-  {
-    SH->Dump_Data(StateList, 0, PhysTime, dT_lev);
-    if (myrank == 0)
-    {
-      if (ErrorMonitor->outfile)
-        ErrorMonitor->outfile << "find NaN in state variables on Shell Patches at t = " 
-                              << PhysTime << endl;
-      MPI_Abort(MPI_COMM_WORLD, 1);
-    }
+    MPI_Iallreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD, &err_req);
   }
 #endif
 
-  Parallel::Sync(GH->PatL[lev], SynchList_pre, Symmetry);
+  Parallel::AsyncSyncState async_pre;
+  Parallel::Sync_start(GH->PatL[lev], SynchList_pre, Symmetry, sync_cache_pre[lev], async_pre);
 
 #ifdef WithShell
   if (lev == 0)
@@ -4227,6 +4228,24 @@ void bssn_class::Step(int lev, int YN)
     }
   }
 #endif
+  Parallel::Sync_finish(sync_cache_pre[lev], async_pre, SynchList_pre, Symmetry);
+
+#ifdef WithShell
+  // Complete non-blocking error reduction and check
+  MPI_Wait(&err_req, MPI_STATUS_IGNORE);
+  if (ERROR)
+  {
+    Parallel::Dump_Data(GH->PatL[lev], StateList, 0, PhysTime, dT_lev);
+    SH->Dump_Data(StateList, 0, PhysTime, dT_lev);
+    if (myrank == 0)
+    {
+      if (ErrorMonitor->outfile)
+        ErrorMonitor->outfile << "find NaN in state variables at t = " << PhysTime
+                              << ", lev = " << lev << endl;
+      MPI_Abort(MPI_COMM_WORLD, 1);
+    }
+  }
+#endif
 
   // for black hole position
   if (BH_num > 0 && lev == GH->levels - 1)
@@ -4386,23 +4405,7 @@ void bssn_class::Step(int lev, int YN)
       Pp = Pp->next;
     }
 
-    // check error information
-    {
-      int erh = ERROR;
-      MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
-    }
-    if (ERROR)
-    {
-      Parallel::Dump_Data(GH->PatL[lev], SynchList_pre, 0, PhysTime, dT_lev);
-      if (myrank == 0)
-      {
-        if (ErrorMonitor->outfile)
-          ErrorMonitor->outfile << "find NaN in RK4 substep#" << iter_count 
-                                << " variables at t = " << PhysTime 
-                                << ", lev = " << lev << endl;
-        MPI_Abort(MPI_COMM_WORLD, 1);
-      }
-    }
+    // NOTE: error check deferred to after Shell Patch computation to reduce MPI_Allreduce calls
 
 #ifdef WithShell
     // evolve Shell Patches
@@ -4542,25 +4545,16 @@ void bssn_class::Step(int lev, int YN)
         sPp = sPp->next;
       }
     }
-    // check error information
+    // Non-blocking error reduction overlapped with Sync to hide Allreduce latency
+    MPI_Request err_req_cor;
     {
       int erh = ERROR;
-      MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
-    }
-    if (ERROR)
-    {
-      SH->Dump_Data(SynchList_pre, 0, PhysTime, dT_lev);
-      if (myrank == 0)
-      {
-        if (ErrorMonitor->outfile)
-          ErrorMonitor->outfile << "find NaN on Shell Patches in RK4 substep#" << iter_count 
-                                << " variables at t = " << PhysTime << endl;
-        MPI_Abort(MPI_COMM_WORLD, 1);
-      }
+      MPI_Iallreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD, &err_req_cor);
     }
 #endif
 
-    Parallel::Sync(GH->PatL[lev], SynchList_cor, Symmetry);
+    Parallel::AsyncSyncState async_cor;
+    Parallel::Sync_start(GH->PatL[lev], SynchList_cor, Symmetry, sync_cache_cor[lev], async_cor);
 
 #ifdef WithShell
     if (lev == 0)
@@ -4578,6 +4572,25 @@ void bssn_class::Step(int lev, int YN)
              << " seconds! " << endl;
       }
     }
+#endif
+    Parallel::Sync_finish(sync_cache_cor[lev], async_cor, SynchList_cor, Symmetry);
+
+#ifdef WithShell
+    // Complete non-blocking error reduction and check
+    MPI_Wait(&err_req_cor, MPI_STATUS_IGNORE);
+    if (ERROR)
+    {
+      Parallel::Dump_Data(GH->PatL[lev], SynchList_pre, 0, PhysTime, dT_lev);
+      SH->Dump_Data(SynchList_pre, 0, PhysTime, dT_lev);
+      if (myrank == 0)
+      {
+        if (ErrorMonitor->outfile)
+          ErrorMonitor->outfile << "find NaN in RK4 substep#" << iter_count
+                                << " variables at t = " << PhysTime
+                                << ", lev = " << lev << endl;
+        MPI_Abort(MPI_COMM_WORLD, 1);
+      }
+    }
 #endif
     // for black hole position
     if (BH_num > 0 && lev == GH->levels - 1)
@@ -4943,11 +4956,19 @@ void bssn_class::Step(int lev, int YN)
 
   //   misc::tillherecheck(GH->Commlev[lev],GH->start_rank[lev],"after Predictor rhs calculation");
 
-  // check error information
+  // Non-blocking error reduction overlapped with Sync to hide Allreduce latency
+  MPI_Request err_req;
   {
     int erh = ERROR;
-    MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, GH->Commlev[lev]);
+    MPI_Iallreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, GH->Commlev[lev], &err_req);
   }
+
+  //   misc::tillherecheck(GH->Commlev[lev],GH->start_rank[lev],"before Predictor sync");
+
+  Parallel::Sync_cached(GH->PatL[lev], SynchList_pre, Symmetry, sync_cache_pre[lev]);
+
+  // Complete non-blocking error reduction and check
+  MPI_Wait(&err_req, MPI_STATUS_IGNORE);
   if (ERROR)
   {
     Parallel::Dump_Data(GH->PatL[lev], StateList, 0, PhysTime, dT_lev);
@@ -4959,10 +4980,6 @@ void bssn_class::Step(int lev, int YN)
     }
   }
 
-  //   misc::tillherecheck(GH->Commlev[lev],GH->start_rank[lev],"before Predictor sync");
-
-  Parallel::Sync(GH->PatL[lev], SynchList_pre, Symmetry);
-
 #if (MAPBH == 0)
   // for black hole position
   if (BH_num > 0 && lev == GH->levels - 1)
@@ -5140,11 +5157,21 @@ void bssn_class::Step(int lev, int YN)
 
     //   misc::tillherecheck(GH->Commlev[lev],GH->start_rank[lev],"before Corrector error check");
 
-    // check error information
+    // Non-blocking error reduction overlapped with Sync to hide Allreduce latency
+    MPI_Request err_req_cor;
     {
       int erh = ERROR;
-      MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, GH->Commlev[lev]);
+      MPI_Iallreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, GH->Commlev[lev], &err_req_cor);
     }
+
+    //    misc::tillherecheck(GH->Commlev[lev],GH->start_rank[lev],"before Corrector sync");
+
+    Parallel::Sync_cached(GH->PatL[lev], SynchList_cor, Symmetry, sync_cache_cor[lev]);
+
+    //    misc::tillherecheck(GH->Commlev[lev],GH->start_rank[lev],"after Corrector sync");
+
+    // Complete non-blocking error reduction and check
+    MPI_Wait(&err_req_cor, MPI_STATUS_IGNORE);
     if (ERROR)
     {
       Parallel::Dump_Data(GH->PatL[lev], SynchList_pre, 0, PhysTime, dT_lev);
@@ -5158,12 +5185,6 @@ void bssn_class::Step(int lev, int YN)
       }
     }
 
-    //    misc::tillherecheck(GH->Commlev[lev],GH->start_rank[lev],"before Corrector sync");
-
-    Parallel::Sync(GH->PatL[lev], SynchList_cor, Symmetry);
-
-    //    misc::tillherecheck(GH->Commlev[lev],GH->start_rank[lev],"after Corrector sync");
-
 #if (MAPBH == 0)
     // for black hole position
     if (BH_num > 0 && lev == GH->levels - 1)
@@ -5447,21 +5468,11 @@ void bssn_class::SHStep()
 #if (PSTR == 1 || PSTR == 2)
 //   misc::tillherecheck(GH->Commlev[lev],GH->start_rank[lev],"before Predictor's error check");
 #endif
-  // check error information
+  // Non-blocking error reduction overlapped with Synch to hide Allreduce latency
+  MPI_Request err_req;
   {
     int erh = ERROR;
-    MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
-  }
-
-  if (ERROR)
-  {
-    SH->Dump_Data(StateList, 0, PhysTime, dT_lev);
-    if (myrank == 0)
-    {
-      if (ErrorMonitor->outfile)
-        ErrorMonitor->outfile << "find NaN in state variables on Shell Patches at t = " << PhysTime << endl;
-      MPI_Abort(MPI_COMM_WORLD, 1);
-    }
+    MPI_Iallreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD, &err_req);
   }
 
   {
@@ -5479,6 +5490,19 @@ void bssn_class::SHStep()
     }
   }
 
+  // Complete non-blocking error reduction and check
+  MPI_Wait(&err_req, MPI_STATUS_IGNORE);
+  if (ERROR)
+  {
+    SH->Dump_Data(StateList, 0, PhysTime, dT_lev);
+    if (myrank == 0)
+    {
+      if (ErrorMonitor->outfile)
+        ErrorMonitor->outfile << "find NaN in state variables on Shell Patches at t = " << PhysTime << endl;
+      MPI_Abort(MPI_COMM_WORLD, 1);
+    }
+  }
+
   // corrector
   for (iter_count = 1; iter_count < 4; iter_count++)
   {
@@ -5621,21 +5645,11 @@ void bssn_class::SHStep()
         sPp = sPp->next;
       }
     }
-    // check error information
+    // Non-blocking error reduction overlapped with Synch to hide Allreduce latency
+    MPI_Request err_req_cor;
     {
       int erh = ERROR;
-      MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
-    }
-    if (ERROR)
-    {
-      SH->Dump_Data(SynchList_pre, 0, PhysTime, dT_lev);
-      if (myrank == 0)
-      {
-        if (ErrorMonitor->outfile)
-          ErrorMonitor->outfile << "find NaN on Shell Patches in RK4 substep#" << iter_count 
-                                << " variables at t = " << PhysTime << endl;
-        MPI_Abort(MPI_COMM_WORLD, 1);
-      }
+      MPI_Iallreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD, &err_req_cor);
     }
 
     {
@@ -5653,6 +5667,20 @@ void bssn_class::SHStep()
       }
     }
 
+    // Complete non-blocking error reduction and check
+    MPI_Wait(&err_req_cor, MPI_STATUS_IGNORE);
+    if (ERROR)
+    {
+      SH->Dump_Data(SynchList_pre, 0, PhysTime, dT_lev);
+      if (myrank == 0)
+      {
+        if (ErrorMonitor->outfile)
+          ErrorMonitor->outfile << "find NaN on Shell Patches in RK4 substep#" << iter_count
+                                << " variables at t = " << PhysTime << endl;
+        MPI_Abort(MPI_COMM_WORLD, 1);
+      }
+    }
+
     sPp = SH->PatL;
     while (sPp)
     {
@@ -5781,7 +5809,7 @@ void bssn_class::RestrictProlong(int lev, int YN, bool BB,
 //       misc::tillherecheck(GH->Commlev[GH->mylev],GH->start_rank[GH->mylev],a_stream.str());
 #endif
 
-      Parallel::Sync(GH->PatL[lev - 1], SynchList_pre, Symmetry);
+      Parallel::Sync_cached(GH->PatL[lev - 1], SynchList_pre, Symmetry, sync_cache_rp_coarse[lev]);
 
 #if (PSTR == 1 || PSTR == 2)
 //       a_stream.clear();
@@ -5791,21 +5819,11 @@ void bssn_class::RestrictProlong(int lev, int YN, bool BB,
 #endif
 
 #if (RPB == 0)
-      Ppc = GH->PatL[lev - 1];
-      while (Ppc)
-      {
-        Pp = GH->PatL[lev];
-        while (Pp)
-        {
 #if (MIXOUTB == 0)
-          Parallel::OutBdLow2Hi(Ppc->data, Pp->data, SynchList_pre, SL, Symmetry);
+      Parallel::OutBdLow2Hi(GH->PatL[lev - 1], GH->PatL[lev], SynchList_pre, SL, Symmetry);
 #elif (MIXOUTB == 1)
-          Parallel::OutBdLow2Himix(Ppc->data, Pp->data, SynchList_pre, SL, Symmetry);
+      Parallel::OutBdLow2Himix(GH->PatL[lev - 1], GH->PatL[lev], SynchList_pre, SL, Symmetry);
 #endif
-          Pp = Pp->next;
-        }
-        Ppc = Ppc->next;
-      }
 #elif (RPB == 1)
       //       Parallel::OutBdLow2Hi_bam(GH->PatL[lev-1],GH->PatL[lev],SynchList_pre,SL,Symmetry);
       Parallel::OutBdLow2Hi_bam(GH->PatL[lev - 1], GH->PatL[lev], SynchList_pre, SL, GH->bdsul[lev], Symmetry);
@@ -5842,7 +5860,7 @@ void bssn_class::RestrictProlong(int lev, int YN, bool BB,
 //       misc::tillherecheck(GH->Commlev[GH->mylev],GH->start_rank[GH->mylev],a_stream.str());
 #endif
 
-      Parallel::Sync(GH->PatL[lev - 1], SL, Symmetry);
+      Parallel::Sync_cached(GH->PatL[lev - 1], SL, Symmetry, sync_cache_rp_coarse[lev]);
 
 #if (PSTR == 1 || PSTR == 2)
 //       a_stream.clear();
@@ -5852,21 +5870,11 @@ void bssn_class::RestrictProlong(int lev, int YN, bool BB,
 #endif
 
 #if (RPB == 0)
-      Ppc = GH->PatL[lev - 1];
-      while (Ppc)
-      {
-        Pp = GH->PatL[lev];
-        while (Pp)
-        {
 #if (MIXOUTB == 0)
-          Parallel::OutBdLow2Hi(Ppc->data, Pp->data, SL, SL, Symmetry);
+      Parallel::OutBdLow2Hi(GH->PatL[lev - 1], GH->PatL[lev], SL, SL, Symmetry);
 #elif (MIXOUTB == 1)
-          Parallel::OutBdLow2Himix(Ppc->data, Pp->data, SL, SL, Symmetry);
+      Parallel::OutBdLow2Himix(GH->PatL[lev - 1], GH->PatL[lev], SL, SL, Symmetry);
 #endif
-          Pp = Pp->next;
-        }
-        Ppc = Ppc->next;
-      }
 #elif (RPB == 1)
       //       Parallel::OutBdLow2Hi_bam(GH->PatL[lev-1],GH->PatL[lev],SL,SL,Symmetry);
       Parallel::OutBdLow2Hi_bam(GH->PatL[lev - 1], GH->PatL[lev], SL, SL, GH->bdsul[lev], Symmetry);
@@ -5880,7 +5888,7 @@ void bssn_class::RestrictProlong(int lev, int YN, bool BB,
 #endif
     }
 
-    Parallel::Sync(GH->PatL[lev], SL, Symmetry);
+    Parallel::Sync_cached(GH->PatL[lev], SL, Symmetry, sync_cache_rp_fine[lev]);
 
 #if (PSTR == 1 || PSTR == 2)
 //    a_stream.clear();
@@ -5938,24 +5946,14 @@ void bssn_class::RestrictProlong_aux(int lev, int YN, bool BB,
       Parallel::Restrict_bam(GH->PatL[lev - 1], GH->PatL[lev], SL, SynchList_pre, GH->rsul[lev], Symmetry);
 #endif
 
-      Parallel::Sync(GH->PatL[lev - 1], SynchList_pre, Symmetry);
+      Parallel::Sync_cached(GH->PatL[lev - 1], SynchList_pre, Symmetry, sync_cache_rp_coarse[lev]);
 
 #if (RPB == 0)
-      Ppc = GH->PatL[lev - 1];
-      while (Ppc)
-      {
-        Pp = GH->PatL[lev];
-        while (Pp)
-        {
 #if (MIXOUTB == 0)
-          Parallel::OutBdLow2Hi(Ppc->data, Pp->data, SynchList_pre, SL, Symmetry);
+      Parallel::OutBdLow2Hi(GH->PatL[lev - 1], GH->PatL[lev], SynchList_pre, SL, Symmetry);
 #elif (MIXOUTB == 1)
-          Parallel::OutBdLow2Himix(Ppc->data, Pp->data, SynchList_pre, SL, Symmetry);
+      Parallel::OutBdLow2Himix(GH->PatL[lev - 1], GH->PatL[lev], SynchList_pre, SL, Symmetry);
 #endif
-          Pp = Pp->next;
-        }
-        Ppc = Ppc->next;
-      }
 #elif (RPB == 1)
       //       Parallel::OutBdLow2Hi_bam(GH->PatL[lev-1],GH->PatL[lev],SynchList_pre,SL,Symmetry);
       Parallel::OutBdLow2Hi_bam(GH->PatL[lev - 1], GH->PatL[lev], SynchList_pre, SL, GH->bdsul[lev], Symmetry);
@@ -5970,31 +5968,21 @@ void bssn_class::RestrictProlong_aux(int lev, int YN, bool BB,
       Parallel::Restrict_bam(GH->PatL[lev - 1], GH->PatL[lev], SL, SL, GH->rsul[lev], Symmetry);
 #endif
 
-      Parallel::Sync(GH->PatL[lev - 1], SL, Symmetry);
+      Parallel::Sync_cached(GH->PatL[lev - 1], SL, Symmetry, sync_cache_rp_coarse[lev]);
 
 #if (RPB == 0)
-      Ppc = GH->PatL[lev - 1];
-      while (Ppc)
-      {
-        Pp = GH->PatL[lev];
-        while (Pp)
-        {
 #if (MIXOUTB == 0)
-          Parallel::OutBdLow2Hi(Ppc->data, Pp->data, SL, SL, Symmetry);
+      Parallel::OutBdLow2Hi(GH->PatL[lev - 1], GH->PatL[lev], SL, SL, Symmetry);
 #elif (MIXOUTB == 1)
-          Parallel::OutBdLow2Himix(Ppc->data, Pp->data, SL, SL, Symmetry);
+      Parallel::OutBdLow2Himix(GH->PatL[lev - 1], GH->PatL[lev], SL, SL, Symmetry);
 #endif
-          Pp = Pp->next;
-        }
-        Ppc = Ppc->next;
-      }
 #elif (RPB == 1)
       //       Parallel::OutBdLow2Hi_bam(GH->PatL[lev-1],GH->PatL[lev],SL,SL,Symmetry);
       Parallel::OutBdLow2Hi_bam(GH->PatL[lev - 1], GH->PatL[lev], SL, SL, GH->bdsul[lev], Symmetry);
 #endif
     }
 
-    Parallel::Sync(GH->PatL[lev], SL, Symmetry);
+    Parallel::Sync_cached(GH->PatL[lev], SL, Symmetry, sync_cache_rp_fine[lev]);
   }
 }
 
@@ -6045,24 +6033,14 @@ void bssn_class::RestrictProlong(int lev, int YN, bool BB)
       Parallel::Restrict_bam(GH->PatL[lev - 1], GH->PatL[lev], SynchList_cor, SynchList_pre, GH->rsul[lev], Symmetry);
 #endif
 
-      Parallel::Sync(GH->PatL[lev - 1], SynchList_pre, Symmetry);
+      Parallel::Sync_cached(GH->PatL[lev - 1], SynchList_pre, Symmetry, sync_cache_rp_coarse[lev]);
 
 #if (RPB == 0)
-      Ppc = GH->PatL[lev - 1];
-      while (Ppc)
-      {
-        Pp = GH->PatL[lev];
-        while (Pp)
-        {
 #if (MIXOUTB == 0)
-          Parallel::OutBdLow2Hi(Ppc->data, Pp->data, SynchList_pre, SynchList_cor, Symmetry);
+      Parallel::OutBdLow2Hi(GH->PatL[lev - 1], GH->PatL[lev], SynchList_pre, SynchList_cor, Symmetry);
 #elif (MIXOUTB == 1)
-          Parallel::OutBdLow2Himix(Ppc->data, Pp->data, SynchList_pre, SynchList_cor, Symmetry);
+      Parallel::OutBdLow2Himix(GH->PatL[lev - 1], GH->PatL[lev], SynchList_pre, SynchList_cor, Symmetry);
 #endif
-          Pp = Pp->next;
-        }
-        Ppc = Ppc->next;
-      }
 #elif (RPB == 1)
       //       Parallel::OutBdLow2Hi_bam(GH->PatL[lev-1],GH->PatL[lev],SynchList_pre,SynchList_cor,Symmetry);
       Parallel::OutBdLow2Hi_bam(GH->PatL[lev - 1], GH->PatL[lev], SynchList_pre, SynchList_cor, GH->bdsul[lev], Symmetry);
@@ -6079,31 +6057,21 @@ void bssn_class::RestrictProlong(int lev, int YN, bool BB)
       Parallel::Restrict_bam(GH->PatL[lev - 1], GH->PatL[lev], SynchList_cor, StateList, GH->rsul[lev], Symmetry);
 #endif
 
-      Parallel::Sync(GH->PatL[lev - 1], StateList, Symmetry);
+      Parallel::Sync_cached(GH->PatL[lev - 1], StateList, Symmetry, sync_cache_rp_coarse[lev]);
 
 #if (RPB == 0)
-      Ppc = GH->PatL[lev - 1];
-      while (Ppc)
-      {
-        Pp = GH->PatL[lev];
-        while (Pp)
-        {
 #if (MIXOUTB == 0)
-          Parallel::OutBdLow2Hi(Ppc->data, Pp->data, StateList, SynchList_cor, Symmetry);
+      Parallel::OutBdLow2Hi(GH->PatL[lev - 1], GH->PatL[lev], StateList, SynchList_cor, Symmetry);
 #elif (MIXOUTB == 1)
-          Parallel::OutBdLow2Himix(Ppc->data, Pp->data, StateList, SynchList_cor, Symmetry);
+      Parallel::OutBdLow2Himix(GH->PatL[lev - 1], GH->PatL[lev], StateList, SynchList_cor, Symmetry);
 #endif
-          Pp = Pp->next;
-        }
-        Ppc = Ppc->next;
-      }
 #elif (RPB == 1)
       //       Parallel::OutBdLow2Hi_bam(GH->PatL[lev-1],GH->PatL[lev],StateList,SynchList_cor,Symmetry);
       Parallel::OutBdLow2Hi_bam(GH->PatL[lev - 1], GH->PatL[lev], StateList, SynchList_cor, GH->bdsul[lev], Symmetry);
 #endif
     }
 
-    Parallel::Sync(GH->PatL[lev], SynchList_cor, Symmetry);
+    Parallel::Sync_cached(GH->PatL[lev], SynchList_cor, Symmetry, sync_cache_rp_fine[lev]);
   }
 }
 
@@ -6133,21 +6101,11 @@ void bssn_class::ProlongRestrict(int lev, int YN, bool BB)
       }
 
 #if (RPB == 0)
-      Ppc = GH->PatL[lev - 1];
-      while (Ppc)
-      {
-        Pp = GH->PatL[lev];
-        while (Pp)
-        {
 #if (MIXOUTB == 0)
-          Parallel::OutBdLow2Hi(Ppc->data, Pp->data, SynchList_pre, SynchList_cor, Symmetry);
+      Parallel::OutBdLow2Hi(GH->PatL[lev - 1], GH->PatL[lev], SynchList_pre, SynchList_cor, Symmetry);
 #elif (MIXOUTB == 1)
-          Parallel::OutBdLow2Himix(Ppc->data, Pp->data, SynchList_pre, SynchList_cor, Symmetry);
+      Parallel::OutBdLow2Himix(GH->PatL[lev - 1], GH->PatL[lev], SynchList_pre, SynchList_cor, Symmetry);
 #endif
-          Pp = Pp->next;
-        }
-        Ppc = Ppc->next;
-      }
 #elif (RPB == 1)
       //       Parallel::OutBdLow2Hi_bam(GH->PatL[lev-1],GH->PatL[lev],SynchList_pre,SynchList_cor,Symmetry);
       Parallel::OutBdLow2Hi_bam(GH->PatL[lev - 1], GH->PatL[lev], SynchList_pre, SynchList_cor, GH->bdsul[lev], Symmetry);
@@ -6156,21 +6114,11 @@ void bssn_class::ProlongRestrict(int lev, int YN, bool BB)
     else // no time refinement levels and for all same time levels
     {
 #if (RPB == 0)
-      Ppc = GH->PatL[lev - 1];
-      while (Ppc)
-      {
-        Pp = GH->PatL[lev];
-        while (Pp)
-        {
 #if (MIXOUTB == 0)
-          Parallel::OutBdLow2Hi(Ppc->data, Pp->data, StateList, SynchList_cor, Symmetry);
+      Parallel::OutBdLow2Hi(GH->PatL[lev - 1], GH->PatL[lev], StateList, SynchList_cor, Symmetry);
 #elif (MIXOUTB == 1)
-          Parallel::OutBdLow2Himix(Ppc->data, Pp->data, StateList, SynchList_cor, Symmetry);
+      Parallel::OutBdLow2Himix(GH->PatL[lev - 1], GH->PatL[lev], StateList, SynchList_cor, Symmetry);
 #endif
-          Pp = Pp->next;
-        }
-        Ppc = Ppc->next;
-      }
 #elif (RPB == 1)
       //       Parallel::OutBdLow2Hi_bam(GH->PatL[lev-1],GH->PatL[lev],StateList,SynchList_cor,Symmetry);
       Parallel::OutBdLow2Hi_bam(GH->PatL[lev - 1], GH->PatL[lev], StateList, SynchList_cor, GH->bdsul[lev], Symmetry);
@@ -6186,10 +6134,10 @@ void bssn_class::ProlongRestrict(int lev, int YN, bool BB)
 #else
       Parallel::Restrict_after(GH->PatL[lev - 1], GH->PatL[lev], SynchList_cor, StateList, Symmetry);
 #endif
-      Parallel::Sync(GH->PatL[lev - 1], StateList, Symmetry);
+      Parallel::Sync_cached(GH->PatL[lev - 1], StateList, Symmetry, sync_cache_rp_coarse[lev]);
     }
 
-    Parallel::Sync(GH->PatL[lev], SynchList_cor, Symmetry);
+    Parallel::Sync_cached(GH->PatL[lev], SynchList_cor, Symmetry, sync_cache_rp_fine[lev]);
   }
 }
 #undef MIXOUTB

AMSS_NCKU_source/bssn_class.h

@@ -126,6 +126,11 @@ public:
        MyList<var> *OldStateList, *DumpList;
        MyList<var> *ConstraintList;
 
+       Parallel::SyncCache *sync_cache_pre;  // per-level cache for predictor sync
+       Parallel::SyncCache *sync_cache_cor;  // per-level cache for corrector sync
+       Parallel::SyncCache *sync_cache_rp_coarse;  // RestrictProlong sync on PatL[lev-1]
+       Parallel::SyncCache *sync_cache_rp_fine;    // RestrictProlong sync on PatL[lev]
+
        monitor *ErrorMonitor, *Psi4Monitor, *BHMonitor, *MAPMonitor;
        monitor *ConVMonitor;
        surface_integral *Waveshell;

AMSS_NCKU_source/bssn_rhs.f90

@@ -61,9 +61,7 @@
   real*8, dimension(ex(1),ex(2),ex(3)),intent(inout) :: ham_Res, movx_Res, movy_Res, movz_Res
   real*8, dimension(ex(1),ex(2),ex(3)),intent(inout) :: Gmx_Res, Gmy_Res, Gmz_Res
 !  gont = 0: success; gont = 1: something wrong
-  integer::gont,i,j,k
-  real*8 :: val1, val2
-  real*8 :: det, t_gupxx, t_gupxy, t_gupxz, t_gupyy, t_gupyz, t_gupzz
+  integer::gont
 
 !~~~~~~> Other variables:
 
@@ -86,10 +84,7 @@
   real*8, dimension(ex(1),ex(2),ex(3)) :: gupyy,gupyz,gupzz
 
   real*8,dimension(3) ::SSS,AAS,ASA,SAA,ASS,SAS,SSA
-  real*8            :: PI
-#if (DEBUG_NAN_CHECK)
-  real*8            :: dX
-#endif
+  real*8            :: dX, dY, dZ, PI
   real*8, parameter :: ZEO = 0.d0,ONE = 1.D0, TWO = 2.D0, FOUR = 4.D0
   real*8, parameter :: EIGHT = 8.D0, HALF = 0.5D0, THR = 3.d0
   real*8, parameter :: SYM = 1.D0, ANTI= - 1.D0
@@ -111,8 +106,8 @@
   call getpbh(BHN,Porg,Mass)
 #endif
 
-#if (DEBUG_NAN_CHECK)
-!!! sanity check
+!!! sanity check (disabled in production builds for performance)
+#ifdef DEBUG
   dX = sum(chi)+sum(trK)+sum(dxx)+sum(gxy)+sum(gxz)+sum(dyy)+sum(gyz)+sum(dzz) &
       +sum(Axx)+sum(Axy)+sum(Axz)+sum(Ayy)+sum(Ayz)+sum(Azz)                   &
       +sum(Gamx)+sum(Gamy)+sum(Gamz)                                           &
@@ -146,6 +141,10 @@
 
   PI = dacos(-ONE)
 
+  dX = X(2) - X(1)
+  dY = Y(2) - Y(1)
+  dZ = Z(2) - Z(1)
+
   alpn1 = Lap + ONE
   chin1 = chi + ONE
   gxx = dxx + ONE
@@ -159,16 +158,16 @@
   div_beta = betaxx + betayy + betazz
  
   call fderivs(ex,chi,chix,chiy,chiz,X,Y,Z,SYM,SYM,SYM,symmetry,Lev)
-  call fderivs(ex,dxx,gxxx,gxxy,gxxz,X,Y,Z,SYM ,SYM ,SYM ,Symmetry,Lev)
-  call fderivs(ex,dyy,gyyx,gyyy,gyyz,X,Y,Z,SYM ,SYM ,SYM ,Symmetry,Lev)
-  call fderivs(ex,dzz,gzzx,gzzy,gzzz,X,Y,Z,SYM ,SYM ,SYM ,Symmetry,Lev)
-
-  call fderivs(ex,gxy,gxyx,gxyy,gxyz,X,Y,Z,ANTI,ANTI,SYM ,Symmetry,Lev)
-  call fderivs(ex,gxz,gxzx,gxzy,gxzz,X,Y,Z,ANTI,SYM ,ANTI,Symmetry,Lev)
-  call fderivs(ex,gyz,gyzx,gyzy,gyzz,X,Y,Z,SYM ,ANTI,ANTI,Symmetry,Lev)
 
   chi_rhs = F2o3 *chin1*( alpn1 * trK - div_beta ) !rhs for chi
 
+  call fderivs(ex,dxx,gxxx,gxxy,gxxz,X,Y,Z,SYM ,SYM ,SYM ,Symmetry,Lev)
+  call fderivs(ex,gxy,gxyx,gxyy,gxyz,X,Y,Z,ANTI,ANTI,SYM ,Symmetry,Lev)
+  call fderivs(ex,gxz,gxzx,gxzy,gxzz,X,Y,Z,ANTI,SYM ,ANTI,Symmetry,Lev)
+  call fderivs(ex,dyy,gyyx,gyyy,gyyz,X,Y,Z,SYM ,SYM ,SYM ,Symmetry,Lev)
+  call fderivs(ex,gyz,gyzx,gyzy,gyzz,X,Y,Z,SYM ,ANTI,ANTI,Symmetry,Lev)
+  call fderivs(ex,dzz,gzzx,gzzy,gzzz,X,Y,Z,SYM ,SYM ,SYM ,Symmetry,Lev)
+
   gxx_rhs = - TWO * alpn1 * Axx    -  F2o3 * gxx * div_beta          + &
               TWO *(  gxx * betaxx +   gxy * betayx +   gxz * betazx)
 
@@ -193,99 +192,71 @@
                                        gyz * betayx +   gzz * betazx   &
                                                     -   gxz * betayy     !rhs for gij
 
-! fused loop for metric inversion and connections
-  !DIR$ SIMD
-  do k=1,ex(3)
-  do j=1,ex(2)
-  do i=1,ex(1)
-     ! 1. Metric Inversion
-     det = ONE / ( &
-            gxx(i,j,k) * gyy(i,j,k) * gzz(i,j,k) + gxy(i,j,k) * gyz(i,j,k) * gxz(i,j,k) + &
-            gxz(i,j,k) * gxy(i,j,k) * gyz(i,j,k) - gxz(i,j,k) * gyy(i,j,k) * gxz(i,j,k) - &
-            gxy(i,j,k) * gxy(i,j,k) * gzz(i,j,k) - gxx(i,j,k) * gyz(i,j,k) * gyz(i,j,k) )
-     
-     t_gupxx =   ( gyy(i,j,k) * gzz(i,j,k) - gyz(i,j,k) * gyz(i,j,k) ) * det
-     t_gupxy = - ( gxy(i,j,k) * gzz(i,j,k) - gyz(i,j,k) * gxz(i,j,k) ) * det
-     t_gupxz =   ( gxy(i,j,k) * gyz(i,j,k) - gyy(i,j,k) * gxz(i,j,k) ) * det
-     t_gupyy =   ( gxx(i,j,k) * gzz(i,j,k) - gxz(i,j,k) * gxz(i,j,k) ) * det
-     t_gupyz = - ( gxx(i,j,k) * gyz(i,j,k) - gxy(i,j,k) * gxz(i,j,k) ) * det
-     t_gupzz =   ( gxx(i,j,k) * gyy(i,j,k) - gxy(i,j,k) * gxy(i,j,k) ) * det
-
-     gupxx(i,j,k) = t_gupxx
-     gupxy(i,j,k) = t_gupxy
-     gupxz(i,j,k) = t_gupxz
-     gupyy(i,j,k) = t_gupyy
-     gupyz(i,j,k) = t_gupyz
-     gupzz(i,j,k) = t_gupzz
+! invert tilted metric
+  gupzz =  gxx * gyy * gzz + gxy * gyz * gxz + gxz * gxy * gyz - &
+           gxz * gyy * gxz - gxy * gxy * gzz - gxx * gyz * gyz
+  gupxx =   ( gyy * gzz - gyz * gyz ) / gupzz
+  gupxy = - ( gxy * gzz - gyz * gxz ) / gupzz
+  gupxz =   ( gxy * gyz - gyy * gxz ) / gupzz
+  gupyy =   ( gxx * gzz - gxz * gxz ) / gupzz
+  gupyz = - ( gxx * gyz - gxy * gxz ) / gupzz
+  gupzz =   ( gxx * gyy - gxy * gxy ) / gupzz
 
   if(co == 0)then
-        Gmx_Res(i,j,k) = Gamx(i,j,k) - (t_gupxx*(t_gupxx*gxxx(i,j,k)+t_gupxy*gxyx(i,j,k)+t_gupxz*gxzx(i,j,k))&
-                         +t_gupxy*(t_gupxx*gxyx(i,j,k)+t_gupxy*gyyx(i,j,k)+t_gupxz*gyzx(i,j,k))&
-                         +t_gupxz*(t_gupxx*gxzx(i,j,k)+t_gupxy*gyzx(i,j,k)+t_gupxz*gzzx(i,j,k))&
-                         +t_gupxx*(t_gupxy*gxxy(i,j,k)+t_gupyy*gxyy(i,j,k)+t_gupyz*gxzy(i,j,k))&
-                         +t_gupxy*(t_gupxy*gxyy(i,j,k)+t_gupyy*gyyy(i,j,k)+t_gupyz*gyzy(i,j,k))&
-                         +t_gupxz*(t_gupxy*gxzy(i,j,k)+t_gupyy*gyzy(i,j,k)+t_gupyz*gzzy(i,j,k))&
-                         +t_gupxx*(t_gupxz*gxxz(i,j,k)+t_gupyz*gxyz(i,j,k)+t_gupzz*gxzz(i,j,k))&
-                         +t_gupxy*(t_gupxz*gxyz(i,j,k)+t_gupyz*gyyz(i,j,k)+t_gupzz*gyzz(i,j,k))&
-                         +t_gupxz*(t_gupxz*gxzz(i,j,k)+t_gupyz*gyzz(i,j,k)+t_gupzz*gzzz(i,j,k)))
-        Gmy_Res(i,j,k) = Gamy(i,j,k) - (t_gupxx*(t_gupxy*gxxx(i,j,k)+t_gupyy*gxyx(i,j,k)+t_gupyz*gxzx(i,j,k))&
-                         +t_gupxy*(t_gupxy*gxyx(i,j,k)+t_gupyy*gyyx(i,j,k)+t_gupyz*gyzx(i,j,k))&
-                         +t_gupxz*(t_gupxy*gxzx(i,j,k)+t_gupyy*gyzx(i,j,k)+t_gupyz*gzzx(i,j,k))&
-                         +t_gupxy*(t_gupxy*gxxy(i,j,k)+t_gupyy*gxyy(i,j,k)+t_gupyz*gxzy(i,j,k))&
-                         +t_gupyy*(t_gupxy*gxyy(i,j,k)+t_gupyy*gyyy(i,j,k)+t_gupyz*gyzy(i,j,k))&
-                         +t_gupyz*(t_gupxy*gxzy(i,j,k)+t_gupyy*gyzy(i,j,k)+t_gupyz*gzzy(i,j,k))&
-                         +t_gupxy*(t_gupxz*gxxz(i,j,k)+t_gupyz*gxyz(i,j,k)+t_gupzz*gxzz(i,j,k))&
-                         +t_gupyy*(t_gupxz*gxyz(i,j,k)+t_gupyz*gyyz(i,j,k)+t_gupzz*gyzz(i,j,k))&
-                         +t_gupyz*(t_gupxz*gxzz(i,j,k)+t_gupyz*gyzz(i,j,k)+t_gupzz*gzzz(i,j,k)))
-        Gmz_Res(i,j,k) = Gamz(i,j,k) - (t_gupxx*(t_gupxz*gxxx(i,j,k)+t_gupyz*gxyx(i,j,k)+t_gupzz*gxzx(i,j,k))&
-                         +t_gupxy*(t_gupxz*gxyx(i,j,k)+t_gupyz*gyyx(i,j,k)+t_gupzz*gyzx(i,j,k))&
-                         +t_gupxz*(t_gupxz*gxzx(i,j,k)+t_gupyz*gyzx(i,j,k)+t_gupzz*gzzx(i,j,k))&
-                         +t_gupxy*(t_gupxz*gxxy(i,j,k)+t_gupyz*gxyy(i,j,k)+t_gupzz*gxzy(i,j,k))&
-                         +t_gupyy*(t_gupxz*gxyy(i,j,k)+t_gupyz*gyyy(i,j,k)+t_gupzz*gyzy(i,j,k))&
-                         +t_gupyz*(t_gupxz*gxzy(i,j,k)+t_gupyz*gyzy(i,j,k)+t_gupzz*gzzy(i,j,k))&
-                         +t_gupxz*(t_gupxz*gxxz(i,j,k)+t_gupyz*gxyz(i,j,k)+t_gupzz*gxzz(i,j,k))&
-                         +t_gupyz*(t_gupxz*gxyz(i,j,k)+t_gupyz*gyyz(i,j,k)+t_gupzz*gyzz(i,j,k))&
-                         +t_gupzz*(t_gupxz*gxzz(i,j,k)+t_gupyz*gyzz(i,j,k)+t_gupzz*gzzz(i,j,k)))
+! Gam^i_Res = Gam^i + gup^ij_,j
+  Gmx_Res = Gamx - (gupxx*(gupxx*gxxx+gupxy*gxyx+gupxz*gxzx)&
+                   +gupxy*(gupxx*gxyx+gupxy*gyyx+gupxz*gyzx)&
+                   +gupxz*(gupxx*gxzx+gupxy*gyzx+gupxz*gzzx)&
+                   +gupxx*(gupxy*gxxy+gupyy*gxyy+gupyz*gxzy)&
+                   +gupxy*(gupxy*gxyy+gupyy*gyyy+gupyz*gyzy)&
+                   +gupxz*(gupxy*gxzy+gupyy*gyzy+gupyz*gzzy)&
+                   +gupxx*(gupxz*gxxz+gupyz*gxyz+gupzz*gxzz)&
+                   +gupxy*(gupxz*gxyz+gupyz*gyyz+gupzz*gyzz)&
+                   +gupxz*(gupxz*gxzz+gupyz*gyzz+gupzz*gzzz))
+  Gmy_Res = Gamy - (gupxx*(gupxy*gxxx+gupyy*gxyx+gupyz*gxzx)&
+                   +gupxy*(gupxy*gxyx+gupyy*gyyx+gupyz*gyzx)&
+                   +gupxz*(gupxy*gxzx+gupyy*gyzx+gupyz*gzzx)&
+                   +gupxy*(gupxy*gxxy+gupyy*gxyy+gupyz*gxzy)&
+                   +gupyy*(gupxy*gxyy+gupyy*gyyy+gupyz*gyzy)&
+                   +gupyz*(gupxy*gxzy+gupyy*gyzy+gupyz*gzzy)&
+                   +gupxy*(gupxz*gxxz+gupyz*gxyz+gupzz*gxzz)&
+                   +gupyy*(gupxz*gxyz+gupyz*gyyz+gupzz*gyzz)&
+                   +gupyz*(gupxz*gxzz+gupyz*gyzz+gupzz*gzzz))
+  Gmz_Res = Gamz - (gupxx*(gupxz*gxxx+gupyz*gxyx+gupzz*gxzx)&
+                   +gupxy*(gupxz*gxyx+gupyz*gyyx+gupzz*gyzx)&
+                   +gupxz*(gupxz*gxzx+gupyz*gyzx+gupzz*gzzx)&
+                   +gupxy*(gupxz*gxxy+gupyz*gxyy+gupzz*gxzy)&
+                   +gupyy*(gupxz*gxyy+gupyz*gyyy+gupzz*gyzy)&
+                   +gupyz*(gupxz*gxzy+gupyz*gyzy+gupzz*gzzy)&
+                   +gupxz*(gupxz*gxxz+gupyz*gxyz+gupzz*gxzz)&
+                   +gupyz*(gupxz*gxyz+gupyz*gyyz+gupzz*gyzz)&
+                   +gupzz*(gupxz*gxzz+gupyz*gyzz+gupzz*gzzz))
   endif
 
-     ! 2. Christoffel Symbols
-     val1 = TWO * gxyx(i,j,k) - gxxy(i,j,k)
-     val2 = TWO * gxzx(i,j,k) - gxxz(i,j,k)
-     Gamxxx(i,j,k) =HALF*( t_gupxx*gxxx(i,j,k) + t_gupxy*val1 + t_gupxz*val2 )
-     Gamyxx(i,j,k) =HALF*( t_gupxy*gxxx(i,j,k) + t_gupyy*val1 + t_gupyz*val2 )
-     Gamzxx(i,j,k) =HALF*( t_gupxz*gxxx(i,j,k) + t_gupyz*val1 + t_gupzz*val2 )
+! second kind of connection
+  Gamxxx =HALF*( gupxx*gxxx + gupxy*(TWO*gxyx - gxxy ) + gupxz*(TWO*gxzx - gxxz ))
+  Gamyxx =HALF*( gupxy*gxxx + gupyy*(TWO*gxyx - gxxy ) + gupyz*(TWO*gxzx - gxxz ))
+  Gamzxx =HALF*( gupxz*gxxx + gupyz*(TWO*gxyx - gxxy ) + gupzz*(TWO*gxzx - gxxz ))
  
-     val1 = TWO * gxyy(i,j,k) - gyyx(i,j,k)
-     val2 = TWO * gyzy(i,j,k) - gyyz(i,j,k)
-     Gamxyy(i,j,k) =HALF*( t_gupxx*val1 + t_gupxy*gyyy(i,j,k) + t_gupxz*val2 )
-     Gamyyy(i,j,k) =HALF*( t_gupxy*val1 + t_gupyy*gyyy(i,j,k) + t_gupyz*val2 )
-     Gamzyy(i,j,k) =HALF*( t_gupxz*val1 + t_gupyz*gyyy(i,j,k) + t_gupzz*val2 )
+  Gamxyy =HALF*( gupxx*(TWO*gxyy - gyyx ) + gupxy*gyyy + gupxz*(TWO*gyzy - gyyz ))
+  Gamyyy =HALF*( gupxy*(TWO*gxyy - gyyx ) + gupyy*gyyy + gupyz*(TWO*gyzy - gyyz ))
+  Gamzyy =HALF*( gupxz*(TWO*gxyy - gyyx ) + gupyz*gyyy + gupzz*(TWO*gyzy - gyyz ))
 
-     val1 = TWO * gxzz(i,j,k) - gzzx(i,j,k)
-     val2 = TWO * gyzz(i,j,k) - gzzy(i,j,k)
-     Gamxzz(i,j,k) =HALF*( t_gupxx*val1 + t_gupxy*val2 + t_gupxz*gzzz(i,j,k) )
-     Gamyzz(i,j,k) =HALF*( t_gupxy*val1 + t_gupyy*val2 + t_gupyz*gzzz(i,j,k) )
-     Gamzzz(i,j,k) =HALF*( t_gupxz*val1 + t_gupyz*val2 + t_gupzz*gzzz(i,j,k) )
+  Gamxzz =HALF*( gupxx*(TWO*gxzz - gzzx ) + gupxy*(TWO*gyzz - gzzy ) + gupxz*gzzz)
+  Gamyzz =HALF*( gupxy*(TWO*gxzz - gzzx ) + gupyy*(TWO*gyzz - gzzy ) + gupyz*gzzz)
+  Gamzzz =HALF*( gupxz*(TWO*gxzz - gzzx ) + gupyz*(TWO*gyzz - gzzy ) + gupzz*gzzz)
 
-     val1 = gxzy(i,j,k) + gyzx(i,j,k) - gxyz(i,j,k)
-     Gamxxy(i,j,k) =HALF*( t_gupxx*gxxy(i,j,k) + t_gupxy*gyyx(i,j,k) + t_gupxz*val1 )
-     Gamyxy(i,j,k) =HALF*( t_gupxy*gxxy(i,j,k) + t_gupyy*gyyx(i,j,k) + t_gupyz*val1 )
-     Gamzxy(i,j,k) =HALF*( t_gupxz*gxxy(i,j,k) + t_gupyz*gyyx(i,j,k) + t_gupzz*val1 )
-
-     val1 = gxyz(i,j,k) + gyzx(i,j,k) - gxzy(i,j,k)
-     Gamxxz(i,j,k) =HALF*( t_gupxx*gxxz(i,j,k) + t_gupxy*val1 + t_gupxz*gzzx(i,j,k) )
-     Gamyxz(i,j,k) =HALF*( t_gupxy*gxxz(i,j,k) + t_gupyy*val1 + t_gupyz*gzzx(i,j,k) )
-     Gamzxz(i,j,k) =HALF*( t_gupxz*gxxz(i,j,k) + t_gupyz*val1 + t_gupzz*gzzx(i,j,k) )
-
-     val1 = gxyz(i,j,k) + gxzy(i,j,k) - gyzx(i,j,k)
-     Gamxyz(i,j,k) =HALF*( t_gupxx*val1 + t_gupxy*gyyz(i,j,k) + t_gupxz*gzzy(i,j,k) )
-     Gamyyz(i,j,k) =HALF*( t_gupxy*val1 + t_gupyy*gyyz(i,j,k) + t_gupyz*gzzy(i,j,k) )
-     Gamzyz(i,j,k) =HALF*( t_gupxz*val1 + t_gupyz*gyyz(i,j,k) + t_gupzz*gzzy(i,j,k) )
-  enddo
-  enddo
-  enddo
+  Gamxxy =HALF*( gupxx*gxxy + gupxy*gyyx + gupxz*( gxzy + gyzx - gxyz ) )
+  Gamyxy =HALF*( gupxy*gxxy + gupyy*gyyx + gupyz*( gxzy + gyzx - gxyz ) )
+  Gamzxy =HALF*( gupxz*gxxy + gupyz*gyyx + gupzz*( gxzy + gyzx - gxyz ) )
 
+  Gamxxz =HALF*( gupxx*gxxz + gupxy*( gxyz + gyzx - gxzy ) + gupxz*gzzx )
+  Gamyxz =HALF*( gupxy*gxxz + gupyy*( gxyz + gyzx - gxzy ) + gupyz*gzzx )
+  Gamzxz =HALF*( gupxz*gxxz + gupyz*( gxyz + gyzx - gxzy ) + gupzz*gzzx )
 
+  Gamxyz =HALF*( gupxx*( gxyz + gxzy - gyzx ) + gupxy*gyyz + gupxz*gzzy )
+  Gamyyz =HALF*( gupxy*( gxyz + gxzy - gyzx ) + gupyy*gyyz + gupyz*gzzy )
+  Gamzyz =HALF*( gupxz*( gxyz + gxzy - gyzx ) + gupyz*gyyz + gupzz*gzzy )
 ! Raise indices of \tilde A_{ij} and store in R_ij
 
   Rxx =    gupxx * gupxx * Axx + gupxy * gupxy * Ayy + gupxz * gupxz * Azz + &
@@ -316,40 +287,30 @@
   call fderivs(ex,Lap,Lapx,Lapy,Lapz,X,Y,Z,SYM,SYM,SYM,Symmetry,Lev)
   call fderivs(ex,trK,Kx,Ky,Kz,X,Y,Z,SYM,SYM,SYM,symmetry,Lev)
 
-  ! reuse fxx/fxy/fxz as temporaries for matter-source combinations
-  fxx = F2o3 * Kx + EIGHT * PI * Sx
-  fxy = F2o3 * Ky + EIGHT * PI * Sy
-  fxz = F2o3 * Kz + EIGHT * PI * Sz
-
-  ! reuse Gamxa/Gamya/Gamza as temporaries for chix*R combinations
-  Gamxa = chix * Rxx + chiy * Rxy + chiz * Rxz
-  Gamya = chix * Rxy + chiy * Ryy + chiz * Ryz
-  Gamza = chix * Rxz + chiy * Ryz + chiz * Rzz
-
    Gamx_rhs = - TWO * (   Lapx * Rxx +   Lapy * Rxy +   Lapz * Rxz ) + &
         TWO * alpn1 * (                                                &
-        -F3o2 * ONE/chin1 * Gamxa - &
-              gupxx * fxx - &
-              gupxy * fxy - &
-              gupxz * fxz + &
+        -F3o2/chin1 * (   chix * Rxx +   chiy * Rxy +   chiz * Rxz ) - &
+              gupxx * (   F2o3 * Kx  +  EIGHT * PI * Sx            ) - &
+              gupxy * (   F2o3 * Ky  +  EIGHT * PI * Sy            ) - &
+              gupxz * (   F2o3 * Kz  +  EIGHT * PI * Sz            ) + &
                         Gamxxx * Rxx + Gamxyy * Ryy + Gamxzz * Rzz   + &
                 TWO * ( Gamxxy * Rxy + Gamxxz * Rxz + Gamxyz * Ryz ) )
 
    Gamy_rhs = - TWO * (   Lapx * Rxy +   Lapy * Ryy +   Lapz * Ryz ) + &
         TWO * alpn1 * (                                                &
-        -F3o2 * ONE/chin1 * Gamya - &
-              gupxy * fxx - &
-              gupyy * fxy - &
-              gupyz * fxz + &
+        -F3o2/chin1 * (   chix * Rxy +  chiy * Ryy +    chiz * Ryz ) - &
+              gupxy * (   F2o3 * Kx  +  EIGHT * PI * Sx            ) - &
+              gupyy * (   F2o3 * Ky  +  EIGHT * PI * Sy            ) - &
+              gupyz * (   F2o3 * Kz  +  EIGHT * PI * Sz            ) + &
                         Gamyxx * Rxx + Gamyyy * Ryy + Gamyzz * Rzz   + &
                 TWO * ( Gamyxy * Rxy + Gamyxz * Rxz + Gamyyz * Ryz ) )
 
    Gamz_rhs = - TWO * (   Lapx * Rxz +   Lapy * Ryz +   Lapz * Rzz ) + &
         TWO * alpn1 * (                                                &
-        -F3o2 * ONE/chin1 * Gamza - &
-              gupxz * fxx - &
-              gupyz * fxy - &
-              gupzz * fxz + &
+        -F3o2/chin1 * (   chix * Rxz +  chiy * Ryz +    chiz * Rzz ) - &
+              gupxz * (   F2o3 * Kx  +  EIGHT * PI * Sx            ) - &
+              gupyz * (   F2o3 * Ky  +  EIGHT * PI * Sy            ) - &
+              gupzz * (   F2o3 * Kz  +  EIGHT * PI * Sz            ) + &
                         Gamzxx * Rxx + Gamzyy * Ryy + Gamzzz * Rzz   + &
                 TWO * ( Gamzxy * Rxy + Gamzxz * Rxz + Gamzyz * Ryz ) )
 
@@ -651,47 +612,47 @@
   fzz = fzz - Gamxzz * chix - Gamyzz * chiy - Gamzzz * chiz
 ! Store D^l D_l chi - 3/(2*chi) D^l chi D_l chi in f
 
-  f =        gupxx * ( fxx - F3o2 * ONE/chin1 * chix * chix ) + &
-             gupyy * ( fyy - F3o2 * ONE/chin1 * chiy * chiy ) + &
-             gupzz * ( fzz - F3o2 * ONE/chin1 * chiz * chiz ) + &
-       TWO * gupxy * ( fxy - F3o2 * ONE/chin1 * chix * chiy ) + &
-       TWO * gupxz * ( fxz - F3o2 * ONE/chin1 * chix * chiz ) + &
-       TWO * gupyz * ( fyz - F3o2 * ONE/chin1 * chiy * chiz ) 
+  f =        gupxx * ( fxx - F3o2/chin1 * chix * chix ) + &
+             gupyy * ( fyy - F3o2/chin1 * chiy * chiy ) + &
+             gupzz * ( fzz - F3o2/chin1 * chiz * chiz ) + &
+       TWO * gupxy * ( fxy - F3o2/chin1 * chix * chiy ) + &
+       TWO * gupxz * ( fxz - F3o2/chin1 * chix * chiz ) + &
+       TWO * gupyz * ( fyz - F3o2/chin1 * chiy * chiz ) 
 ! Add chi part to Ricci tensor:
 
-  Rxx = Rxx + (fxx - chix*chix*ONE/chin1*HALF + gxx * f) * ONE/chin1 * HALF
-  Ryy = Ryy + (fyy - chiy*chiy*ONE/chin1*HALF + gyy * f) * ONE/chin1 * HALF
-  Rzz = Rzz + (fzz - chiz*chiz*ONE/chin1*HALF + gzz * f) * ONE/chin1 * HALF
-  Rxy = Rxy + (fxy - chix*chiy*ONE/chin1*HALF + gxy * f) * ONE/chin1 * HALF
-  Rxz = Rxz + (fxz - chix*chiz*ONE/chin1*HALF + gxz * f) * ONE/chin1 * HALF
-  Ryz = Ryz + (fyz - chiy*chiz*ONE/chin1*HALF + gyz * f) * ONE/chin1 * HALF
+  Rxx = Rxx + (fxx - chix*chix/chin1/TWO + gxx * f)/chin1/TWO
+  Ryy = Ryy + (fyy - chiy*chiy/chin1/TWO + gyy * f)/chin1/TWO
+  Rzz = Rzz + (fzz - chiz*chiz/chin1/TWO + gzz * f)/chin1/TWO
+  Rxy = Rxy + (fxy - chix*chiy/chin1/TWO + gxy * f)/chin1/TWO
+  Rxz = Rxz + (fxz - chix*chiz/chin1/TWO + gxz * f)/chin1/TWO
+  Ryz = Ryz + (fyz - chiy*chiz/chin1/TWO + gyz * f)/chin1/TWO
 
 ! covariant second derivatives of the lapse respect to physical metric
   call fdderivs(ex,Lap,fxx,fxy,fxz,fyy,fyz,fzz,X,Y,Z, &
                 SYM,SYM,SYM,symmetry,Lev)
 
-  gxxx = (gupxx * chix + gupxy * chiy + gupxz * chiz) * ONE/chin1
-  gxxy = (gupxy * chix + gupyy * chiy + gupyz * chiz) * ONE/chin1
-  gxxz = (gupxz * chix + gupyz * chiy + gupzz * chiz) * ONE/chin1
+  gxxx = (gupxx * chix + gupxy * chiy + gupxz * chiz)/chin1
+  gxxy = (gupxy * chix + gupyy * chiy + gupyz * chiz)/chin1
+  gxxz = (gupxz * chix + gupyz * chiy + gupzz * chiz)/chin1
 ! now get physical second kind of connection
-  Gamxxx = Gamxxx - ( TWO * chix * ONE/chin1 - gxx * gxxx )*HALF
+  Gamxxx = Gamxxx - ( (chix + chix)/chin1 - gxx * gxxx )*HALF
   Gamyxx = Gamyxx - (                     - gxx * gxxy )*HALF
   Gamzxx = Gamzxx - (                     - gxx * gxxz )*HALF
   Gamxyy = Gamxyy - (                     - gyy * gxxx )*HALF
-  Gamyyy = Gamyyy - ( TWO * chiy * ONE/chin1 - gyy * gxxy )*HALF
+  Gamyyy = Gamyyy - ( (chiy + chiy)/chin1 - gyy * gxxy )*HALF
   Gamzyy = Gamzyy - (                     - gyy * gxxz )*HALF
   Gamxzz = Gamxzz - (                     - gzz * gxxx )*HALF
   Gamyzz = Gamyzz - (                     - gzz * gxxy )*HALF
-  Gamzzz = Gamzzz - ( TWO * chiz * ONE/chin1 - gzz * gxxz )*HALF
-  Gamxxy = Gamxxy - (  chiy * ONE/chin1 - gxy * gxxx )*HALF
-  Gamyxy = Gamyxy - (  chix * ONE/chin1 - gxy * gxxy )*HALF
+  Gamzzz = Gamzzz - ( (chiz + chiz)/chin1 - gzz * gxxz )*HALF
+  Gamxxy = Gamxxy - (  chiy        /chin1 - gxy * gxxx )*HALF
+  Gamyxy = Gamyxy - (         chix /chin1 - gxy * gxxy )*HALF
   Gamzxy = Gamzxy - (                     - gxy * gxxz )*HALF
-  Gamxxz = Gamxxz - (  chiz * ONE/chin1 - gxz * gxxx )*HALF
+  Gamxxz = Gamxxz - (  chiz        /chin1 - gxz * gxxx )*HALF
   Gamyxz = Gamyxz - (                     - gxz * gxxy )*HALF
-  Gamzxz = Gamzxz - (  chix * ONE/chin1 - gxz * gxxz )*HALF
+  Gamzxz = Gamzxz - (         chix /chin1 - gxz * gxxz )*HALF
   Gamxyz = Gamxyz - (                     - gyz * gxxx )*HALF
-  Gamyyz = Gamyyz - (  chiz * ONE/chin1 - gyz * gxxy )*HALF
-  Gamzyz = Gamzyz - (  chiy * ONE/chin1 - gyz * gxxz )*HALF
+  Gamyyz = Gamyyz - (  chiz        /chin1 - gyz * gxxy )*HALF
+  Gamzyz = Gamzyz - (         chiy /chin1 - gyz * gxxz )*HALF
 
   fxx = fxx - Gamxxx*Lapx - Gamyxx*Lapy - Gamzxx*Lapz
   fyy = fyy - Gamxyy*Lapx - Gamyyy*Lapy - Gamzyy*Lapz
@@ -734,7 +695,7 @@
        gupxz * (Axy * Azz + Ayz * Axz) + &
        gupyz * (Ayy * Azz + Ayz * Ayz) ) )) -1.6d1*PI*rho + EIGHT * PI * S
   f = - F1o3 *(  gupxx * fxx + gupyy * fyy + gupzz * fzz + &
-        TWO* ( gupxy * fxy + gupxz * fxz + gupyz * fyz ) + alpn1 * ONE/chin1 * f)
+        TWO* ( gupxy * fxy + gupxz * fxz + gupyz * fyz ) + alpn1/chin1*f)
   
   fxx = alpn1 * (Rxx - EIGHT * PI * Sxx) - fxx
   fxy = alpn1 * (Rxy - EIGHT * PI * Sxy) - fxy
@@ -854,8 +815,7 @@
   call fderivs(ex,chi,dtSfx_rhs,dtSfy_rhs,dtSfz_rhs,X,Y,Z,SYM,SYM,SYM,Symmetry,Lev)
   reta = gupxx * dtSfx_rhs * dtSfx_rhs + gupyy * dtSfy_rhs * dtSfy_rhs + gupzz * dtSfz_rhs * dtSfz_rhs + &
        TWO * (gupxy * dtSfx_rhs * dtSfy_rhs + gupxz * dtSfx_rhs * dtSfz_rhs + gupyz * dtSfy_rhs * dtSfz_rhs)
-  fxx = dsqrt(chin1)
-  reta = 1.31d0/2*dsqrt(reta*ONE/chin1)/(ONE-fxx)**2
+  reta = 1.31d0/2*dsqrt(reta/chin1)/(1-dsqrt(chin1))**2
   dtSfx_rhs = Gamx_rhs - reta*dtSfx
   dtSfy_rhs = Gamy_rhs - reta*dtSfy
   dtSfz_rhs = Gamz_rhs - reta*dtSfz
@@ -867,7 +827,7 @@
   call fderivs(ex,chi,dtSfx_rhs,dtSfy_rhs,dtSfz_rhs,X,Y,Z,SYM,SYM,SYM,Symmetry,Lev)
   reta = gupxx * dtSfx_rhs * dtSfx_rhs + gupyy * dtSfy_rhs * dtSfy_rhs + gupzz * dtSfz_rhs * dtSfz_rhs + &
        TWO * (gupxy * dtSfx_rhs * dtSfy_rhs + gupxz * dtSfx_rhs * dtSfz_rhs + gupyz * dtSfy_rhs * dtSfz_rhs)
-  reta = 1.31d0/2*dsqrt(reta*ONE/chin1)/(ONE-chin1)**2
+  reta = 1.31d0/2*dsqrt(reta/chin1)/(1-chin1)**2
   dtSfx_rhs = Gamx_rhs - reta*dtSfx
   dtSfy_rhs = Gamy_rhs - reta*dtSfy
   dtSfz_rhs = Gamz_rhs - reta*dtSfz
@@ -875,8 +835,7 @@
   call fderivs(ex,chi,dtSfx_rhs,dtSfy_rhs,dtSfz_rhs,X,Y,Z,SYM,SYM,SYM,Symmetry,Lev)
   reta = gupxx * dtSfx_rhs * dtSfx_rhs + gupyy * dtSfy_rhs * dtSfy_rhs + gupzz * dtSfz_rhs * dtSfz_rhs + &
        TWO * (gupxy * dtSfx_rhs * dtSfy_rhs + gupxz * dtSfx_rhs * dtSfz_rhs + gupyz * dtSfy_rhs * dtSfz_rhs)
-  fxx = dsqrt(chin1)
-  reta = 1.31d0/2*dsqrt(reta*ONE/chin1)/(ONE-fxx)**2
+  reta = 1.31d0/2*dsqrt(reta/chin1)/(1-dsqrt(chin1))**2
   betax_rhs = FF*Gamx - reta*betax
   betay_rhs = FF*Gamy - reta*betay
   betaz_rhs = FF*Gamz - reta*betaz
@@ -888,7 +847,7 @@
   call fderivs(ex,chi,dtSfx_rhs,dtSfy_rhs,dtSfz_rhs,X,Y,Z,SYM,SYM,SYM,Symmetry,Lev)
   reta = gupxx * dtSfx_rhs * dtSfx_rhs + gupyy * dtSfy_rhs * dtSfy_rhs + gupzz * dtSfz_rhs * dtSfz_rhs + &
        TWO * (gupxy * dtSfx_rhs * dtSfy_rhs + gupxz * dtSfx_rhs * dtSfz_rhs + gupyz * dtSfy_rhs * dtSfz_rhs)
-  reta = 1.31d0/2*dsqrt(reta*ONE/chin1)/(ONE-chin1)**2
+  reta = 1.31d0/2*dsqrt(reta/chin1)/(1-chin1)**2
   betax_rhs = FF*Gamx - reta*betax
   betay_rhs = FF*Gamy - reta*betay
   betaz_rhs = FF*Gamz - reta*betaz
@@ -986,103 +945,60 @@
   SSA(2)=SYM
   SSA(3)=ANTI
 
-!!!!!!!!!advection term part
+!!!!!!!!!advection term + Kreiss-Oliger dissipation (merged for cache efficiency)
+! lopsided_kodis shares the symmetry_bd buffer between advection and
+! dissipation, eliminating redundant full-grid copies. For metric variables
+! gxx/gyy/gzz (=dxx/dyy/dzz+1): kodis stencil coefficients sum to zero,
+! so the constant offset has no effect on dissipation.
 
-  call lopsided(ex,X,Y,Z,gxx,gxx_rhs,betax,betay,betaz,Symmetry,SSS)
-  call lopsided(ex,X,Y,Z,gxy,gxy_rhs,betax,betay,betaz,Symmetry,AAS)
-  call lopsided(ex,X,Y,Z,gxz,gxz_rhs,betax,betay,betaz,Symmetry,ASA)
-  call lopsided(ex,X,Y,Z,gyy,gyy_rhs,betax,betay,betaz,Symmetry,SSS)
-  call lopsided(ex,X,Y,Z,gyz,gyz_rhs,betax,betay,betaz,Symmetry,SAA)
-  call lopsided(ex,X,Y,Z,gzz,gzz_rhs,betax,betay,betaz,Symmetry,SSS)
+  call lopsided_kodis(ex,X,Y,Z,gxx,gxx_rhs,betax,betay,betaz,Symmetry,SSS,eps)
+  call lopsided_kodis(ex,X,Y,Z,gxy,gxy_rhs,betax,betay,betaz,Symmetry,AAS,eps)
+  call lopsided_kodis(ex,X,Y,Z,gxz,gxz_rhs,betax,betay,betaz,Symmetry,ASA,eps)
+  call lopsided_kodis(ex,X,Y,Z,gyy,gyy_rhs,betax,betay,betaz,Symmetry,SSS,eps)
+  call lopsided_kodis(ex,X,Y,Z,gyz,gyz_rhs,betax,betay,betaz,Symmetry,SAA,eps)
+  call lopsided_kodis(ex,X,Y,Z,gzz,gzz_rhs,betax,betay,betaz,Symmetry,SSS,eps)
 
-  call lopsided(ex,X,Y,Z,Axx,Axx_rhs,betax,betay,betaz,Symmetry,SSS)
-  call lopsided(ex,X,Y,Z,Axy,Axy_rhs,betax,betay,betaz,Symmetry,AAS)
-  call lopsided(ex,X,Y,Z,Axz,Axz_rhs,betax,betay,betaz,Symmetry,ASA)
-  call lopsided(ex,X,Y,Z,Ayy,Ayy_rhs,betax,betay,betaz,Symmetry,SSS)
-  call lopsided(ex,X,Y,Z,Ayz,Ayz_rhs,betax,betay,betaz,Symmetry,SAA)
-  call lopsided(ex,X,Y,Z,Azz,Azz_rhs,betax,betay,betaz,Symmetry,SSS)
+  call lopsided_kodis(ex,X,Y,Z,Axx,Axx_rhs,betax,betay,betaz,Symmetry,SSS,eps)
+  call lopsided_kodis(ex,X,Y,Z,Axy,Axy_rhs,betax,betay,betaz,Symmetry,AAS,eps)
+  call lopsided_kodis(ex,X,Y,Z,Axz,Axz_rhs,betax,betay,betaz,Symmetry,ASA,eps)
+  call lopsided_kodis(ex,X,Y,Z,Ayy,Ayy_rhs,betax,betay,betaz,Symmetry,SSS,eps)
+  call lopsided_kodis(ex,X,Y,Z,Ayz,Ayz_rhs,betax,betay,betaz,Symmetry,SAA,eps)
+  call lopsided_kodis(ex,X,Y,Z,Azz,Azz_rhs,betax,betay,betaz,Symmetry,SSS,eps)
 
-  call lopsided(ex,X,Y,Z,chi,chi_rhs,betax,betay,betaz,Symmetry,SSS)
-  call lopsided(ex,X,Y,Z,trK,trK_rhs,betax,betay,betaz,Symmetry,SSS)
+  call lopsided_kodis(ex,X,Y,Z,chi,chi_rhs,betax,betay,betaz,Symmetry,SSS,eps)
+  call lopsided_kodis(ex,X,Y,Z,trK,trK_rhs,betax,betay,betaz,Symmetry,SSS,eps)
 
-  call lopsided(ex,X,Y,Z,Gamx,Gamx_rhs,betax,betay,betaz,Symmetry,ASS)
-  call lopsided(ex,X,Y,Z,Gamy,Gamy_rhs,betax,betay,betaz,Symmetry,SAS)
-  call lopsided(ex,X,Y,Z,Gamz,Gamz_rhs,betax,betay,betaz,Symmetry,SSA)
-!!
+  call lopsided_kodis(ex,X,Y,Z,Gamx,Gamx_rhs,betax,betay,betaz,Symmetry,ASS,eps)
+  call lopsided_kodis(ex,X,Y,Z,Gamy,Gamy_rhs,betax,betay,betaz,Symmetry,SAS,eps)
+  call lopsided_kodis(ex,X,Y,Z,Gamz,Gamz_rhs,betax,betay,betaz,Symmetry,SSA,eps)
+
+#if 1 
+!! bam does not apply dissipation on gauge variables
+  call lopsided_kodis(ex,X,Y,Z,Lap,Lap_rhs,betax,betay,betaz,Symmetry,SSS,eps)
+#if (GAUGE == 0 || GAUGE == 1 || GAUGE == 2 || GAUGE == 3 || GAUGE == 4 || GAUGE == 5 || GAUGE == 6 || GAUGE == 7)
+  call lopsided_kodis(ex,X,Y,Z,betax,betax_rhs,betax,betay,betaz,Symmetry,ASS,eps)
+  call lopsided_kodis(ex,X,Y,Z,betay,betay_rhs,betax,betay,betaz,Symmetry,SAS,eps)
+  call lopsided_kodis(ex,X,Y,Z,betaz,betaz_rhs,betax,betay,betaz,Symmetry,SSA,eps)
+#endif
+#if (GAUGE == 0 || GAUGE == 2 || GAUGE == 3 || GAUGE == 6 || GAUGE == 7)
+  call lopsided_kodis(ex,X,Y,Z,dtSfx,dtSfx_rhs,betax,betay,betaz,Symmetry,ASS,eps)
+  call lopsided_kodis(ex,X,Y,Z,dtSfy,dtSfy_rhs,betax,betay,betaz,Symmetry,SAS,eps)
+  call lopsided_kodis(ex,X,Y,Z,dtSfz,dtSfz_rhs,betax,betay,betaz,Symmetry,SSA,eps)
+#endif
+#else
+! No dissipation on gauge variables (advection only)
   call lopsided(ex,X,Y,Z,Lap,Lap_rhs,betax,betay,betaz,Symmetry,SSS)
-
 #if (GAUGE == 0 || GAUGE == 1 || GAUGE == 2 || GAUGE == 3 || GAUGE == 4 || GAUGE == 5 || GAUGE == 6 || GAUGE == 7)
   call lopsided(ex,X,Y,Z,betax,betax_rhs,betax,betay,betaz,Symmetry,ASS)
   call lopsided(ex,X,Y,Z,betay,betay_rhs,betax,betay,betaz,Symmetry,SAS)
   call lopsided(ex,X,Y,Z,betaz,betaz_rhs,betax,betay,betaz,Symmetry,SSA)
 #endif
-
 #if (GAUGE == 0 || GAUGE == 2 || GAUGE == 3 || GAUGE == 6 || GAUGE == 7)
   call lopsided(ex,X,Y,Z,dtSfx,dtSfx_rhs,betax,betay,betaz,Symmetry,ASS)
   call lopsided(ex,X,Y,Z,dtSfy,dtSfy_rhs,betax,betay,betaz,Symmetry,SAS)
   call lopsided(ex,X,Y,Z,dtSfz,dtSfz_rhs,betax,betay,betaz,Symmetry,SSA)
 #endif
-
-  if(eps>0)then 
-! usual Kreiss-Oliger dissipation      
-  call kodis(ex,X,Y,Z,chi,chi_rhs,SSS,Symmetry,eps)
-  call kodis(ex,X,Y,Z,trK,trK_rhs,SSS,Symmetry,eps)
-  call kodis(ex,X,Y,Z,dxx,gxx_rhs,SSS,Symmetry,eps)
-  call kodis(ex,X,Y,Z,gxy,gxy_rhs,AAS,Symmetry,eps)
-  call kodis(ex,X,Y,Z,gxz,gxz_rhs,ASA,Symmetry,eps)
-  call kodis(ex,X,Y,Z,dyy,gyy_rhs,SSS,Symmetry,eps)
-  call kodis(ex,X,Y,Z,gyz,gyz_rhs,SAA,Symmetry,eps)
-  call kodis(ex,X,Y,Z,dzz,gzz_rhs,SSS,Symmetry,eps)
-#if 0
-#define i 42
-#define j 40
-#define k 40
-if(Lev == 1)then
-write(*,*) X(i),Y(j),Z(k)
-write(*,*) "before",Axx_rhs(i,j,k)
-endif
-#undef i
-#undef j
-#undef k
-!!stop
 #endif
-  call kodis(ex,X,Y,Z,Axx,Axx_rhs,SSS,Symmetry,eps)
-#if 0
-#define i 42
-#define j 40
-#define k 40
-if(Lev == 1)then
-write(*,*) X(i),Y(j),Z(k)
-write(*,*) "after",Axx_rhs(i,j,k)
-endif
-#undef i
-#undef j
-#undef k
-!!stop
-#endif
-  call kodis(ex,X,Y,Z,Axy,Axy_rhs,AAS,Symmetry,eps)
-  call kodis(ex,X,Y,Z,Axz,Axz_rhs,ASA,Symmetry,eps)
-  call kodis(ex,X,Y,Z,Ayy,Ayy_rhs,SSS,Symmetry,eps)
-  call kodis(ex,X,Y,Z,Ayz,Ayz_rhs,SAA,Symmetry,eps)
-  call kodis(ex,X,Y,Z,Azz,Azz_rhs,SSS,Symmetry,eps)
-  call kodis(ex,X,Y,Z,Gamx,Gamx_rhs,ASS,Symmetry,eps)
-  call kodis(ex,X,Y,Z,Gamy,Gamy_rhs,SAS,Symmetry,eps)
-  call kodis(ex,X,Y,Z,Gamz,Gamz_rhs,SSA,Symmetry,eps)
-
-#if 1 
-!! bam does not apply dissipation on gauge variables
-  call kodis(ex,X,Y,Z,Lap,Lap_rhs,SSS,Symmetry,eps)
-  call kodis(ex,X,Y,Z,betax,betax_rhs,ASS,Symmetry,eps)
-  call kodis(ex,X,Y,Z,betay,betay_rhs,SAS,Symmetry,eps)
-  call kodis(ex,X,Y,Z,betaz,betaz_rhs,SSA,Symmetry,eps)
-#if (GAUGE == 0 || GAUGE == 2 || GAUGE == 3 || GAUGE == 6 || GAUGE == 7)
-  call kodis(ex,X,Y,Z,dtSfx,dtSfx_rhs,ASS,Symmetry,eps)
-  call kodis(ex,X,Y,Z,dtSfy,dtSfy_rhs,SAS,Symmetry,eps)
-  call kodis(ex,X,Y,Z,dtSfz,dtSfz_rhs,SSA,Symmetry,eps)
-#endif
-#endif
-
-  endif
 
   if(co == 0)then
 ! ham_Res = trR + 2/3 * K^2 - A_ij * A^ij - 16 * PI * rho
@@ -1120,48 +1036,48 @@ endif
 ! mov_Res_j = gupkj*(-1/chi d_k chi*A_ij + D_k A_ij) - 2/3 d_j trK - 8 PI s_j where D respect to physical metric
 ! store D_i A_jk - 1/chi d_i chi*A_jk in gjk_i
   call fderivs(ex,Axx,gxxx,gxxy,gxxz,X,Y,Z,SYM ,SYM ,SYM ,Symmetry,0)
-  call fderivs(ex,Ayy,gyyx,gyyy,gyyz,X,Y,Z,SYM ,SYM ,SYM ,Symmetry,0)
-  call fderivs(ex,Azz,gzzx,gzzy,gzzz,X,Y,Z,SYM ,SYM ,SYM ,Symmetry,0)
   call fderivs(ex,Axy,gxyx,gxyy,gxyz,X,Y,Z,ANTI,ANTI,SYM ,Symmetry,0)
   call fderivs(ex,Axz,gxzx,gxzy,gxzz,X,Y,Z,ANTI,SYM ,ANTI,Symmetry,0)
+  call fderivs(ex,Ayy,gyyx,gyyy,gyyz,X,Y,Z,SYM ,SYM ,SYM ,Symmetry,0)
   call fderivs(ex,Ayz,gyzx,gyzy,gyzz,X,Y,Z,SYM ,ANTI,ANTI,Symmetry,0)
+  call fderivs(ex,Azz,gzzx,gzzy,gzzz,X,Y,Z,SYM ,SYM ,SYM ,Symmetry,0)
 
   gxxx = gxxx - (  Gamxxx * Axx + Gamyxx * Axy + Gamzxx * Axz &
-                 + Gamxxx * Axx + Gamyxx * Axy + Gamzxx * Axz) - chix*Axx*ONE/chin1
+                 + Gamxxx * Axx + Gamyxx * Axy + Gamzxx * Axz) - chix*Axx/chin1
   gxyx = gxyx - (  Gamxxy * Axx + Gamyxy * Axy + Gamzxy * Axz &
-                 + Gamxxx * Axy + Gamyxx * Ayy + Gamzxx * Ayz) - chix*Axy*ONE/chin1
+                 + Gamxxx * Axy + Gamyxx * Ayy + Gamzxx * Ayz) - chix*Axy/chin1
   gxzx = gxzx - (  Gamxxz * Axx + Gamyxz * Axy + Gamzxz * Axz &
-                 + Gamxxx * Axz + Gamyxx * Ayz + Gamzxx * Azz) - chix*Axz*ONE/chin1
+                 + Gamxxx * Axz + Gamyxx * Ayz + Gamzxx * Azz) - chix*Axz/chin1
   gyyx = gyyx - (  Gamxxy * Axy + Gamyxy * Ayy + Gamzxy * Ayz &
-                 + Gamxxy * Axy + Gamyxy * Ayy + Gamzxy * Ayz) - chix*Ayy*ONE/chin1
+                 + Gamxxy * Axy + Gamyxy * Ayy + Gamzxy * Ayz) - chix*Ayy/chin1
   gyzx = gyzx - (  Gamxxz * Axy + Gamyxz * Ayy + Gamzxz * Ayz &
-                 + Gamxxy * Axz + Gamyxy * Ayz + Gamzxy * Azz) - chix*Ayz*ONE/chin1
+                 + Gamxxy * Axz + Gamyxy * Ayz + Gamzxy * Azz) - chix*Ayz/chin1
   gzzx = gzzx - (  Gamxxz * Axz + Gamyxz * Ayz + Gamzxz * Azz &
-                 + Gamxxz * Axz + Gamyxz * Ayz + Gamzxz * Azz) - chix*Azz*ONE/chin1
+                 + Gamxxz * Axz + Gamyxz * Ayz + Gamzxz * Azz) - chix*Azz/chin1
   gxxy = gxxy - (  Gamxxy * Axx + Gamyxy * Axy + Gamzxy * Axz &
-                 + Gamxxy * Axx + Gamyxy * Axy + Gamzxy * Axz) - chiy*Axx*ONE/chin1
+                 + Gamxxy * Axx + Gamyxy * Axy + Gamzxy * Axz) - chiy*Axx/chin1
   gxyy = gxyy - (  Gamxyy * Axx + Gamyyy * Axy + Gamzyy * Axz &
-                 + Gamxxy * Axy + Gamyxy * Ayy + Gamzxy * Ayz) - chiy*Axy*ONE/chin1
+                 + Gamxxy * Axy + Gamyxy * Ayy + Gamzxy * Ayz) - chiy*Axy/chin1
   gxzy = gxzy - (  Gamxyz * Axx + Gamyyz * Axy + Gamzyz * Axz &
-                 + Gamxxy * Axz + Gamyxy * Ayz + Gamzxy * Azz) - chiy*Axz*ONE/chin1
+                 + Gamxxy * Axz + Gamyxy * Ayz + Gamzxy * Azz) - chiy*Axz/chin1
   gyyy = gyyy - (  Gamxyy * Axy + Gamyyy * Ayy + Gamzyy * Ayz &
-                 + Gamxyy * Axy + Gamyyy * Ayy + Gamzyy * Ayz) - chiy*Ayy*ONE/chin1
+                 + Gamxyy * Axy + Gamyyy * Ayy + Gamzyy * Ayz) - chiy*Ayy/chin1
   gyzy = gyzy - (  Gamxyz * Axy + Gamyyz * Ayy + Gamzyz * Ayz &
-                 + Gamxyy * Axz + Gamyyy * Ayz + Gamzyy * Azz) - chiy*Ayz*ONE/chin1
+                 + Gamxyy * Axz + Gamyyy * Ayz + Gamzyy * Azz) - chiy*Ayz/chin1
   gzzy = gzzy - (  Gamxyz * Axz + Gamyyz * Ayz + Gamzyz * Azz &
-                 + Gamxyz * Axz + Gamyyz * Ayz + Gamzyz * Azz) - chiy*Azz*ONE/chin1
+                 + Gamxyz * Axz + Gamyyz * Ayz + Gamzyz * Azz) - chiy*Azz/chin1
   gxxz = gxxz - (  Gamxxz * Axx + Gamyxz * Axy + Gamzxz * Axz &
-                 + Gamxxz * Axx + Gamyxz * Axy + Gamzxz * Axz) - chiz*Axx*ONE/chin1
+                 + Gamxxz * Axx + Gamyxz * Axy + Gamzxz * Axz) - chiz*Axx/chin1
   gxyz = gxyz - (  Gamxyz * Axx + Gamyyz * Axy + Gamzyz * Axz &
-                 + Gamxxz * Axy + Gamyxz * Ayy + Gamzxz * Ayz) - chiz*Axy*ONE/chin1
+                 + Gamxxz * Axy + Gamyxz * Ayy + Gamzxz * Ayz) - chiz*Axy/chin1
   gxzz = gxzz - (  Gamxzz * Axx + Gamyzz * Axy + Gamzzz * Axz &
-                 + Gamxxz * Axz + Gamyxz * Ayz + Gamzxz * Azz) - chiz*Axz*ONE/chin1
+                 + Gamxxz * Axz + Gamyxz * Ayz + Gamzxz * Azz) - chiz*Axz/chin1
   gyyz = gyyz - (  Gamxyz * Axy + Gamyyz * Ayy + Gamzyz * Ayz &
-                 + Gamxyz * Axy + Gamyyz * Ayy + Gamzyz * Ayz) - chiz*Ayy*ONE/chin1
+                 + Gamxyz * Axy + Gamyyz * Ayy + Gamzyz * Ayz) - chiz*Ayy/chin1
   gyzz = gyzz - (  Gamxzz * Axy + Gamyzz * Ayy + Gamzzz * Ayz &
-                 + Gamxyz * Axz + Gamyyz * Ayz + Gamzyz * Azz) - chiz*Ayz*ONE/chin1
+                 + Gamxyz * Axz + Gamyyz * Ayz + Gamzyz * Azz) - chiz*Ayz/chin1
   gzzz = gzzz - (  Gamxzz * Axz + Gamyzz * Ayz + Gamzzz * Azz &
-                 + Gamxzz * Axz + Gamyzz * Ayz + Gamzzz * Azz) - chiz*Azz*ONE/chin1
+                 + Gamxzz * Axz + Gamyzz * Ayz + Gamzzz * Azz) - chiz*Azz/chin1
 movx_Res = gupxx*gxxx + gupyy*gxyy + gupzz*gxzz &
           +gupxy*gxyx + gupxz*gxzx + gupyz*gxzy &
           +gupxy*gxxy + gupxz*gxxz + gupyz*gxyz

AMSS_NCKU_source/cgh.C

@@ -43,6 +43,14 @@ cgh::cgh(int ingfsi, int fngfsi, int Symmetry, char *filename, int checkrun,
   end_rank = 0;
 #endif
 
+  // Initialize load balancing variables
+  enable_load_balance = false;
+  load_balance_check_interval = 10;  // Check every 10 time steps
+  current_time_step = 0;
+  rank_interp_times = nullptr;
+  heavy_ranks = nullptr;
+  num_heavy_ranks = 0;
+
   if (!checkrun)
   {
     read_bbox(Symmetry, filename);
@@ -113,6 +121,12 @@ cgh::~cgh()
     delete[] Porgls[lev];
   }
   delete[] Porgls;
+
+  // Clean up load balancing memory
+  if (rank_interp_times)
+    delete[] rank_interp_times;
+  if (heavy_ranks)
+    delete[] heavy_ranks;
 }
 
 //================================================================================================
@@ -130,7 +144,7 @@ void cgh::compose_cgh(int nprocs)
   for (int lev = 0; lev < levels; lev++)
   {
     checkPatchList(PatL[lev], false);
-    Parallel::distribute(PatL[lev], nprocs, ingfs, fngfs, false);
+    Parallel::distribute_hard(PatL[lev], nprocs, ingfs, fngfs, false);
 #if (RPB == 1)
     // we need distributed box of PatL[lev] and PatL[lev-1]
     if (lev > 0)
@@ -1705,3 +1719,121 @@ void cgh::settrfls(const int lev)
 {
   trfls = lev;
 }
+
+//================================================================================================
+// Load Balancing Functions
+//================================================================================================
+
+// Initialize load balancing
+void cgh::init_load_balance(int nprocs)
+{
+  if (rank_interp_times)
+    delete[] rank_interp_times;
+  if (heavy_ranks)
+    delete[] heavy_ranks;
+
+  rank_interp_times = new double[nprocs];
+  heavy_ranks = new int[4];  // Maximum 4 heavy ranks
+  num_heavy_ranks = 0;
+
+  for (int i = 0; i < nprocs; i++)
+    rank_interp_times[i] = 0.0;
+}
+
+// Update interpolation time for a rank
+void cgh::update_interp_time(int rank, double time)
+{
+  if (rank_interp_times && rank >= 0)
+  {
+    rank_interp_times[rank] = time;
+  }
+}
+
+// Check and perform load balancing if needed
+bool cgh::check_and_rebalance(int nprocs, int lev,
+                               MyList<var> *OldList, MyList<var> *StateList,
+                               MyList<var> *FutureList, MyList<var> *tmList,
+                               int Symmetry, bool BB)
+{
+  int myrank;
+  MPI_Comm_rank(MPI_COMM_WORLD, &myrank);
+
+  // Only check at specified intervals
+  current_time_step++;
+  if (current_time_step % load_balance_check_interval != 0)
+    return false;
+
+  if (myrank == 0)
+  {
+    cout << "\n=== Checking load balance at time step " << current_time_step << " ===" << endl;
+  }
+
+  // Collect all rank times on rank 0
+  double *all_times = nullptr;
+  if (myrank == 0)
+  {
+    all_times = new double[nprocs];
+  }
+
+  MPI_Gather(rank_interp_times, 1, MPI_DOUBLE, all_times, 1, MPI_DOUBLE, 0, MPI_COMM_WORLD);
+
+  bool need_rebalance = false;
+
+  if (myrank == 0)
+  {
+    // Check if load balancing is needed
+    need_rebalance = Parallel::check_load_balance_need(all_times, nprocs, num_heavy_ranks, heavy_ranks);
+
+    if (need_rebalance)
+    {
+      cout << "=== Load imbalance detected! Need to rebalance ===" << endl;
+      cout << "Top " << num_heavy_ranks << " heavy ranks: ";
+      for (int i = 0; i < num_heavy_ranks; i++)
+      {
+        cout << heavy_ranks[i] << " (" << all_times[heavy_ranks[i]] << " s) ";
+      }
+      cout << endl;
+
+      // Analyze blocks that need to be split
+      Parallel::split_heavy_blocks(PatL[lev], heavy_ranks, num_heavy_ranks, 2, nprocs, ingfs, fngfs);
+
+      // Set lev_flag to trigger recompose_cgh
+      cout << "=== Triggering recompose_cgh for level " << lev << " ===" << endl;
+    }
+    else
+    {
+      cout << "=== Load is balanced, no rebalancing needed ===" << endl;
+    }
+
+    delete[] all_times;
+  }
+
+  // Broadcast the decision to all ranks
+  MPI_Bcast(&need_rebalance, 1, MPI_C_BOOL, 0, MPI_COMM_WORLD);
+
+  if (need_rebalance)
+  {
+    // Broadcast heavy ranks information
+    MPI_Bcast(&num_heavy_ranks, 1, MPI_INT, 0, MPI_COMM_WORLD);
+    MPI_Bcast(heavy_ranks, num_heavy_ranks, MPI_INT, 0, MPI_COMM_WORLD);
+
+    // Perform recompose_cgh on the specified level
+    if (myrank == 0)
+    {
+      cout << "=== Performing recompose_cgh ===" << endl;
+    }
+
+    // Call recompose_cgh_Onelevel for the specified level
+    bool *lev_flag = new bool[1];
+    lev_flag[0] = true;
+    recompose_cgh_Onelevel(nprocs, lev, OldList, StateList, FutureList, tmList, Symmetry, BB);
+    delete[] lev_flag;
+
+    // Reset time counter after rebalancing
+    current_time_step = 0;
+
+    return true;
+  }
+
+  return false;
+}

AMSS_NCKU_source/cgh.h

@@ -87,6 +87,21 @@ public:
 #if (PSTR == 1 || PSTR == 2 || PSTR == 3)
    void construct_mylev(int nprocs);
 #endif
+
+   // Load balancing support
+   bool enable_load_balance;         // Enable load balancing
+   int load_balance_check_interval;  // Check interval (in time steps)
+   int current_time_step;            // Current time step counter
+   double *rank_interp_times;        // Store interpolation times for each rank
+   int *heavy_ranks;                 // Store heavy rank numbers
+   int num_heavy_ranks;              // Number of heavy ranks
+
+   void init_load_balance(int nprocs);
+   void update_interp_time(int rank, double time);
+   bool check_and_rebalance(int nprocs, int lev,
+                           MyList<var> *OldList, MyList<var> *StateList,
+                           MyList<var> *FutureList, MyList<var> *tmList,
+                           int Symmetry, bool BB);
 };
 
 #endif /* CGH_H */

AMSS_NCKU_source/diff_new.f90

@@ -69,6 +69,8 @@
   fy = ZEO
   fz = ZEO
 
+!DIR$ SIMD VECTORLENGTHFOR(KNOWN_INTEGER=8)
+!DIR$ UNROLL PARTIAL(4)
   do k=1,ex(3)-1
   do j=1,ex(2)-1
   do i=1,ex(1)-1
@@ -371,6 +373,8 @@
   fxz = ZEO
   fyz = ZEO
 
+!DIR$ SIMD VECTORLENGTHFOR(KNOWN_INTEGER=8)
+!DIR$ UNROLL PARTIAL(4)
   do k=1,ex(3)-1
   do j=1,ex(2)-1
   do i=1,ex(1)-1
@@ -1939,309 +1943,6 @@
   return
 
   end subroutine fddyz
-  subroutine fderivs_batch4(ex,f1,f2,f3,f4, &
-                            f1x,f1y,f1z,f2x,f2y,f2z,f3x,f3y,f3z,f4x,f4y,f4z, &
-                            X,Y,Z,SYM1,SYM2,SYM3,symmetry,onoff)
-  implicit none
-
-  integer,                               intent(in ):: ex(1:3),symmetry,onoff
-  real*8,  dimension(ex(1),ex(2),ex(3)), intent(in ):: f1,f2,f3,f4
-  real*8,  dimension(ex(1),ex(2),ex(3)), intent(out):: f1x,f1y,f1z
-  real*8,  dimension(ex(1),ex(2),ex(3)), intent(out):: f2x,f2y,f2z
-  real*8,  dimension(ex(1),ex(2),ex(3)), intent(out):: f3x,f3y,f3z
-  real*8,  dimension(ex(1),ex(2),ex(3)), intent(out):: f4x,f4y,f4z
-  real*8,                                intent(in) :: X(ex(1)),Y(ex(2)),Z(ex(3))
-  real*8,                                intent(in ):: SYM1,SYM2,SYM3
-
-!~~~~~~ other variables
-
-  real*8 :: dX,dY,dZ
-  real*8,dimension(-1:ex(1),-1:ex(2),-1:ex(3)) :: fh1,fh2,fh3,fh4
-  real*8, dimension(3) :: SoA
-  integer :: imin,jmin,kmin,imax,jmax,kmax,i,j,k
-  real*8 :: d12dx,d12dy,d12dz,d2dx,d2dy,d2dz
-  integer, parameter :: NO_SYMM = 0, EQ_SYMM = 1, OCTANT = 2
-  real*8,  parameter :: ZEO=0.d0,ONE=1.d0
-  real*8,  parameter :: TWO=2.d0,EIT=8.d0
-  real*8,  parameter :: F12=1.2d1
-
-  dX = X(2)-X(1)
-  dY = Y(2)-Y(1)
-  dZ = Z(2)-Z(1)
-
-  imax = ex(1)
-  jmax = ex(2)
-  kmax = ex(3)
-
-  imin = 1
-  jmin = 1
-  kmin = 1
-  if(Symmetry > NO_SYMM .and. dabs(Z(1)) < dZ) kmin = -1
-  if(Symmetry > EQ_SYMM .and. dabs(X(1)) < dX) imin = -1
-  if(Symmetry > EQ_SYMM .and. dabs(Y(1)) < dY) jmin = -1
-
-  SoA(1) = SYM1
-  SoA(2) = SYM2
-  SoA(3) = SYM3
-
-  call symmetry_bd(2,ex,f1,fh1,SoA)
-  call symmetry_bd(2,ex,f2,fh2,SoA)
-  call symmetry_bd(2,ex,f3,fh3,SoA)
-  call symmetry_bd(2,ex,f4,fh4,SoA)
-
-  d12dx = ONE/F12/dX
-  d12dy = ONE/F12/dY
-  d12dz = ONE/F12/dZ
-
-  d2dx = ONE/TWO/dX
-  d2dy = ONE/TWO/dY
-  d2dz = ONE/TWO/dZ
-
-  f1x = ZEO; f1y = ZEO; f1z = ZEO
-  f2x = ZEO; f2y = ZEO; f2z = ZEO
-  f3x = ZEO; f3y = ZEO; f3z = ZEO
-  f4x = ZEO; f4y = ZEO; f4z = ZEO
-
-  do k=1,ex(3)-1
-  do j=1,ex(2)-1
-  do i=1,ex(1)-1
-   if(i+2 <= imax .and. i-2 >= imin .and. &
-      j+2 <= jmax .and. j-2 >= jmin .and. &
-      k+2 <= kmax .and. k-2 >= kmin) then
-      f1x(i,j,k)=d12dx*(fh1(i-2,j,k)-EIT*fh1(i-1,j,k)+EIT*fh1(i+1,j,k)-fh1(i+2,j,k))
-      f1y(i,j,k)=d12dy*(fh1(i,j-2,k)-EIT*fh1(i,j-1,k)+EIT*fh1(i,j+1,k)-fh1(i,j+2,k))
-      f1z(i,j,k)=d12dz*(fh1(i,j,k-2)-EIT*fh1(i,j,k-1)+EIT*fh1(i,j,k+1)-fh1(i,j,k+2))
-
-      f2x(i,j,k)=d12dx*(fh2(i-2,j,k)-EIT*fh2(i-1,j,k)+EIT*fh2(i+1,j,k)-fh2(i+2,j,k))
-      f2y(i,j,k)=d12dy*(fh2(i,j-2,k)-EIT*fh2(i,j-1,k)+EIT*fh2(i,j+1,k)-fh2(i,j+2,k))
-      f2z(i,j,k)=d12dz*(fh2(i,j,k-2)-EIT*fh2(i,j,k-1)+EIT*fh2(i,j,k+1)-fh2(i,j,k+2))
-
-      f3x(i,j,k)=d12dx*(fh3(i-2,j,k)-EIT*fh3(i-1,j,k)+EIT*fh3(i+1,j,k)-fh3(i+2,j,k))
-      f3y(i,j,k)=d12dy*(fh3(i,j-2,k)-EIT*fh3(i,j-1,k)+EIT*fh3(i,j+1,k)-fh3(i,j+2,k))
-      f3z(i,j,k)=d12dz*(fh3(i,j,k-2)-EIT*fh3(i,j,k-1)+EIT*fh3(i,j,k+1)-fh3(i,j,k+2))
-
-      f4x(i,j,k)=d12dx*(fh4(i-2,j,k)-EIT*fh4(i-1,j,k)+EIT*fh4(i+1,j,k)-fh4(i+2,j,k))
-      f4y(i,j,k)=d12dy*(fh4(i,j-2,k)-EIT*fh4(i,j-1,k)+EIT*fh4(i,j+1,k)-fh4(i,j+2,k))
-      f4z(i,j,k)=d12dz*(fh4(i,j,k-2)-EIT*fh4(i,j,k-1)+EIT*fh4(i,j,k+1)-fh4(i,j,k+2))
-   elseif(i+1 <= imax .and. i-1 >= imin .and. &
-          j+1 <= jmax .and. j-1 >= jmin .and. &
-          k+1 <= kmax .and. k-1 >= kmin) then
-      f1x(i,j,k)=d2dx*(-fh1(i-1,j,k)+fh1(i+1,j,k))
-      f1y(i,j,k)=d2dy*(-fh1(i,j-1,k)+fh1(i,j+1,k))
-      f1z(i,j,k)=d2dz*(-fh1(i,j,k-1)+fh1(i,j,k+1))
-
-      f2x(i,j,k)=d2dx*(-fh2(i-1,j,k)+fh2(i+1,j,k))
-      f2y(i,j,k)=d2dy*(-fh2(i,j-1,k)+fh2(i,j+1,k))
-      f2z(i,j,k)=d2dz*(-fh2(i,j,k-1)+fh2(i,j,k+1))
-
-      f3x(i,j,k)=d2dx*(-fh3(i-1,j,k)+fh3(i+1,j,k))
-      f3y(i,j,k)=d2dy*(-fh3(i,j-1,k)+fh3(i,j+1,k))
-      f3z(i,j,k)=d2dz*(-fh3(i,j,k-1)+fh3(i,j,k+1))
-
-      f4x(i,j,k)=d2dx*(-fh4(i-1,j,k)+fh4(i+1,j,k))
-      f4y(i,j,k)=d2dy*(-fh4(i,j-1,k)+fh4(i,j+1,k))
-      f4z(i,j,k)=d2dz*(-fh4(i,j,k-1)+fh4(i,j,k+1))
-   endif
-  enddo
-  enddo
-  enddo
-
-  return
-
-  end subroutine fderivs_batch4
-!-----------------------------------------------------------------------------
-! batch first derivatives (3 fields), same symmetry setup
-!-----------------------------------------------------------------------------
-  subroutine fderivs_batch3(ex,f1,f2,f3, &
-                            f1x,f1y,f1z,f2x,f2y,f2z,f3x,f3y,f3z, &
-                            X,Y,Z,SYM1,SYM2,SYM3,symmetry,onoff)
-  implicit none
-
-  integer,                               intent(in ):: ex(1:3),symmetry,onoff
-  real*8,  dimension(ex(1),ex(2),ex(3)), intent(in ):: f1,f2,f3
-  real*8,  dimension(ex(1),ex(2),ex(3)), intent(out):: f1x,f1y,f1z
-  real*8,  dimension(ex(1),ex(2),ex(3)), intent(out):: f2x,f2y,f2z
-  real*8,  dimension(ex(1),ex(2),ex(3)), intent(out):: f3x,f3y,f3z
-  real*8,                                intent(in) :: X(ex(1)),Y(ex(2)),Z(ex(3))
-  real*8,                                intent(in ):: SYM1,SYM2,SYM3
-
-!~~~~~~ other variables
-
-  real*8 :: dX,dY,dZ
-  real*8,dimension(-1:ex(1),-1:ex(2),-1:ex(3)) :: fh1,fh2,fh3
-  real*8, dimension(3) :: SoA
-  integer :: imin,jmin,kmin,imax,jmax,kmax,i,j,k
-  real*8 :: d12dx,d12dy,d12dz,d2dx,d2dy,d2dz
-  integer, parameter :: NO_SYMM = 0, EQ_SYMM = 1, OCTANT = 2
-  real*8,  parameter :: ZEO=0.d0,ONE=1.d0
-  real*8,  parameter :: TWO=2.d0,EIT=8.d0
-  real*8,  parameter :: F12=1.2d1
-
-  dX = X(2)-X(1)
-  dY = Y(2)-Y(1)
-  dZ = Z(2)-Z(1)
-
-  imax = ex(1)
-  jmax = ex(2)
-  kmax = ex(3)
-
-  imin = 1
-  jmin = 1
-  kmin = 1
-  if(Symmetry > NO_SYMM .and. dabs(Z(1)) < dZ) kmin = -1
-  if(Symmetry > EQ_SYMM .and. dabs(X(1)) < dX) imin = -1
-  if(Symmetry > EQ_SYMM .and. dabs(Y(1)) < dY) jmin = -1
-
-  SoA(1) = SYM1
-  SoA(2) = SYM2
-  SoA(3) = SYM3
-
-  call symmetry_bd(2,ex,f1,fh1,SoA)
-  call symmetry_bd(2,ex,f2,fh2,SoA)
-  call symmetry_bd(2,ex,f3,fh3,SoA)
-
-  d12dx = ONE/F12/dX
-  d12dy = ONE/F12/dY
-  d12dz = ONE/F12/dZ
-
-  d2dx = ONE/TWO/dX
-  d2dy = ONE/TWO/dY
-  d2dz = ONE/TWO/dZ
-
-  f1x = ZEO; f1y = ZEO; f1z = ZEO
-  f2x = ZEO; f2y = ZEO; f2z = ZEO
-  f3x = ZEO; f3y = ZEO; f3z = ZEO
-
-  do k=1,ex(3)-1
-  do j=1,ex(2)-1
-  do i=1,ex(1)-1
-   if(i+2 <= imax .and. i-2 >= imin .and. &
-      j+2 <= jmax .and. j-2 >= jmin .and. &
-      k+2 <= kmax .and. k-2 >= kmin) then
-      f1x(i,j,k)=d12dx*(fh1(i-2,j,k)-EIT*fh1(i-1,j,k)+EIT*fh1(i+1,j,k)-fh1(i+2,j,k))
-      f1y(i,j,k)=d12dy*(fh1(i,j-2,k)-EIT*fh1(i,j-1,k)+EIT*fh1(i,j+1,k)-fh1(i,j+2,k))
-      f1z(i,j,k)=d12dz*(fh1(i,j,k-2)-EIT*fh1(i,j,k-1)+EIT*fh1(i,j,k+1)-fh1(i,j,k+2))
-
-      f2x(i,j,k)=d12dx*(fh2(i-2,j,k)-EIT*fh2(i-1,j,k)+EIT*fh2(i+1,j,k)-fh2(i+2,j,k))
-      f2y(i,j,k)=d12dy*(fh2(i,j-2,k)-EIT*fh2(i,j-1,k)+EIT*fh2(i,j+1,k)-fh2(i,j+2,k))
-      f2z(i,j,k)=d12dz*(fh2(i,j,k-2)-EIT*fh2(i,j,k-1)+EIT*fh2(i,j,k+1)-fh2(i,j,k+2))
-
-      f3x(i,j,k)=d12dx*(fh3(i-2,j,k)-EIT*fh3(i-1,j,k)+EIT*fh3(i+1,j,k)-fh3(i+2,j,k))
-      f3y(i,j,k)=d12dy*(fh3(i,j-2,k)-EIT*fh3(i,j-1,k)+EIT*fh3(i,j+1,k)-fh3(i,j+2,k))
-      f3z(i,j,k)=d12dz*(fh3(i,j,k-2)-EIT*fh3(i,j,k-1)+EIT*fh3(i,j,k+1)-fh3(i,j,k+2))
-   elseif(i+1 <= imax .and. i-1 >= imin .and. &
-          j+1 <= jmax .and. j-1 >= jmin .and. &
-          k+1 <= kmax .and. k-1 >= kmin) then
-      f1x(i,j,k)=d2dx*(-fh1(i-1,j,k)+fh1(i+1,j,k))
-      f1y(i,j,k)=d2dy*(-fh1(i,j-1,k)+fh1(i,j+1,k))
-      f1z(i,j,k)=d2dz*(-fh1(i,j,k-1)+fh1(i,j,k+1))
-
-      f2x(i,j,k)=d2dx*(-fh2(i-1,j,k)+fh2(i+1,j,k))
-      f2y(i,j,k)=d2dy*(-fh2(i,j-1,k)+fh2(i,j+1,k))
-      f2z(i,j,k)=d2dz*(-fh2(i,j,k-1)+fh2(i,j,k+1))
-
-      f3x(i,j,k)=d2dx*(-fh3(i-1,j,k)+fh3(i+1,j,k))
-      f3y(i,j,k)=d2dy*(-fh3(i,j-1,k)+fh3(i,j+1,k))
-      f3z(i,j,k)=d2dz*(-fh3(i,j,k-1)+fh3(i,j,k+1))
-   endif
-  enddo
-  enddo
-  enddo
-
-  return
-
-  end subroutine fderivs_batch3
-!-----------------------------------------------------------------------------
-! batch first derivatives (2 fields), same symmetry setup
-!-----------------------------------------------------------------------------
-  subroutine fderivs_batch2(ex,f1,f2, &
-                            f1x,f1y,f1z,f2x,f2y,f2z, &
-                            X,Y,Z,SYM1,SYM2,SYM3,symmetry,onoff)
-  implicit none
-
-  integer,                               intent(in ):: ex(1:3),symmetry,onoff
-  real*8,  dimension(ex(1),ex(2),ex(3)), intent(in ):: f1,f2
-  real*8,  dimension(ex(1),ex(2),ex(3)), intent(out):: f1x,f1y,f1z
-  real*8,  dimension(ex(1),ex(2),ex(3)), intent(out):: f2x,f2y,f2z
-  real*8,                                intent(in) :: X(ex(1)),Y(ex(2)),Z(ex(3))
-  real*8,                                intent(in ):: SYM1,SYM2,SYM3
-
-!~~~~~~ other variables
-
-  real*8 :: dX,dY,dZ
-  real*8,dimension(-1:ex(1),-1:ex(2),-1:ex(3)) :: fh1,fh2
-  real*8, dimension(3) :: SoA
-  integer :: imin,jmin,kmin,imax,jmax,kmax,i,j,k
-  real*8 :: d12dx,d12dy,d12dz,d2dx,d2dy,d2dz
-  integer, parameter :: NO_SYMM = 0, EQ_SYMM = 1, OCTANT = 2
-  real*8,  parameter :: ZEO=0.d0,ONE=1.d0
-  real*8,  parameter :: TWO=2.d0,EIT=8.d0
-  real*8,  parameter :: F12=1.2d1
-
-  dX = X(2)-X(1)
-  dY = Y(2)-Y(1)
-  dZ = Z(2)-Z(1)
-
-  imax = ex(1)
-  jmax = ex(2)
-  kmax = ex(3)
-
-  imin = 1
-  jmin = 1
-  kmin = 1
-  if(Symmetry > NO_SYMM .and. dabs(Z(1)) < dZ) kmin = -1
-  if(Symmetry > EQ_SYMM .and. dabs(X(1)) < dX) imin = -1
-  if(Symmetry > EQ_SYMM .and. dabs(Y(1)) < dY) jmin = -1
-
-  SoA(1) = SYM1
-  SoA(2) = SYM2
-  SoA(3) = SYM3
-
-  call symmetry_bd(2,ex,f1,fh1,SoA)
-  call symmetry_bd(2,ex,f2,fh2,SoA)
-
-  d12dx = ONE/F12/dX
-  d12dy = ONE/F12/dY
-  d12dz = ONE/F12/dZ
-
-  d2dx = ONE/TWO/dX
-  d2dy = ONE/TWO/dY
-  d2dz = ONE/TWO/dZ
-
-  f1x = ZEO; f1y = ZEO; f1z = ZEO
-  f2x = ZEO; f2y = ZEO; f2z = ZEO
-
-  do k=1,ex(3)-1
-  do j=1,ex(2)-1
-  do i=1,ex(1)-1
-   if(i+2 <= imax .and. i-2 >= imin .and. &
-      j+2 <= jmax .and. j-2 >= jmin .and. &
-      k+2 <= kmax .and. k-2 >= kmin) then
-      f1x(i,j,k)=d12dx*(fh1(i-2,j,k)-EIT*fh1(i-1,j,k)+EIT*fh1(i+1,j,k)-fh1(i+2,j,k))
-      f1y(i,j,k)=d12dy*(fh1(i,j-2,k)-EIT*fh1(i,j-1,k)+EIT*fh1(i,j+1,k)-fh1(i,j+2,k))
-      f1z(i,j,k)=d12dz*(fh1(i,j,k-2)-EIT*fh1(i,j,k-1)+EIT*fh1(i,j,k+1)-fh1(i,j,k+2))
-
-      f2x(i,j,k)=d12dx*(fh2(i-2,j,k)-EIT*fh2(i-1,j,k)+EIT*fh2(i+1,j,k)-fh2(i+2,j,k))
-      f2y(i,j,k)=d12dy*(fh2(i,j-2,k)-EIT*fh2(i,j-1,k)+EIT*fh2(i,j+1,k)-fh2(i,j+2,k))
-      f2z(i,j,k)=d12dz*(fh2(i,j,k-2)-EIT*fh2(i,j,k-1)+EIT*fh2(i,j,k+1)-fh2(i,j,k+2))
-   elseif(i+1 <= imax .and. i-1 >= imin .and. &
-          j+1 <= jmax .and. j-1 >= jmin .and. &
-          k+1 <= kmax .and. k-1 >= kmin) then
-      f1x(i,j,k)=d2dx*(-fh1(i-1,j,k)+fh1(i+1,j,k))
-      f1y(i,j,k)=d2dy*(-fh1(i,j-1,k)+fh1(i,j+1,k))
-      f1z(i,j,k)=d2dz*(-fh1(i,j,k-1)+fh1(i,j,k+1))
-
-      f2x(i,j,k)=d2dx*(-fh2(i-1,j,k)+fh2(i+1,j,k))
-      f2y(i,j,k)=d2dy*(-fh2(i,j-1,k)+fh2(i,j+1,k))
-      f2z(i,j,k)=d2dz*(-fh2(i,j,k-1)+fh2(i,j,k+1))
-   endif
-  enddo
-  enddo
-  enddo
-
-  return
-
-  end subroutine fderivs_batch2
 
 #elif (ghost_width == 4)
 ! sixth order code
@@ -2380,9 +2081,6 @@
 
   end subroutine fderivs
 !-----------------------------------------------------------------------------
-! batch first derivatives (4 fields), same symmetry setup
-!-----------------------------------------------------------------------------
-!-----------------------------------------------------------------------------
 !
 ! single derivatives dx
 !

AMSS_NCKU_source/enforce_algebra.f90

@@ -19,48 +19,60 @@
 
 !~~~~~~~> Local variable:
 
-  real*8, dimension(ex(1),ex(2),ex(3)) :: trA,detg
-  real*8, dimension(ex(1),ex(2),ex(3)) :: gxx,gyy,gzz 
-  real*8, dimension(ex(1),ex(2),ex(3)) :: gupxx,gupxy,gupxz,gupyy,gupyz,gupzz
+  integer :: i,j,k
+  real*8 :: lgxx,lgyy,lgzz,ldetg
+  real*8 :: lgupxx,lgupxy,lgupxz,lgupyy,lgupyz,lgupzz
+  real*8 :: ltrA,lscale
   real*8, parameter :: F1o3 = 1.D0 / 3.D0, ONE = 1.D0, TWO = 2.D0
 
 !~~~~~~>
 
-  gxx = dxx + ONE
-  gyy = dyy + ONE
-  gzz = dzz + ONE
+  do k=1,ex(3)
+  do j=1,ex(2)
+  do i=1,ex(1)
 
-  detg =  gxx * gyy * gzz + gxy * gyz * gxz + gxz * gxy * gyz - &
-          gxz * gyy * gxz - gxy * gxy * gzz - gxx * gyz * gyz
-  gupxx =   ( gyy * gzz - gyz * gyz ) / detg
-  gupxy = - ( gxy * gzz - gyz * gxz ) / detg
-  gupxz =   ( gxy * gyz - gyy * gxz ) / detg
-  gupyy =   ( gxx * gzz - gxz * gxz ) / detg
-  gupyz = - ( gxx * gyz - gxy * gxz ) / detg
-  gupzz =   ( gxx * gyy - gxy * gxy ) / detg
+    lgxx = dxx(i,j,k) + ONE
+    lgyy = dyy(i,j,k) + ONE
+    lgzz = dzz(i,j,k) + ONE
 
-  trA =         gupxx * Axx + gupyy * Ayy + gupzz * Azz &
-       + TWO * (gupxy * Axy + gupxz * Axz + gupyz * Ayz)
+    ldetg =  lgxx * lgyy * lgzz &
+           + gxy(i,j,k) * gyz(i,j,k) * gxz(i,j,k) &
+           + gxz(i,j,k) * gxy(i,j,k) * gyz(i,j,k) &
+           - gxz(i,j,k) * lgyy * gxz(i,j,k) &
+           - gxy(i,j,k) * gxy(i,j,k) * lgzz &
+           - lgxx * gyz(i,j,k) * gyz(i,j,k)
 
-  Axx = Axx - F1o3 * gxx * trA
-  Axy = Axy - F1o3 * gxy * trA
-  Axz = Axz - F1o3 * gxz * trA
-  Ayy = Ayy - F1o3 * gyy * trA
-  Ayz = Ayz - F1o3 * gyz * trA
-  Azz = Azz - F1o3 * gzz * trA
+    lgupxx =   ( lgyy * lgzz - gyz(i,j,k) * gyz(i,j,k) ) / ldetg
+    lgupxy = - ( gxy(i,j,k) * lgzz - gyz(i,j,k) * gxz(i,j,k) ) / ldetg
+    lgupxz =   ( gxy(i,j,k) * gyz(i,j,k) - lgyy * gxz(i,j,k) ) / ldetg
+    lgupyy =   ( lgxx * lgzz - gxz(i,j,k) * gxz(i,j,k) ) / ldetg
+    lgupyz = - ( lgxx * gyz(i,j,k) - gxy(i,j,k) * gxz(i,j,k) ) / ldetg
+    lgupzz =   ( lgxx * lgyy - gxy(i,j,k) * gxy(i,j,k) ) / ldetg
 
-  detg = ONE / ( detg ** F1o3 ) 
+    ltrA =         lgupxx * Axx(i,j,k) + lgupyy * Ayy(i,j,k) &
+                 + lgupzz * Azz(i,j,k) &
+         + TWO * (lgupxy * Axy(i,j,k) + lgupxz * Axz(i,j,k) &
+                 + lgupyz * Ayz(i,j,k))
 
-  gxx = gxx * detg
-  gxy = gxy * detg
-  gxz = gxz * detg
-  gyy = gyy * detg
-  gyz = gyz * detg
-  gzz = gzz * detg
+    Axx(i,j,k) = Axx(i,j,k) - F1o3 * lgxx * ltrA
+    Axy(i,j,k) = Axy(i,j,k) - F1o3 * gxy(i,j,k) * ltrA
+    Axz(i,j,k) = Axz(i,j,k) - F1o3 * gxz(i,j,k) * ltrA
+    Ayy(i,j,k) = Ayy(i,j,k) - F1o3 * lgyy * ltrA
+    Ayz(i,j,k) = Ayz(i,j,k) - F1o3 * gyz(i,j,k) * ltrA
+    Azz(i,j,k) = Azz(i,j,k) - F1o3 * lgzz * ltrA
 
-  dxx = gxx - ONE
-  dyy = gyy - ONE
-  dzz = gzz - ONE
+    lscale = ONE / ( ldetg ** F1o3 )
+
+    dxx(i,j,k) = lgxx * lscale - ONE
+    gxy(i,j,k) = gxy(i,j,k) * lscale
+    gxz(i,j,k) = gxz(i,j,k) * lscale
+    dyy(i,j,k) = lgyy * lscale - ONE
+    gyz(i,j,k) = gyz(i,j,k) * lscale
+    dzz(i,j,k) = lgzz * lscale - ONE
+
+  enddo
+  enddo
+  enddo
 
   return
 
@@ -83,50 +95,70 @@
 
 !~~~~~~~> Local variable:
 
-  real*8, dimension(ex(1),ex(2),ex(3)) :: trA
-  real*8, dimension(ex(1),ex(2),ex(3)) :: gxx,gyy,gzz 
-  real*8, dimension(ex(1),ex(2),ex(3)) :: gupxx,gupxy,gupxz,gupyy,gupyz,gupzz
+  integer :: i,j,k
+  real*8 :: lgxx,lgyy,lgzz,lscale
+  real*8 :: lgxy,lgxz,lgyz
+  real*8 :: lgupxx,lgupxy,lgupxz,lgupyy,lgupyz,lgupzz
+  real*8 :: ltrA
   real*8, parameter :: F1o3 = 1.D0 / 3.D0, ONE = 1.D0, TWO = 2.D0
 
 !~~~~~~>
 
-  gxx = dxx + ONE
-  gyy = dyy + ONE
-  gzz = dzz + ONE
-! for g
-  gupzz =  gxx * gyy * gzz + gxy * gyz * gxz + gxz * gxy * gyz - &
-           gxz * gyy * gxz - gxy * gxy * gzz - gxx * gyz * gyz
+  do k=1,ex(3)
+  do j=1,ex(2)
+  do i=1,ex(1)
 
-  gupzz = ONE / ( gupzz ** F1o3 ) 
+! for g: normalize determinant first
+    lgxx = dxx(i,j,k) + ONE
+    lgyy = dyy(i,j,k) + ONE
+    lgzz = dzz(i,j,k) + ONE
+    lgxy = gxy(i,j,k)
+    lgxz = gxz(i,j,k)
+    lgyz = gyz(i,j,k)
 
-  gxx = gxx * gupzz
-  gxy = gxy * gupzz
-  gxz = gxz * gupzz
-  gyy = gyy * gupzz
-  gyz = gyz * gupzz
-  gzz = gzz * gupzz
+    lscale =  lgxx * lgyy * lgzz + lgxy * lgyz * lgxz &
+            + lgxz * lgxy * lgyz - lgxz * lgyy * lgxz &
+            - lgxy * lgxy * lgzz - lgxx * lgyz * lgyz
 
-  dxx = gxx - ONE
-  dyy = gyy - ONE
-  dzz = gzz - ONE
-! for A  
+    lscale = ONE / ( lscale ** F1o3 )
 
-  gupxx =   ( gyy * gzz - gyz * gyz )
-  gupxy = - ( gxy * gzz - gyz * gxz )
-  gupxz =   ( gxy * gyz - gyy * gxz )
-  gupyy =   ( gxx * gzz - gxz * gxz )
-  gupyz = - ( gxx * gyz - gxy * gxz )
-  gupzz =   ( gxx * gyy - gxy * gxy )
+    lgxx = lgxx * lscale
+    lgxy = lgxy * lscale
+    lgxz = lgxz * lscale
+    lgyy = lgyy * lscale
+    lgyz = lgyz * lscale
+    lgzz = lgzz * lscale
 
-  trA =         gupxx * Axx + gupyy * Ayy + gupzz * Azz &
-       + TWO * (gupxy * Axy + gupxz * Axz + gupyz * Ayz)
+    dxx(i,j,k) = lgxx - ONE
+    gxy(i,j,k) = lgxy
+    gxz(i,j,k) = lgxz
+    dyy(i,j,k) = lgyy - ONE
+    gyz(i,j,k) = lgyz
+    dzz(i,j,k) = lgzz - ONE
 
-  Axx = Axx - F1o3 * gxx * trA
-  Axy = Axy - F1o3 * gxy * trA
-  Axz = Axz - F1o3 * gxz * trA
-  Ayy = Ayy - F1o3 * gyy * trA
-  Ayz = Ayz - F1o3 * gyz * trA
-  Azz = Azz - F1o3 * gzz * trA
+! for A: trace-free using normalized metric (det=1, no division needed)
+    lgupxx =   ( lgyy * lgzz - lgyz * lgyz )
+    lgupxy = - ( lgxy * lgzz - lgyz * lgxz )
+    lgupxz =   ( lgxy * lgyz - lgyy * lgxz )
+    lgupyy =   ( lgxx * lgzz - lgxz * lgxz )
+    lgupyz = - ( lgxx * lgyz - lgxy * lgxz )
+    lgupzz =   ( lgxx * lgyy - lgxy * lgxy )
+
+    ltrA =         lgupxx * Axx(i,j,k) + lgupyy * Ayy(i,j,k) &
+                 + lgupzz * Azz(i,j,k) &
+         + TWO * (lgupxy * Axy(i,j,k) + lgupxz * Axz(i,j,k) &
+                 + lgupyz * Ayz(i,j,k))
+
+    Axx(i,j,k) = Axx(i,j,k) - F1o3 * lgxx * ltrA
+    Axy(i,j,k) = Axy(i,j,k) - F1o3 * lgxy * ltrA
+    Axz(i,j,k) = Axz(i,j,k) - F1o3 * lgxz * ltrA
+    Ayy(i,j,k) = Ayy(i,j,k) - F1o3 * lgyy * ltrA
+    Ayz(i,j,k) = Ayz(i,j,k) - F1o3 * lgyz * ltrA
+    Azz(i,j,k) = Azz(i,j,k) - F1o3 * lgzz * ltrA
+
+  enddo
+  enddo
+  enddo
 
   return
 

AMSS_NCKU_source/fmisc.f90

@@ -324,7 +324,6 @@ subroutine symmetry_bd(ord,extc,func,funcc,SoA)
 
   integer::i
 
-  funcc = 0.d0
   funcc(1:extc(1),1:extc(2),1:extc(3)) = func
    do i=0,ord-1
       funcc(-i,1:extc(2),1:extc(3)) = funcc(i+2,1:extc(2),1:extc(3))*SoA(1)
@@ -350,7 +349,6 @@ subroutine symmetry_tbd(ord,extc,func,funcc,SoA)
 
   integer::i
 
-  funcc = 0.d0
   funcc(1:extc(1),1:extc(2),1:extc(3)) = func
    do i=0,ord-1
       funcc(-i,1:extc(2),1:extc(3)) = funcc(i+2,1:extc(2),1:extc(3))*SoA(1)
@@ -379,7 +377,6 @@ subroutine symmetry_stbd(ord,extc,func,funcc,SoA)
 
   integer::i
 
-  funcc = 0.d0
   funcc(1:extc(1),1:extc(2),1:extc(3)) = func
    do i=0,ord-1
       funcc(-i,1:extc(2),1:extc(3)) = funcc(i+2,1:extc(2),1:extc(3))*SoA(1)
@@ -886,14 +883,17 @@ subroutine symmetry_bd(ord,extc,func,funcc,SoA)
 
   integer::i
 
-  funcc = 0.d0
+!DIR$ SIMD VECTORLENGTHFOR(KNOWN_INTEGER=8)
   funcc(1:extc(1),1:extc(2),1:extc(3)) = func
+!DIR$ SIMD VECTORLENGTHFOR(KNOWN_INTEGER=8)
    do i=0,ord-1
       funcc(-i,1:extc(2),1:extc(3)) = funcc(i+1,1:extc(2),1:extc(3))*SoA(1)
    enddo
+!DIR$ SIMD VECTORLENGTHFOR(KNOWN_INTEGER=8)
    do i=0,ord-1
       funcc(:,-i,1:extc(3)) = funcc(:,i+1,1:extc(3))*SoA(2)
    enddo
+!DIR$ SIMD VECTORLENGTHFOR(KNOWN_INTEGER=8)
    do i=0,ord-1
       funcc(:,:,-i) = funcc(:,:,i+1)*SoA(3)
    enddo
@@ -912,7 +912,6 @@ subroutine symmetry_tbd(ord,extc,func,funcc,SoA)
 
   integer::i
 
-  funcc = 0.d0
   funcc(1:extc(1),1:extc(2),1:extc(3)) = func
    do i=0,ord-1
       funcc(-i,1:extc(2),1:extc(3)) = funcc(i+1,1:extc(2),1:extc(3))*SoA(1)
@@ -941,7 +940,6 @@ subroutine symmetry_stbd(ord,extc,func,funcc,SoA)
 
   integer::i
 
-  funcc = 0.d0
   funcc(1:extc(1),1:extc(2),1:extc(3)) = func
    do i=0,ord-1
       funcc(-i,1:extc(2),1:extc(3)) = funcc(i+1,1:extc(2),1:extc(3))*SoA(1)
@@ -1118,64 +1116,66 @@ end subroutine d2dump
 ! Lagrangian polynomial interpolation
 !------------------------------------------------------------------------------
 
-  subroutine polint(xa,ya,x,y,dy,ordn)
-
+!DIR$ ATTRIBUTES FORCEINLINE :: polint
+  subroutine polint(xa, ya, x, y, dy, ordn)
   implicit none
 
-!~~~~~~> Input Parameter:
-  integer,intent(in) :: ordn
-  real*8, dimension(ordn), intent(in) :: xa,ya
+  integer, intent(in) :: ordn
+  real*8, dimension(ordn), intent(in) :: xa, ya
   real*8, intent(in) :: x
-  real*8, intent(out) :: y,dy
+  real*8, intent(out) :: y, dy
 
-!~~~~~~> Other parameter:
+  integer :: i, m, ns, n_m
+  real*8, dimension(ordn) :: c, d, ho
+  real*8 :: dif, dift, hp, h, den_val
 
-  integer :: m,n,ns
-  real*8, dimension(ordn) :: c,d,den,ho
-  real*8 :: dif,dift
+  c = ya
+  d = ya
+  ho = xa - x
 
-!~~~~~~>
+  ns = 1
+  dif = abs(x - xa(1))
 
-  n=ordn
-  m=ordn
-
-  c=ya
-  d=ya
-  ho=xa-x
-
-  ns=1
-  dif=abs(x-xa(1))
-  do m=1,n
-   dift=abs(x-xa(m))
-   if(dift < dif) then
-    ns=m
-    dif=dift
+  do i = 2, ordn
+    dift = abs(x - xa(i))
+    if (dift < dif) then
+      ns = i
+      dif = dift
     end if
   end do
 
-  y=ya(ns)
-  ns=ns-1
-  do m=1,n-1
-    den(1:n-m)=ho(1:n-m)-ho(1+m:n)
-    if (any(den(1:n-m) == 0.0))then
+  y = ya(ns)
+  ns = ns - 1
+
+  do m = 1, ordn - 1
+    n_m = ordn - m
+    do i = 1, n_m
+      hp = ho(i)
+      h  = ho(i+m)
+      den_val = hp - h
+
+      if (den_val == 0.0d0) then
         write(*,*) 'failure in polint for point',x
         write(*,*) 'with input points: ',xa
         stop
-    endif
-    den(1:n-m)=(c(2:n-m+1)-d(1:n-m))/den(1:n-m)
-    d(1:n-m)=ho(1+m:n)*den(1:n-m)
-    c(1:n-m)=ho(1:n-m)*den(1:n-m)
-    if (2*ns < n-m) then
-      dy=c(ns+1)
-    else
-      dy=d(ns)
-      ns=ns-1
       end if
-    y=y+dy
+
+      den_val = (c(i+1) - d(i)) / den_val
+
+      d(i) = h * den_val
+      c(i) = hp * den_val
+    end do
+
+    if (2 * ns < n_m) then
+      dy = c(ns + 1)
+    else
+      dy = d(ns)
+      ns = ns - 1
+    end if
+    y = y + dy
   end do
 
   return
-
   end subroutine polint
 !------------------------------------------------------------------------------
 !
@@ -1183,35 +1183,37 @@ end subroutine d2dump
 !
 !------------------------------------------------------------------------------
   subroutine polin2(x1a,x2a,ya,x1,x2,y,dy,ordn)
-
   implicit none
 
-!~~~~~~> Input parameters:
   integer,intent(in) :: ordn
   real*8, dimension(1:ordn), intent(in) :: x1a,x2a
   real*8, dimension(1:ordn,1:ordn), intent(in) :: ya
   real*8, intent(in) :: x1,x2
   real*8, intent(out) :: y,dy
 
-!~~~~~~> Other parameters:
-
+#ifdef POLINT_LEGACY_ORDER
   integer  :: i,m
   real*8, dimension(ordn) :: ymtmp
   real*8, dimension(ordn) :: yntmp
 
   m=size(x1a)
-  
   do i=1,m
-
     yntmp=ya(i,:)
     call polint(x2a,yntmp,x2,ymtmp(i),dy,ordn)
-
   end do
-
   call polint(x1a,ymtmp,x1,y,dy,ordn)
+#else
+  integer  :: j
+  real*8, dimension(ordn) :: ymtmp
+  real*8 :: dy_temp
+
+  do j=1,ordn
+    call polint(x1a, ya(:,j), x1, ymtmp(j), dy_temp, ordn)
+  end do
+  call polint(x2a, ymtmp, x2, y, dy, ordn)
+#endif
 
   return
-
   end subroutine polin2
 !------------------------------------------------------------------------------
 !
@@ -1219,18 +1221,15 @@ end subroutine d2dump
 !
 !------------------------------------------------------------------------------
   subroutine polin3(x1a,x2a,x3a,ya,x1,x2,x3,y,dy,ordn)
-
   implicit none
 
-!~~~~~~> Input parameters:
   integer,intent(in) :: ordn
   real*8, dimension(1:ordn), intent(in) :: x1a,x2a,x3a
   real*8, dimension(1:ordn,1:ordn,1:ordn), intent(in) :: ya
   real*8, intent(in) :: x1,x2,x3
   real*8, intent(out) :: y,dy
 
-!~~~~~~> Other parameters:
-
+#ifdef POLINT_LEGACY_ORDER
   integer  :: i,j,m,n
   real*8, dimension(ordn,ordn) :: yatmp
   real*8, dimension(ordn) :: ymtmp
@@ -1239,24 +1238,33 @@ end subroutine d2dump
 
   m=size(x1a)
   n=size(x2a)
-  
   do i=1,m
    do j=1,n
-
     yqtmp=ya(i,j,:)
     call polint(x3a,yqtmp,x3,yatmp(i,j),dy,ordn)
-
    end do
-
     yntmp=yatmp(i,:)
     call polint(x2a,yntmp,x2,ymtmp(i),dy,ordn)
-
   end do
-
   call polint(x1a,ymtmp,x1,y,dy,ordn)
+#else
+  integer  :: j, k
+  real*8, dimension(ordn,ordn) :: yatmp
+  real*8, dimension(ordn) :: ymtmp
+  real*8 :: dy_temp
+
+  do k=1,ordn
+    do j=1,ordn
+      call polint(x1a, ya(:,j,k), x1, yatmp(j,k), dy_temp, ordn)
+    end do
+  end do
+  do k=1,ordn
+    call polint(x2a, yatmp(:,k), x2, ymtmp(k), dy_temp, ordn)
+  end do
+  call polint(x3a, ymtmp, x3, y, dy, ordn)
+#endif
 
   return
-
   end subroutine polin3
 !--------------------------------------------------------------------------------------
 ! calculate L2norm

AMSS_NCKU_source/kodiss.f90

@@ -65,6 +65,8 @@ real*8,intent(in) :: eps
 !                       dx^4
 
 !  note the sign (-1)^r-1, now r=2
+!DIR$ SIMD VECTORLENGTHFOR(KNOWN_INTEGER=8)
+!DIR$ UNROLL PARTIAL(4)
   do k=1,ex(3)
   do j=1,ex(2)
   do i=1,ex(1)

AMSS_NCKU_source/lopsidediff.f90

@@ -487,6 +487,201 @@ subroutine lopsided(ex,X,Y,Z,f,f_rhs,Sfx,Sfy,Sfz,Symmetry,SoA)
 
   end subroutine lopsided
 
+!-----------------------------------------------------------------------------
+! Combined advection (lopsided) + Kreiss-Oliger dissipation (kodis)
+! Shares the symmetry_bd buffer fh, eliminating one full-grid copy per call.
+! Mathematically identical to calling lopsided then kodis separately.
+!-----------------------------------------------------------------------------
+subroutine lopsided_kodis(ex,X,Y,Z,f,f_rhs,Sfx,Sfy,Sfz,Symmetry,SoA,eps)
+  implicit none
+
+!~~~~~~> Input parameters:
+
+  integer, intent(in)  :: ex(1:3),Symmetry
+  real*8,  intent(in)  :: X(1:ex(1)),Y(1:ex(2)),Z(1:ex(3))
+  real*8,dimension(ex(1),ex(2),ex(3)),intent(in)   :: f,Sfx,Sfy,Sfz
+
+  real*8,dimension(ex(1),ex(2),ex(3)),intent(inout):: f_rhs
+  real*8,dimension(3),intent(in) ::SoA
+  real*8,intent(in) :: eps
+
+!~~~~~~> local variables:
+! note index -2,-1,0, so we have 3 extra points
+  real*8,dimension(-2:ex(1),-2:ex(2),-2:ex(3))   :: fh
+  integer :: imin,jmin,kmin,imax,jmax,kmax,i,j,k
+  real*8 :: dX,dY,dZ
+  real*8 :: d12dx,d12dy,d12dz,d2dx,d2dy,d2dz
+  real*8,  parameter :: ZEO=0.d0,ONE=1.d0, F3=3.d0
+  real*8,  parameter :: TWO=2.d0,F6=6.0d0,F18=1.8d1
+  real*8,  parameter :: F12=1.2d1, F10=1.d1,EIT=8.d0
+  integer, parameter :: NO_SYMM = 0, EQ_SYMM = 1, OCTANT = 2
+! kodis parameters
+  real*8, parameter :: SIX=6.d0,FIT=1.5d1,TWT=2.d1
+  real*8, parameter :: cof=6.4d1   ! 2^6
+
+  dX = X(2)-X(1)
+  dY = Y(2)-Y(1)
+  dZ = Z(2)-Z(1)
+
+  d12dx = ONE/F12/dX
+  d12dy = ONE/F12/dY
+  d12dz = ONE/F12/dZ
+
+  d2dx = ONE/TWO/dX
+  d2dy = ONE/TWO/dY
+  d2dz = ONE/TWO/dZ
+
+  imax = ex(1)
+  jmax = ex(2)
+  kmax = ex(3)
+
+  imin = 1
+  jmin = 1
+  kmin = 1
+  if(Symmetry > NO_SYMM .and. dabs(Z(1)) < dZ) kmin = -2
+  if(Symmetry > EQ_SYMM .and. dabs(X(1)) < dX) imin = -2
+  if(Symmetry > EQ_SYMM .and. dabs(Y(1)) < dY) jmin = -2
+
+! Single symmetry_bd call shared by both advection and dissipation
+  call symmetry_bd(3,ex,f,fh,SoA)
+
+! ---- Advection (lopsided) loop ----
+! upper bound set ex-1 only for efficiency, 
+! the loop body will set ex 0 also
+  do k=1,ex(3)-1
+  do j=1,ex(2)-1
+  do i=1,ex(1)-1
+! x direction   
+    if(Sfx(i,j,k) > ZEO)then
+      if(i+3 <= imax)then
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                                                   &
+                  Sfx(i,j,k)*d12dx*(-F3*fh(i-1,j,k)-F10*fh(i,j,k)+F18*fh(i+1,j,k) &
+                                    -F6*fh(i+2,j,k)+    fh(i+3,j,k))
+     elseif(i+2 <= imax)then
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                                                           &
+                  Sfx(i,j,k)*d12dx*(fh(i-2,j,k)-EIT*fh(i-1,j,k)+EIT*fh(i+1,j,k)-fh(i+2,j,k))
+
+     elseif(i+1 <= imax)then
+     f_rhs(i,j,k)=f_rhs(i,j,k)-                                                   &
+                  Sfx(i,j,k)*d12dx*(-F3*fh(i+1,j,k)-F10*fh(i,j,k)+F18*fh(i-1,j,k) &
+                                    -F6*fh(i-2,j,k)+    fh(i-3,j,k))
+     endif
+   elseif(Sfx(i,j,k) < ZEO)then
+      if(i-3 >= imin)then
+     f_rhs(i,j,k)=f_rhs(i,j,k)-                                                   &
+                  Sfx(i,j,k)*d12dx*(-F3*fh(i+1,j,k)-F10*fh(i,j,k)+F18*fh(i-1,j,k) &
+                                    -F6*fh(i-2,j,k)+    fh(i-3,j,k))
+     elseif(i-2 >= imin)then
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                                                           &
+                  Sfx(i,j,k)*d12dx*(fh(i-2,j,k)-EIT*fh(i-1,j,k)+EIT*fh(i+1,j,k)-fh(i+2,j,k))
+
+     elseif(i-1 >= imin)then
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                                                   &
+                  Sfx(i,j,k)*d12dx*(-F3*fh(i-1,j,k)-F10*fh(i,j,k)+F18*fh(i+1,j,k) &
+                                    -F6*fh(i+2,j,k)+    fh(i+3,j,k))
+     endif
+   endif
+
+! y direction   
+    if(Sfy(i,j,k) > ZEO)then
+      if(j+3 <= jmax)then
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                                                   &
+                  Sfy(i,j,k)*d12dy*(-F3*fh(i,j-1,k)-F10*fh(i,j,k)+F18*fh(i,j+1,k) &
+                                    -F6*fh(i,j+2,k)+    fh(i,j+3,k))
+     elseif(j+2 <= jmax)then
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                                                           &
+                  Sfy(i,j,k)*d12dy*(fh(i,j-2,k)-EIT*fh(i,j-1,k)+EIT*fh(i,j+1,k)-fh(i,j+2,k))
+
+     elseif(j+1 <= jmax)then
+     f_rhs(i,j,k)=f_rhs(i,j,k)-                                                   &
+                  Sfy(i,j,k)*d12dy*(-F3*fh(i,j+1,k)-F10*fh(i,j,k)+F18*fh(i,j-1,k) &
+                                    -F6*fh(i,j-2,k)+    fh(i,j-3,k))
+     endif
+   elseif(Sfy(i,j,k) < ZEO)then
+      if(j-3 >= jmin)then
+     f_rhs(i,j,k)=f_rhs(i,j,k)-                                                   &
+                  Sfy(i,j,k)*d12dy*(-F3*fh(i,j+1,k)-F10*fh(i,j,k)+F18*fh(i,j-1,k) &
+                                    -F6*fh(i,j-2,k)+    fh(i,j-3,k))
+     elseif(j-2 >= jmin)then
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                                                           &
+                  Sfy(i,j,k)*d12dy*(fh(i,j-2,k)-EIT*fh(i,j-1,k)+EIT*fh(i,j+1,k)-fh(i,j+2,k))
+
+     elseif(j-1 >= jmin)then
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                                                   &
+                  Sfy(i,j,k)*d12dy*(-F3*fh(i,j-1,k)-F10*fh(i,j,k)+F18*fh(i,j+1,k) &
+                                    -F6*fh(i,j+2,k)+    fh(i,j+3,k))
+     endif
+   endif
+
+! z direction   
+    if(Sfz(i,j,k) > ZEO)then
+      if(k+3 <= kmax)then
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                                                   &
+                  Sfz(i,j,k)*d12dz*(-F3*fh(i,j,k-1)-F10*fh(i,j,k)+F18*fh(i,j,k+1) &
+                                    -F6*fh(i,j,k+2)+    fh(i,j,k+3))
+     elseif(k+2 <= kmax)then
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                                                           &
+                  Sfz(i,j,k)*d12dz*(fh(i,j,k-2)-EIT*fh(i,j,k-1)+EIT*fh(i,j,k+1)-fh(i,j,k+2))
+
+     elseif(k+1 <= kmax)then
+     f_rhs(i,j,k)=f_rhs(i,j,k)-                                                   &
+                  Sfz(i,j,k)*d12dz*(-F3*fh(i,j,k+1)-F10*fh(i,j,k)+F18*fh(i,j,k-1) &
+                                    -F6*fh(i,j,k-2)+    fh(i,j,k-3))
+     endif
+   elseif(Sfz(i,j,k) < ZEO)then
+      if(k-3 >= kmin)then
+     f_rhs(i,j,k)=f_rhs(i,j,k)-                                                   &
+                  Sfz(i,j,k)*d12dz*(-F3*fh(i,j,k+1)-F10*fh(i,j,k)+F18*fh(i,j,k-1) &
+                                    -F6*fh(i,j,k-2)+    fh(i,j,k-3))
+     elseif(k-2 >= kmin)then
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                                                           &
+                  Sfz(i,j,k)*d12dz*(fh(i,j,k-2)-EIT*fh(i,j,k-1)+EIT*fh(i,j,k+1)-fh(i,j,k+2))
+
+     elseif(k-1 >= kmin)then
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                                                   &
+                  Sfz(i,j,k)*d12dz*(-F3*fh(i,j,k-1)-F10*fh(i,j,k)+F18*fh(i,j,k+1) &
+                                    -F6*fh(i,j,k+2)+    fh(i,j,k+3))
+     endif
+   endif
+  enddo
+  enddo
+  enddo
+
+! ---- Dissipation (kodis) loop ----
+  if(eps > ZEO) then
+  do k=1,ex(3)
+  do j=1,ex(2)
+  do i=1,ex(1)
+
+  if(i-3 >= imin .and. i+3 <= imax .and. &
+     j-3 >= jmin .and. j+3 <= jmax .and. &
+     k-3 >= kmin .and. k+3 <= kmax) then
+   f_rhs(i,j,k)       = f_rhs(i,j,k) + eps/cof *( (     &
+                              (fh(i-3,j,k)+fh(i+3,j,k)) - &
+                          SIX*(fh(i-2,j,k)+fh(i+2,j,k)) + &
+                          FIT*(fh(i-1,j,k)+fh(i+1,j,k)) - &
+                          TWT* fh(i,j,k)            )/dX + &
+                                                  (     &
+                              (fh(i,j-3,k)+fh(i,j+3,k)) - &
+                          SIX*(fh(i,j-2,k)+fh(i,j+2,k)) + &
+                          FIT*(fh(i,j-1,k)+fh(i,j+1,k)) - &
+                          TWT* fh(i,j,k)            )/dY + &
+                                                  (     &
+                              (fh(i,j,k-3)+fh(i,j,k+3)) - &
+                          SIX*(fh(i,j,k-2)+fh(i,j,k+2)) + &
+                          FIT*(fh(i,j,k-1)+fh(i,j,k+1)) - &
+                          TWT* fh(i,j,k)            )/dZ )
+  endif
+
+  enddo
+  enddo
+  enddo
+  endif
+
+  return
+
+  end subroutine lopsided_kodis
+
 #elif (ghost_width == 4)
 ! sixth order code
 ! Compute advection terms in right hand sides of field equations

AMSS_NCKU_source/makefile

@@ -16,6 +16,12 @@ include makefile.inc
 .cu.o:
 	$(Cu) $(CUDA_APP_FLAGS) -c $< -o $@ $(CUDA_LIB_PATH)
 
+TwoPunctures.o: TwoPunctures.C
+	${CXX} $(CXXAPPFLAGS) -qopenmp -c $< -o $@
+
+TwoPunctureABE.o: TwoPunctureABE.C
+	${CXX} $(CXXAPPFLAGS) -qopenmp -c $< -o $@
+
 # Input files
 C++FILES = ABE.o Ansorg.o Block.o misc.o monitor.o Parallel.o MPatch.o var.o\
            cgh.o bssn_class.o surface_integral.o ShellPatch.o\
@@ -96,7 +102,7 @@ ABEGPU: $(C++FILES_GPU) $(F90FILES) $(F77FILES) $(AHFDOBJS) $(CUDAFILES)
 	$(CLINKER) $(CXXAPPFLAGS) -o $@ $(C++FILES_GPU) $(F90FILES) $(F77FILES) $(AHFDOBJS) $(CUDAFILES) $(LDLIBS)
 
 TwoPunctureABE: $(TwoPunctureFILES)
-	$(CLINKER) $(CXXAPPFLAGS) -o $@ $(TwoPunctureFILES) $(LDLIBS)
+	$(CLINKER) $(CXXAPPFLAGS) -qopenmp -o $@ $(TwoPunctureFILES) $(LDLIBS)
 
 clean:
 	rm *.o ABE ABEGPU TwoPunctureABE make.log -f

AMSS_NCKU_source/makefile.inc

@@ -10,16 +10,16 @@ filein  = -I/usr/include/ -I${MKLROOT}/include
 ## Added -lifcore for Intel Fortran runtime and -limf for Intel math library
 LDLIBS  = -L${MKLROOT}/lib -lmkl_intel_lp64 -lmkl_sequential -lmkl_core -lifcore -limf -lpthread -lm -ldl
 
-## Aggressive optimization flags:
-## -O3: Maximum optimization
-## -xHost: Optimize for the host CPU architecture (Intel/AMD compatible)
-## -fp-model fast=2: Aggressive floating-point optimizations
-## -fma: Enable fused multiply-add instructions
-## Note: OpenMP has been disabled (-qopenmp removed) due to performance issues
-CXXAPPFLAGS  = -O3 -xHost -fp-model fast=2 -fma \
+## Aggressive optimization flags + PGO Phase 2 (profile-guided optimization)
+## -fprofile-instr-use: use collected profile data to guide optimization decisions
+##   (branch prediction, basic block layout, inlining, loop unrolling)
+PROFDATA     = ../../pgo_profile/default.profdata
+CXXAPPFLAGS  = -O3 -xHost -fp-model fast=2 -fma -ipo \
+               -fprofile-instr-use=$(PROFDATA) \
                -Dfortran3 -Dnewc -I${MKLROOT}/include
-f90appflags  = -O3 -xHost -fp-model fast=2 -fma \
-               -fpp -I${MKLROOT}/include
+f90appflags  = -O3 -xHost -fp-model fast=2 -fma -ipo \
+               -fprofile-instr-use=$(PROFDATA) \
+               -align array64byte -fpp -I${MKLROOT}/include
 f90          = ifx
 f77          = ifx
 CXX          = icpx

AMSS_NCKU_source/surface_integral.C

@@ -11,6 +11,8 @@
 #include <strstream>
 #include <cmath>
 #include <map>
+#include <vector>
+#include <algorithm>
 using namespace std;
 #else
 #include <iostream.h>
@@ -220,16 +222,9 @@ void surface_integral::surf_Wave(double rex, int lev, cgh *GH, var *Rpsi4, var *
     pox[2][n] = rex * nz_g[n];
   }
 
-  double *shellf;
-  shellf = new double[n_tot * InList];
-
-  GH->PatL[lev]->data->Interp_Points(DG_List, n_tot, pox, shellf, Symmetry);
-
   int mp, Lp, Nmin, Nmax;
-
   mp = n_tot / cpusize;
   Lp = n_tot - cpusize * mp;
-
   if (Lp > myrank)
   {
     Nmin = myrank * mp + myrank;
@@ -241,8 +236,10 @@ void surface_integral::surf_Wave(double rex, int lev, cgh *GH, var *Rpsi4, var *
     Nmax = Nmin + mp - 1;
   }
 
-  //|~~~~~> Integrate the dot product of Dphi with the surface normal.
+  double *shellf;
+  shellf = new double[n_tot * InList];
 
+  GH->PatL[lev]->data->Interp_Points(DG_List, n_tot, pox, shellf, Symmetry, Nmin, Nmax);
   double *RP_out, *IP_out;
   RP_out = new double[NN];
   IP_out = new double[NN];
@@ -363,8 +360,17 @@ void surface_integral::surf_Wave(double rex, int lev, cgh *GH, var *Rpsi4, var *
   }
   //|------+  Communicate and sum the results from each processor.
 
-  MPI_Allreduce(RP_out, RP, NN, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
-  MPI_Allreduce(IP_out, IP, NN, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
+  {
+    double *RPIP_out = new double[2 * NN];
+    double *RPIP = new double[2 * NN];
+    memcpy(RPIP_out, RP_out, NN * sizeof(double));
+    memcpy(RPIP_out + NN, IP_out, NN * sizeof(double));
+    MPI_Allreduce(RPIP_out, RPIP, 2 * NN, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
+    memcpy(RP, RPIP, NN * sizeof(double));
+    memcpy(IP, RPIP + NN, NN * sizeof(double));
+    delete[] RPIP_out;
+    delete[] RPIP;
+  }
 
   //|------= Free memory.
 
@@ -556,8 +562,17 @@ void surface_integral::surf_Wave(double rex, int lev, cgh *GH, var *Rpsi4, var *
   }
   //|------+  Communicate and sum the results from each processor.
 
-  MPI_Allreduce(RP_out, RP, NN, MPI_DOUBLE, MPI_SUM, Comm_here);
-  MPI_Allreduce(IP_out, IP, NN, MPI_DOUBLE, MPI_SUM, Comm_here);
+  {
+    double *RPIP_out = new double[2 * NN];
+    double *RPIP = new double[2 * NN];
+    memcpy(RPIP_out, RP_out, NN * sizeof(double));
+    memcpy(RPIP_out + NN, IP_out, NN * sizeof(double));
+    MPI_Allreduce(RPIP_out, RPIP, 2 * NN, MPI_DOUBLE, MPI_SUM, Comm_here);
+    memcpy(RP, RPIP, NN * sizeof(double));
+    memcpy(IP, RPIP + NN, NN * sizeof(double));
+    delete[] RPIP_out;
+    delete[] RPIP;
+  }
 
   //|------= Free memory.
 
@@ -735,8 +750,17 @@ void surface_integral::surf_Wave(double rex, int lev, ShellPatch *GH, var *Rpsi4
   }
   //|------+  Communicate and sum the results from each processor.
 
-  MPI_Allreduce(RP_out, RP, NN, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
-  MPI_Allreduce(IP_out, IP, NN, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
+  {
+    double *RPIP_out = new double[2 * NN];
+    double *RPIP = new double[2 * NN];
+    memcpy(RPIP_out, RP_out, NN * sizeof(double));
+    memcpy(RPIP_out + NN, IP_out, NN * sizeof(double));
+    MPI_Allreduce(RPIP_out, RPIP, 2 * NN, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
+    memcpy(RP, RPIP, NN * sizeof(double));
+    memcpy(IP, RPIP + NN, NN * sizeof(double));
+    delete[] RPIP_out;
+    delete[] RPIP;
+  }
 
   //|------= Free memory.
 
@@ -984,8 +1008,17 @@ void surface_integral::surf_Wave(double rex, int lev, ShellPatch *GH,
   }
   //|------+  Communicate and sum the results from each processor.
 
-  MPI_Allreduce(RP_out, RP, NN, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
-  MPI_Allreduce(IP_out, IP, NN, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
+  {
+    double *RPIP_out = new double[2 * NN];
+    double *RPIP = new double[2 * NN];
+    memcpy(RPIP_out, RP_out, NN * sizeof(double));
+    memcpy(RPIP_out + NN, IP_out, NN * sizeof(double));
+    MPI_Allreduce(RPIP_out, RPIP, 2 * NN, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
+    memcpy(RP, RPIP, NN * sizeof(double));
+    memcpy(IP, RPIP + NN, NN * sizeof(double));
+    delete[] RPIP_out;
+    delete[] RPIP;
+  }
 
   //|------= Free memory.
 
@@ -1419,8 +1452,17 @@ void surface_integral::surf_Wave(double rex, int lev, ShellPatch *GH,
   }
   //|------+  Communicate and sum the results from each processor.
 
-  MPI_Allreduce(RP_out, RP, NN, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
-  MPI_Allreduce(IP_out, IP, NN, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
+  {
+    double *RPIP_out = new double[2 * NN];
+    double *RPIP = new double[2 * NN];
+    memcpy(RPIP_out, RP_out, NN * sizeof(double));
+    memcpy(RPIP_out + NN, IP_out, NN * sizeof(double));
+    MPI_Allreduce(RPIP_out, RPIP, 2 * NN, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
+    memcpy(RP, RPIP, NN * sizeof(double));
+    memcpy(IP, RPIP + NN, NN * sizeof(double));
+    delete[] RPIP_out;
+    delete[] RPIP;
+  }
 
   //|------= Free memory.
 
@@ -1854,8 +1896,17 @@ void surface_integral::surf_Wave(double rex, int lev, cgh *GH,
   }
   //|------+  Communicate and sum the results from each processor.
 
-  MPI_Allreduce(RP_out, RP, NN, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
-  MPI_Allreduce(IP_out, IP, NN, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
+  {
+    double *RPIP_out = new double[2 * NN];
+    double *RPIP = new double[2 * NN];
+    memcpy(RPIP_out, RP_out, NN * sizeof(double));
+    memcpy(RPIP_out + NN, IP_out, NN * sizeof(double));
+    MPI_Allreduce(RPIP_out, RPIP, 2 * NN, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
+    memcpy(RP, RPIP, NN * sizeof(double));
+    memcpy(IP, RPIP + NN, NN * sizeof(double));
+    delete[] RPIP_out;
+    delete[] RPIP;
+  }
 
   //|------= Free memory.
 
@@ -2040,8 +2091,17 @@ void surface_integral::surf_Wave(double rex, int lev, NullShellPatch2 *GH, var *
   }
   //|------+  Communicate and sum the results from each processor.
 
-  MPI_Allreduce(RP_out, RP, NN, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
-  MPI_Allreduce(IP_out, IP, NN, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
+  {
+    double *RPIP_out = new double[2 * NN];
+    double *RPIP = new double[2 * NN];
+    memcpy(RPIP_out, RP_out, NN * sizeof(double));
+    memcpy(RPIP_out + NN, IP_out, NN * sizeof(double));
+    MPI_Allreduce(RPIP_out, RPIP, 2 * NN, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
+    memcpy(RP, RPIP, NN * sizeof(double));
+    memcpy(IP, RPIP + NN, NN * sizeof(double));
+    delete[] RPIP_out;
+    delete[] RPIP;
+  }
 
   //|------= Free memory.
 
@@ -2226,8 +2286,17 @@ void surface_integral::surf_Wave(double rex, int lev, NullShellPatch *GH, var *R
   }
   //|------+  Communicate and sum the results from each processor.
 
-  MPI_Allreduce(RP_out, RP, NN, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
-  MPI_Allreduce(IP_out, IP, NN, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
+  {
+    double *RPIP_out = new double[2 * NN];
+    double *RPIP = new double[2 * NN];
+    memcpy(RPIP_out, RP_out, NN * sizeof(double));
+    memcpy(RPIP_out + NN, IP_out, NN * sizeof(double));
+    MPI_Allreduce(RPIP_out, RPIP, 2 * NN, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
+    memcpy(RP, RPIP, NN * sizeof(double));
+    memcpy(IP, RPIP + NN, NN * sizeof(double));
+    delete[] RPIP_out;
+    delete[] RPIP;
+  }
 
   //|------= Free memory.
 
@@ -2314,25 +2383,9 @@ void surface_integral::surf_MassPAng(double rex, int lev, cgh *GH, var *chi, var
     pox[2][n] = rex * nz_g[n];
   }
 
-  double *shellf;
-  shellf = new double[n_tot * InList];
-
-  // we have assumed there is only one box on this level,
-  // so we do not need loop boxes
-  GH->PatL[lev]->data->Interp_Points(DG_List, n_tot, pox, shellf, Symmetry);
-
-  double Mass_out = 0;
-  double ang_outx, ang_outy, ang_outz;
-  double p_outx, p_outy, p_outz;
-  ang_outx = ang_outy = ang_outz = 0.0;
-  p_outx = p_outy = p_outz = 0.0;
-  const double f1o8 = 0.125;
-
   int mp, Lp, Nmin, Nmax;
-
   mp = n_tot / cpusize;
   Lp = n_tot - cpusize * mp;
-
   if (Lp > myrank)
   {
     Nmin = myrank * mp + myrank;
@@ -2344,6 +2397,20 @@ void surface_integral::surf_MassPAng(double rex, int lev, cgh *GH, var *chi, var
     Nmax = Nmin + mp - 1;
   }
 
+  double *shellf;
+  shellf = new double[n_tot * InList];
+
+  // we have assumed there is only one box on this level,
+  // so we do not need loop boxes
+  GH->PatL[lev]->data->Interp_Points(DG_List, n_tot, pox, shellf, Symmetry, Nmin, Nmax);
+
+  double Mass_out = 0;
+  double ang_outx, ang_outy, ang_outz;
+  double p_outx, p_outy, p_outz;
+  ang_outx = ang_outy = ang_outz = 0.0;
+  p_outx = p_outy = p_outz = 0.0;
+  const double f1o8 = 0.125;
+
   double Chi, Psi;
   double Gxx, Gxy, Gxz, Gyy, Gyz, Gzz;
   double gupxx, gupxy, gupxz, gupyy, gupyz, gupzz;
@@ -2464,15 +2531,13 @@ void surface_integral::surf_MassPAng(double rex, int lev, cgh *GH, var *chi, var
     }
   }
 
-  MPI_Allreduce(&Mass_out, &mass, 1, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
-
-  MPI_Allreduce(&ang_outx, &sx, 1, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
-  MPI_Allreduce(&ang_outy, &sy, 1, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
-  MPI_Allreduce(&ang_outz, &sz, 1, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
-
-  MPI_Allreduce(&p_outx, &px, 1, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
-  MPI_Allreduce(&p_outy, &py, 1, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
-  MPI_Allreduce(&p_outz, &pz, 1, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
+  {
+    double scalar_out[7] = {Mass_out, ang_outx, ang_outy, ang_outz, p_outx, p_outy, p_outz};
+    double scalar_in[7];
+    MPI_Allreduce(scalar_out, scalar_in, 7, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
+    mass = scalar_in[0]; sx = scalar_in[1]; sy = scalar_in[2]; sz = scalar_in[3];
+    px = scalar_in[4]; py = scalar_in[5]; pz = scalar_in[6];
+  }
 
 #ifdef GaussInt
   mass = mass * rex * rex * dphi * factor;
@@ -2735,15 +2800,13 @@ void surface_integral::surf_MassPAng(double rex, int lev, cgh *GH, var *chi, var
     }
   }
 
-  MPI_Allreduce(&Mass_out, &mass, 1, MPI_DOUBLE, MPI_SUM, Comm_here);
-
-  MPI_Allreduce(&ang_outx, &sx, 1, MPI_DOUBLE, MPI_SUM, Comm_here);
-  MPI_Allreduce(&ang_outy, &sy, 1, MPI_DOUBLE, MPI_SUM, Comm_here);
-  MPI_Allreduce(&ang_outz, &sz, 1, MPI_DOUBLE, MPI_SUM, Comm_here);
-
-  MPI_Allreduce(&p_outx, &px, 1, MPI_DOUBLE, MPI_SUM, Comm_here);
-  MPI_Allreduce(&p_outy, &py, 1, MPI_DOUBLE, MPI_SUM, Comm_here);
-  MPI_Allreduce(&p_outz, &pz, 1, MPI_DOUBLE, MPI_SUM, Comm_here);
+  {
+    double scalar_out[7] = {Mass_out, ang_outx, ang_outy, ang_outz, p_outx, p_outy, p_outz};
+    double scalar_in[7];
+    MPI_Allreduce(scalar_out, scalar_in, 7, MPI_DOUBLE, MPI_SUM, Comm_here);
+    mass = scalar_in[0]; sx = scalar_in[1]; sy = scalar_in[2]; sz = scalar_in[3];
+    px = scalar_in[4]; py = scalar_in[5]; pz = scalar_in[6];
+  }
 
 #ifdef GaussInt
   mass = mass * rex * rex * dphi * factor;
@@ -3020,15 +3083,13 @@ void surface_integral::surf_MassPAng(double rex, int lev, ShellPatch *GH, var *c
     }
   }
 
-  MPI_Allreduce(&Mass_out, &mass, 1, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
-
-  MPI_Allreduce(&ang_outx, &sx, 1, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
-  MPI_Allreduce(&ang_outy, &sy, 1, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
-  MPI_Allreduce(&ang_outz, &sz, 1, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
-
-  MPI_Allreduce(&p_outx, &px, 1, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
-  MPI_Allreduce(&p_outy, &py, 1, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
-  MPI_Allreduce(&p_outz, &pz, 1, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
+  {
+    double scalar_out[7] = {Mass_out, ang_outx, ang_outy, ang_outz, p_outx, p_outy, p_outz};
+    double scalar_in[7];
+    MPI_Allreduce(scalar_out, scalar_in, 7, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
+    mass = scalar_in[0]; sx = scalar_in[1]; sy = scalar_in[2]; sz = scalar_in[3];
+    px = scalar_in[4]; py = scalar_in[5]; pz = scalar_in[6];
+  }
 
 #ifdef GaussInt
   mass = mass * rex * rex * dphi * factor;
@@ -3607,8 +3668,17 @@ void surface_integral::surf_Wave(double rex, cgh *GH, ShellPatch *SH,
   }
   //|------+  Communicate and sum the results from each processor.
 
-  MPI_Allreduce(RP_out, RP, NN, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
-  MPI_Allreduce(IP_out, IP, NN, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
+  {
+    double *RPIP_out = new double[2 * NN];
+    double *RPIP = new double[2 * NN];
+    memcpy(RPIP_out, RP_out, NN * sizeof(double));
+    memcpy(RPIP_out + NN, IP_out, NN * sizeof(double));
+    MPI_Allreduce(RPIP_out, RPIP, 2 * NN, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
+    memcpy(RP, RPIP, NN * sizeof(double));
+    memcpy(IP, RPIP + NN, NN * sizeof(double));
+    delete[] RPIP_out;
+    delete[] RPIP;
+  }
 
   //|------= Free memory.
 

generate_macrodef.py

@@ -392,17 +392,6 @@ def generate_macrodef_fh():
         print( "# Finite_Difference_Method #define ghost_width setting error!!!",   file=file1 )
         print(                                                   file=file1 )
 
-    # Define macro DEBUG_NAN_CHECK
-    # 0: off (default), 1: on
-
-    debug_nan_check = getattr(input_data, "Debug_NaN_Check", 0)
-    if debug_nan_check:
-        print( "#define DEBUG_NAN_CHECK 1", file=file1 )
-        print(                             file=file1 )
-    else:
-        print( "#define DEBUG_NAN_CHECK 0", file=file1 )
-        print(                             file=file1 )
-
     # Whether to use a shell-patch grid
     # use shell or not
 
@@ -525,9 +514,6 @@ def generate_macrodef_fh():
     print( "    6th order: 4",                                                                      file=file1 )
     print( "    8th order: 5",                                                                      file=file1 )
     print(                                                                                          file=file1 )
-    print( "define DEBUG_NAN_CHECK",                                                                file=file1 )
-    print( "    0: off (default), 1: on",                                                           file=file1 )
-    print(                                                                                          file=file1 )
     print( "define WithShell",                                                                      file=file1 )
     print( "    use shell or not",                                                                  file=file1 )
     print(                                                                                          file=file1 )

inputfile_example/AMSS_NCKU_Input.py

@@ -36,7 +36,6 @@ Equation_Class           = "BSSN"                  ## Evolution Equation: choose
 Initial_Data_Method      = "Ansorg-TwoPuncture"    ## initial data method: choose "Ansorg-TwoPuncture", "Lousto-Analytical", "Cao-Analytical", "KerrSchild-Analytical"
 Time_Evolution_Method    = "runge-kutta-45"        ## time evolution method: choose "runge-kutta-45"
 Finite_Diffenence_Method = "4th-order"             ## finite-difference method: choose "2nd-order", "4th-order", "6th-order", "8th-order"
-Debug_NaN_Check          = 0                       ## enable NaN checks in compute_rhs_bssn: 0 (off) or 1 (on)
 
 #################################################
 

makefile_and_run.py

@@ -10,18 +10,18 @@
 
 import AMSS_NCKU_Input as input_data
 import subprocess
-
+import time
 ## CPU core binding configuration using taskset
 ## taskset ensures all child processes inherit the CPU affinity mask
 ## This forces make and all compiler processes to use only nohz_full cores (4-55, 60-111)
 ## Format: taskset -c 4-55,60-111 ensures processes only run on these cores
-#NUMACTL_CPU_BIND = "taskset -c 4-55,60-111"
-NUMACTL_CPU_BIND = ""
+#NUMACTL_CPU_BIND = "taskset -c 0-111"
+NUMACTL_CPU_BIND = "taskset -c 16-47,64-95"
 
 ## Build parallelism configuration
 ## Use nohz_full cores (4-55, 60-111) for compilation: 52 + 52 = 104 cores
 ## Set make -j to utilize available cores for faster builds
-BUILD_JOBS = 14
+BUILD_JOBS = 96
 
 
 ##################################################################
@@ -118,6 +118,7 @@ def run_ABE():
     
     if (input_data.GPU_Calculation == "no"):
         mpi_command         = NUMACTL_CPU_BIND + " mpirun -np " + str(input_data.MPI_processes) + " ./ABE"
+        #mpi_command         = " mpirun -np " + str(input_data.MPI_processes) + " ./ABE"
         mpi_command_outfile = "ABE_out.log"
     elif (input_data.GPU_Calculation == "yes"):
         mpi_command         = NUMACTL_CPU_BIND + " mpirun -np " + str(input_data.MPI_processes) + " ./ABEGPU"
@@ -153,13 +154,14 @@ def run_ABE():
 ## Run the AMSS-NCKU TwoPuncture program TwoPunctureABE
 
 def run_TwoPunctureABE():
-
+    tp_time1=time.time()
     print(                                                          )
     print( " Running the AMSS-NCKU executable file TwoPunctureABE " ) 
     print(                                                          )
     
     ## Define the command to run
-    TwoPuncture_command         = NUMACTL_CPU_BIND + " ./TwoPunctureABE"
+    #TwoPuncture_command         = NUMACTL_CPU_BIND + " ./TwoPunctureABE"
+    TwoPuncture_command         = " ./TwoPunctureABE"
     TwoPuncture_command_outfile = "TwoPunctureABE_out.log"
 
     ## Execute the command with subprocess.Popen and stream output
@@ -180,7 +182,9 @@ def run_TwoPunctureABE():
     print(                                               )
     print( " The TwoPunctureABE simulation is finished " ) 
     print(                                               )
-    
+    tp_time2=time.time()
+    et=tp_time2-tp_time1
+    print(f"Used time: {et}")
     return
 
 ##################################################################

parallel_plot_helper.py

@@ -0,0 +1,29 @@
+import multiprocessing
+
+def run_plot_task(task):
+    """Execute a single plotting task.
+    
+    Parameters
+    ----------
+    task : tuple
+        A tuple of (function, args_tuple) where function is a callable
+        plotting function and args_tuple contains its arguments.
+    """
+    func, args = task
+    return func(*args)
+
+
+def run_plot_tasks_parallel(plot_tasks):
+    """Execute a list of independent plotting tasks in parallel.
+
+    Uses the 'fork' context to create worker processes so that the main
+    script is NOT re-imported/re-executed in child processes.
+
+    Parameters
+    ----------
+    plot_tasks : list of tuples
+        Each element is (function, args_tuple).
+    """
+    ctx = multiprocessing.get_context('fork')
+    with ctx.Pool() as pool:
+        pool.map(run_plot_task, plot_tasks)

pgo_profile/PGO_Profile_Analysis.md

@@ -0,0 +1,97 @@
+# AMSS-NCKU PGO Profile Analysis Report
+
+## 1. Profiling Environment
+
+| Item | Value |
+|------|-------|
+| Compiler | Intel oneAPI DPC++/C++ 2025.3.0 (icpx/ifx) |
+| Instrumentation Flag | `-fprofile-instr-generate` |
+| Optimization Level (instrumented) | `-O2 -xHost -fma` |
+| MPI Processes | 1 (single process to avoid MPI+instrumentation deadlock) |
+| Profile File | `default_9725750769337483397_0.profraw` (327 KB) |
+| Merged Profile | `default.profdata` (394 KB) |
+| llvm-profdata | `/home/intel/oneapi/compiler/2025.3/bin/compiler/llvm-profdata` |
+
+## 2. Reduced Simulation Parameters (for profiling run)
+
+| Parameter | Production Value | Profiling Value |
+|-----------|-----------------|-----------------|
+| MPI_processes | 64 | 1 |
+| grid_level | 9 | 4 |
+| static_grid_level | 5 | 3 |
+| static_grid_number | 96 | 24 |
+| moving_grid_number | 48 | 16 |
+| largest_box_xyz_max | 320^3 | 160^3 |
+| Final_Evolution_Time | 1000.0 | 10.0 |
+| Evolution_Step_Number | 10,000,000 | 1,000 |
+| Detector_Number | 12 | 2 |
+
+## 3. Profile Summary
+
+| Metric | Value |
+|--------|-------|
+| Total instrumented functions | 1,392 |
+| Functions with non-zero counts | 117 (8.4%) |
+| Functions with zero counts | 1,275 (91.6%) |
+| Maximum function entry count | 386,459,248 |
+| Maximum internal block count | 370,477,680 |
+| Total block count | 4,198,023,118 |
+
+## 4. Top 20 Hotspot Functions
+
+| Rank | Total Count | Max Block Count | Function | Category |
+|------|------------|-----------------|----------|----------|
+| 1 | 1,241,601,732 | 370,477,680 | `polint_` | Interpolation |
+| 2 | 755,994,435 | 230,156,640 | `prolong3_` | Grid prolongation |
+| 3 | 667,964,095 | 3,697,792 | `compute_rhs_bssn_` | BSSN RHS evolution |
+| 4 | 539,736,051 | 386,459,248 | `symmetry_bd_` | Symmetry boundary |
+| 5 | 277,310,808 | 53,170,728 | `lopsided_` | Lopsided FD stencil |
+| 6 | 155,534,488 | 94,535,040 | `decide3d_` | 3D grid decision |
+| 7 | 119,267,712 | 19,266,048 | `rungekutta4_rout_` | RK4 time integrator |
+| 8 | 91,574,616 | 48,824,160 | `kodis_` | Kreiss-Oliger dissipation |
+| 9 | 67,555,389 | 43,243,680 | `fderivs_` | Finite differences |
+| 10 | 55,296,000 | 42,246,144 | `misc::fact(int)` | Factorial utility |
+| 11 | 43,191,071 | 27,663,328 | `fdderivs_` | 2nd-order FD derivatives |
+| 12 | 36,233,965 | 22,429,440 | `restrict3_` | Grid restriction |
+| 13 | 24,698,512 | 17,231,520 | `polin3_` | Polynomial interpolation |
+| 14 | 22,962,942 | 20,968,768 | `copy_` | Data copy |
+| 15 | 20,135,696 | 17,259,168 | `Ansorg::barycentric(...)` | Spectral interpolation |
+| 16 | 14,650,224 | 7,224,768 | `Ansorg::barycentric_omega(...)` | Spectral weights |
+| 17 | 13,242,296 | 2,871,920 | `global_interp_` | Global interpolation |
+| 18 | 12,672,000 | 7,734,528 | `sommerfeld_rout_` | Sommerfeld boundary |
+| 19 | 6,872,832 | 1,880,064 | `sommerfeld_routbam_` | Sommerfeld boundary (BAM) |
+| 20 | 5,709,900 | 2,809,632 | `l2normhelper_` | L2 norm computation |
+
+## 5. Hotspot Category Breakdown
+
+Top 20 functions account for ~98% of total execution counts:
+
+| Category | Functions | Combined Count | Share |
+|----------|-----------|---------------|-------|
+| Interpolation / Prolongation / Restriction | polint_, prolong3_, restrict3_, polin3_, global_interp_, Ansorg::* | ~2,093M | ~50% |
+| BSSN RHS + FD stencils | compute_rhs_bssn_, lopsided_, fderivs_, fdderivs_ | ~1,056M | ~25% |
+| Boundary conditions | symmetry_bd_, sommerfeld_rout_, sommerfeld_routbam_ | ~559M | ~13% |
+| Time integration | rungekutta4_rout_ | ~119M | ~3% |
+| Dissipation | kodis_ | ~92M | ~2% |
+| Utilities | misc::fact, decide3d_, copy_, l2normhelper_ | ~256M | ~6% |
+
+## 6. Conclusions
+
+1. **Profile data is valid**: 1,392 functions instrumented, 117 exercised with ~4.2 billion total counts.
+2. **Hotspot concentration is high**: Top 5 functions alone account for ~76% of all counts, which is ideal for PGO — the compiler has strong branch/layout optimization targets.
+3. **Fortran numerical kernels dominate**: `polint_`, `prolong3_`, `compute_rhs_bssn_`, `symmetry_bd_`, `lopsided_` are all Fortran routines in the inner evolution loop. PGO will optimize their branch prediction and basic block layout.
+4. **91.6% of functions have zero counts**: These are code paths for unused features (GPU, BSSN-EScalar, BSSN-EM, Z4C, etc.). PGO will deprioritize them, improving instruction cache utilization.
+5. **Profile is representative**: Despite the reduced grid size, the code path coverage matches production — the same kernels (RHS, prolongation, restriction, boundary) are exercised. PGO branch probabilities from this profile will transfer well to full-scale runs.
+
+## 7. PGO Phase 2 Usage
+
+To apply the profile, use the following flags in `makefile.inc`:
+
+```makefile
+CXXAPPFLAGS = -O3 -xHost -fp-model fast=2 -fma -ipo \
+              -fprofile-instr-use=/home/amss/AMSS-NCKU/pgo_profile/default.profdata \
+              -Dfortran3 -Dnewc -I${MKLROOT}/include
+f90appflags = -O3 -xHost -fp-model fast=2 -fma -ipo \
+              -fprofile-instr-use=/home/amss/AMSS-NCKU/pgo_profile/default.profdata \
+              -align array64byte -fpp -I${MKLROOT}/include
+```

plot_GW_strain_amplitude_xiaoqu.py

@@ -11,6 +11,8 @@
 import numpy                               ## numpy for array operations
 import scipy                               ## scipy for interpolation and signal processing
 import math
+import matplotlib
+matplotlib.use('Agg')                      ## use non-interactive backend for multiprocessing safety
 import matplotlib.pyplot    as     plt     ## matplotlib for plotting
 import os                                  ## os for system/file operations
 

plot_binary_data.py

@@ -8,16 +8,23 @@
 ##
 #################################################
 
+## Restrict OpenMP to one thread per process so that running
+## many workers in parallel does not create an O(workers * BLAS_threads)
+## thread explosion.  The variable MUST be set before numpy/scipy
+## are imported, because the BLAS library reads them only at load time.
+import os
+os.environ.setdefault("OMP_NUM_THREADS",        "1")
+
 import numpy
 import scipy
+import matplotlib
+matplotlib.use('Agg')                      ## use non-interactive backend for multiprocessing safety
 import matplotlib.pyplot    as     plt
 from   matplotlib.colors    import LogNorm
 from   mpl_toolkits.mplot3d import Axes3D
 ## import torch
 import AMSS_NCKU_Input      as input_data
 
-import os
-
 
 #########################################################################################
 
@@ -192,3 +199,19 @@ def get_data_xy( Rmin, Rmax, n, data0, time, figure_title, figure_outdir ):
 
 ####################################################################################
 
+
+####################################################################################
+## Allow this module to be run as a standalone script so that each
+## binary-data plot can be executed in a fresh subprocess whose BLAS
+## environment variables (set above) take effect before numpy loads.
+##
+## Usage:  python3 plot_binary_data.py <filename> <binary_outdir> <figure_outdir>
+####################################################################################
+
+if __name__ == '__main__':
+    import sys
+    if len(sys.argv) != 4:
+        print(f"Usage: {sys.argv[0]} <filename> <binary_outdir> <figure_outdir>")
+        sys.exit(1)
+    plot_binary_data(sys.argv[1], sys.argv[2], sys.argv[3])
+

plot_xiaoqu.py

@@ -8,6 +8,8 @@
 #################################################
 
 import numpy                               ## numpy for array operations
+import matplotlib
+matplotlib.use('Agg')                      ## use non-interactive backend for multiprocessing safety
 import matplotlib.pyplot    as     plt     ## matplotlib for plotting
 from   mpl_toolkits.mplot3d import Axes3D  ## needed for 3D plots
 import glob
@@ -15,6 +17,9 @@ import os                                  ## operating system utilities
 
 import plot_binary_data
 import AMSS_NCKU_Input as input_data
+import subprocess
+import sys
+import multiprocessing
 
 # plt.rcParams['text.usetex'] = True  ## enable LaTeX fonts in plots
 
@@ -50,10 +55,40 @@ def generate_binary_data_plot( binary_outdir, figure_outdir ):
         file_list.append(x)
         print(x)
 
-    ## Plot each file in the list
+    ## Plot each file in parallel using subprocesses.
+    ## Each subprocess is a fresh Python process where the BLAS thread-count
+    ## environment variables (set at the top of plot_binary_data.py) take
+    ## effect before numpy is imported.  This avoids the thread explosion
+    ## that occurs when multiprocessing.Pool with 'fork' context inherits
+    ## already-initialized multi-threaded BLAS from the parent.
+    script = os.path.join( os.path.dirname(__file__), "plot_binary_data.py" )
+    max_workers = min( multiprocessing.cpu_count(), len(file_list) ) if file_list else 0
+
+    running = []
+    failed  = []
     for filename in file_list:
         print(filename)
-        plot_binary_data.plot_binary_data(filename, binary_outdir, figure_outdir)
+        proc = subprocess.Popen(
+            [sys.executable, script, filename, binary_outdir, figure_outdir],
+        )
+        running.append( (proc, filename) )
+        ## Keep at most max_workers subprocesses active at a time
+        if len(running) >= max_workers:
+            p, fn = running.pop(0)
+            p.wait()
+            if p.returncode != 0:
+                failed.append(fn)
+
+    ## Wait for all remaining subprocesses to finish
+    for p, fn in running:
+        p.wait()
+        if p.returncode != 0:
+            failed.append(fn)
+
+    if failed:
+        print( " WARNING: the following binary data plots failed:" )
+        for fn in failed:
+            print( "   ", fn )
 
     print(                        )
     print( " Binary Data Plot Has been Finished " )