fixup! Fix load explosion: use subprocess for binary data plots to avoid thread conflict

* Seems we don't have to set so many variables, `OMP_NUM_THREADS` is enough. Test: Annotate the code for setting other environment variables. It runs normally.
Fix load explosion: use subprocess for binary data plots to avoid thread conflict
2026-02-09 23:00:17 +08:00 · 2026-02-09 21:40:27 +08:00 · 2026-02-09 21:36:45 +08:00 · 2026-02-09 15:13:18 +08:00 · 2026-02-08 16:46:44 +08:00 · 2026-02-08 16:19:13 +08:00
25 changed files with 3072 additions and 2415 deletions
--- a/.gitignore
+++ b/.gitignore
@@ -1,7 +1,6 @@
 __pycache__
 GW150914
 GW150914-origin
-GW150914-mini
 docs
 *.tmp

--- a/AMSS_NCKU_Input.py
+++ b/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)
-

 #################################################

--- a/AMSS_NCKU_Input_Mini.py
+++ b/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
-                                   
-#################################################
--- a/AMSS_NCKU_MiniProgram.py
+++ b/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()
--- a/AMSS_NCKU_Program.py
+++ b/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)
+

 ##################################################################

@@ -424,26 +432,31 @@ 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 )
--- a/AMSS_NCKU_source/Parallel.C
+++ b/AMSS_NCKU_source/Parallel.C
@@ -3756,6 +3756,358 @@ void Parallel::Sync(MyList<Patch> *PatL, MyList<var> *VarList, int Symmetry)
  delete[] transfer_src;
  delete[] transfer_dst;
 }
+//
+// Async Sync: split into SyncBegin (initiate MPI) and SyncEnd (wait + unpack)
+// This allows overlapping MPI communication with computation.
+//
+static void transfer_begin(Parallel::TransferState *ts)
+{
+  int myrank;
+  MPI_Comm_rank(MPI_COMM_WORLD, &myrank);
+  int cpusize = ts->cpusize;
+
+  ts->reqs = new MPI_Request[2 * cpusize];
+  ts->stats = new MPI_Status[2 * cpusize];
+  ts->req_no = 0;
+  ts->send_data = new double *[cpusize];
+  ts->rec_data = new double *[cpusize];
+  int length;
+
+  for (int node = 0; node < cpusize; node++)
+  {
+    ts->send_data[node] = ts->rec_data[node] = 0;
+    if (node == myrank)
+    {
+      // Local copy: pack then immediately unpack (no MPI needed)
+      if ((length = Parallel::data_packer(0, ts->transfer_src[myrank], ts->transfer_dst[myrank],
+                                          node, PACK, ts->VarList1, ts->VarList2, ts->Symmetry)))
+      {
+        double *local_data = new double[length];
+        if (!local_data)
+        {
+          cout << "out of memory in transfer_begin, local copy" << endl;
+          MPI_Abort(MPI_COMM_WORLD, 1);
+        }
+        Parallel::data_packer(local_data, ts->transfer_src[myrank], ts->transfer_dst[myrank],
+                              node, PACK, ts->VarList1, ts->VarList2, ts->Symmetry);
+        Parallel::data_packer(local_data, ts->transfer_src[node], ts->transfer_dst[node],
+                              node, UNPACK, ts->VarList1, ts->VarList2, ts->Symmetry);
+        delete[] local_data;
+      }
+    }
+    else
+    {
+      // send from this cpu to cpu#node
+      if ((length = Parallel::data_packer(0, ts->transfer_src[myrank], ts->transfer_dst[myrank],
+                                          node, PACK, ts->VarList1, ts->VarList2, ts->Symmetry)))
+      {
+        ts->send_data[node] = new double[length];
+        if (!ts->send_data[node])
+        {
+          cout << "out of memory in transfer_begin, send" << endl;
+          MPI_Abort(MPI_COMM_WORLD, 1);
+        }
+        Parallel::data_packer(ts->send_data[node], ts->transfer_src[myrank], ts->transfer_dst[myrank],
+                              node, PACK, ts->VarList1, ts->VarList2, ts->Symmetry);
+        MPI_Isend((void *)ts->send_data[node], length, MPI_DOUBLE, node, 1, MPI_COMM_WORLD,
+                  ts->reqs + ts->req_no++);
+      }
+      // receive from cpu#node to this cpu
+      if ((length = Parallel::data_packer(0, ts->transfer_src[node], ts->transfer_dst[node],
+                                          node, UNPACK, ts->VarList1, ts->VarList2, ts->Symmetry)))
+      {
+        ts->rec_data[node] = new double[length];
+        if (!ts->rec_data[node])
+        {
+          cout << "out of memory in transfer_begin, recv" << endl;
+          MPI_Abort(MPI_COMM_WORLD, 1);
+        }
+        MPI_Irecv((void *)ts->rec_data[node], length, MPI_DOUBLE, node, 1, MPI_COMM_WORLD,
+                  ts->reqs + ts->req_no++);
+      }
+    }
+  }
+  // NOTE: MPI_Waitall is NOT called here - that happens in transfer_end
+}
+//
+static void transfer_end(Parallel::TransferState *ts)
+{
+  // Wait for all pending MPI operations
+  MPI_Waitall(ts->req_no, ts->reqs, ts->stats);
+
+  // Unpack received data from remote ranks
+  for (int node = 0; node < ts->cpusize; node++)
+    if (ts->rec_data[node])
+      Parallel::data_packer(ts->rec_data[node], ts->transfer_src[node], ts->transfer_dst[node],
+                            node, UNPACK, ts->VarList1, ts->VarList2, ts->Symmetry);
+
+  // Cleanup MPI buffers
+  for (int node = 0; node < ts->cpusize; node++)
+  {
+    if (ts->send_data[node])
+      delete[] ts->send_data[node];
+    if (ts->rec_data[node])
+      delete[] ts->rec_data[node];
+  }
+  delete[] ts->reqs;
+  delete[] ts->stats;
+  delete[] ts->send_data;
+  delete[] ts->rec_data;
+}
+//
+Parallel::SyncHandle *Parallel::SyncBegin(Patch *Pat, MyList<var> *VarList, int Symmetry)
+{
+  int cpusize;
+  MPI_Comm_size(MPI_COMM_WORLD, &cpusize);
+
+  SyncHandle *handle = new SyncHandle;
+  handle->num_states = 1;
+  handle->states = new TransferState[1];
+
+  TransferState *ts = &handle->states[0];
+  ts->cpusize = cpusize;
+  ts->VarList1 = VarList;
+  ts->VarList2 = VarList;
+  ts->Symmetry = Symmetry;
+  ts->owns_gsl = true;
+
+  ts->dst = build_ghost_gsl(Pat);
+  ts->src = new MyList<Parallel::gridseg> *[cpusize];
+  ts->transfer_src = new MyList<Parallel::gridseg> *[cpusize];
+  ts->transfer_dst = new MyList<Parallel::gridseg> *[cpusize];
+  for (int node = 0; node < cpusize; node++)
+  {
+    ts->src[node] = build_owned_gsl0(Pat, node);
+    build_gstl(ts->src[node], ts->dst, &ts->transfer_src[node], &ts->transfer_dst[node]);
+  }
+
+  transfer_begin(ts);
+
+  return handle;
+}
+//
+Parallel::SyncHandle *Parallel::SyncBegin(MyList<Patch> *PatL, MyList<var> *VarList, int Symmetry)
+{
+  int cpusize;
+  MPI_Comm_size(MPI_COMM_WORLD, &cpusize);
+
+  // Count patches
+  int num_patches = 0;
+  MyList<Patch> *Pp = PatL;
+  while (Pp) { num_patches++; Pp = Pp->next; }
+
+  SyncHandle *handle = new SyncHandle;
+  handle->num_states = num_patches + 1; // intra-patch transfers + 1 inter-patch transfer
+  handle->states = new TransferState[handle->num_states];
+
+  // Intra-patch sync: for each patch, build ghost lists and initiate transfer
+  int idx = 0;
+  Pp = PatL;
+  while (Pp)
+  {
+    TransferState *ts = &handle->states[idx];
+    ts->cpusize = cpusize;
+    ts->VarList1 = VarList;
+    ts->VarList2 = VarList;
+    ts->Symmetry = Symmetry;
+    ts->owns_gsl = true;
+
+    ts->dst = build_ghost_gsl(Pp->data);
+    ts->src = new MyList<Parallel::gridseg> *[cpusize];
+    ts->transfer_src = new MyList<Parallel::gridseg> *[cpusize];
+    ts->transfer_dst = new MyList<Parallel::gridseg> *[cpusize];
+    for (int node = 0; node < cpusize; node++)
+    {
+      ts->src[node] = build_owned_gsl0(Pp->data, node);
+      build_gstl(ts->src[node], ts->dst, &ts->transfer_src[node], &ts->transfer_dst[node]);
+    }
+
+    transfer_begin(ts);
+
+    idx++;
+    Pp = Pp->next;
+  }
+
+  // Inter-patch sync: buffer zone exchange between patches
+  {
+    TransferState *ts = &handle->states[idx];
+    ts->cpusize = cpusize;
+    ts->VarList1 = VarList;
+    ts->VarList2 = VarList;
+    ts->Symmetry = Symmetry;
+    ts->owns_gsl = true;
+
+    ts->dst = build_buffer_gsl(PatL);
+    ts->src = new MyList<Parallel::gridseg> *[cpusize];
+    ts->transfer_src = new MyList<Parallel::gridseg> *[cpusize];
+    ts->transfer_dst = new MyList<Parallel::gridseg> *[cpusize];
+    for (int node = 0; node < cpusize; node++)
+    {
+      ts->src[node] = build_owned_gsl(PatL, node, 5, Symmetry);
+      build_gstl(ts->src[node], ts->dst, &ts->transfer_src[node], &ts->transfer_dst[node]);
+    }
+
+    transfer_begin(ts);
+  }
+
+  return handle;
+}
+//
+void Parallel::SyncEnd(SyncHandle *handle)
+{
+  if (!handle)
+    return;
+
+  // Wait for all pending transfers and unpack
+  for (int i = 0; i < handle->num_states; i++)
+  {
+    TransferState *ts = &handle->states[i];
+    transfer_end(ts);
+
+    // Cleanup grid segment lists only if this state owns them
+    if (ts->owns_gsl)
+    {
+      if (ts->dst)
+        ts->dst->destroyList();
+      for (int node = 0; node < ts->cpusize; node++)
+      {
+        if (ts->src[node])
+          ts->src[node]->destroyList();
+        if (ts->transfer_src[node])
+          ts->transfer_src[node]->destroyList();
+        if (ts->transfer_dst[node])
+          ts->transfer_dst[node]->destroyList();
+      }
+      delete[] ts->src;
+      delete[] ts->transfer_src;
+      delete[] ts->transfer_dst;
+    }
+  }
+
+  delete[] handle->states;
+  delete handle;
+}
+//
+// SyncPreparePlan: Pre-build grid segment lists for a patch list.
+// The plan can be reused across multiple SyncBeginWithPlan calls
+// as long as the mesh topology does not change (no regridding).
+//
+Parallel::SyncPlan *Parallel::SyncPreparePlan(MyList<Patch> *PatL, int Symmetry)
+{
+  int cpusize;
+  MPI_Comm_size(MPI_COMM_WORLD, &cpusize);
+
+  // Count patches
+  int num_patches = 0;
+  MyList<Patch> *Pp = PatL;
+  while (Pp) { num_patches++; Pp = Pp->next; }
+
+  SyncPlan *plan = new SyncPlan;
+  plan->num_entries = num_patches + 1; // intra-patch + 1 inter-patch
+  plan->Symmetry = Symmetry;
+  plan->entries = new SyncPlanEntry[plan->num_entries];
+
+  // Intra-patch entries: ghost zone exchange within each patch
+  int idx = 0;
+  Pp = PatL;
+  while (Pp)
+  {
+    SyncPlanEntry *pe = &plan->entries[idx];
+    pe->cpusize = cpusize;
+    pe->dst = build_ghost_gsl(Pp->data);
+    pe->src = new MyList<Parallel::gridseg> *[cpusize];
+    pe->transfer_src = new MyList<Parallel::gridseg> *[cpusize];
+    pe->transfer_dst = new MyList<Parallel::gridseg> *[cpusize];
+    for (int node = 0; node < cpusize; node++)
+    {
+      pe->src[node] = build_owned_gsl0(Pp->data, node);
+      build_gstl(pe->src[node], pe->dst, &pe->transfer_src[node], &pe->transfer_dst[node]);
+    }
+    idx++;
+    Pp = Pp->next;
+  }
+
+  // Inter-patch entry: buffer zone exchange between patches
+  {
+    SyncPlanEntry *pe = &plan->entries[idx];
+    pe->cpusize = cpusize;
+    pe->dst = build_buffer_gsl(PatL);
+    pe->src = new MyList<Parallel::gridseg> *[cpusize];
+    pe->transfer_src = new MyList<Parallel::gridseg> *[cpusize];
+    pe->transfer_dst = new MyList<Parallel::gridseg> *[cpusize];
+    for (int node = 0; node < cpusize; node++)
+    {
+      pe->src[node] = build_owned_gsl(PatL, node, 5, Symmetry);
+      build_gstl(pe->src[node], pe->dst, &pe->transfer_src[node], &pe->transfer_dst[node]);
+    }
+  }
+
+  return plan;
+}
+//
+void Parallel::SyncFreePlan(SyncPlan *plan)
+{
+  if (!plan)
+    return;
+
+  for (int i = 0; i < plan->num_entries; i++)
+  {
+    SyncPlanEntry *pe = &plan->entries[i];
+    if (pe->dst)
+      pe->dst->destroyList();
+    for (int node = 0; node < pe->cpusize; node++)
+    {
+      if (pe->src[node])
+        pe->src[node]->destroyList();
+      if (pe->transfer_src[node])
+        pe->transfer_src[node]->destroyList();
+      if (pe->transfer_dst[node])
+        pe->transfer_dst[node]->destroyList();
+    }
+    delete[] pe->src;
+    delete[] pe->transfer_src;
+    delete[] pe->transfer_dst;
+  }
+  delete[] plan->entries;
+  delete plan;
+}
+//
+// SyncBeginWithPlan: Use pre-built GSLs from a SyncPlan to initiate async transfer.
+// This avoids the O(cpusize * blocks^2) cost of rebuilding GSLs on every call.
+//
+Parallel::SyncHandle *Parallel::SyncBeginWithPlan(SyncPlan *plan, MyList<var> *VarList)
+{
+  return SyncBeginWithPlan(plan, VarList, VarList);
+}
+//
+Parallel::SyncHandle *Parallel::SyncBeginWithPlan(SyncPlan *plan, MyList<var> *VarList1, MyList<var> *VarList2)
+{
+  SyncHandle *handle = new SyncHandle;
+  handle->num_states = plan->num_entries;
+  handle->states = new TransferState[handle->num_states];
+
+  for (int i = 0; i < plan->num_entries; i++)
+  {
+    SyncPlanEntry *pe = &plan->entries[i];
+    TransferState *ts = &handle->states[i];
+
+    ts->cpusize = pe->cpusize;
+    ts->VarList1 = VarList1;
+    ts->VarList2 = VarList2;
+    ts->Symmetry = plan->Symmetry;
+    ts->owns_gsl = false; // GSLs are owned by the plan, not this handle
+
+    // Borrow GSL pointers from the plan (do NOT free them in SyncEnd)
+    ts->transfer_src = pe->transfer_src;
+    ts->transfer_dst = pe->transfer_dst;
+    ts->src = pe->src;
+    ts->dst = pe->dst;
+
+    transfer_begin(ts);
+  }
+
+  return handle;
+}
 // collect buffer grid segments or blocks for the periodic boundary condition of given patch
 // ---------------------------------------------------
 // |con |                                       |con |
--- a/AMSS_NCKU_source/Parallel.h
+++ b/AMSS_NCKU_source/Parallel.h
@@ -81,6 +81,53 @@ namespace Parallel
                   int Symmetry);
  void Sync(Patch *Pat, MyList<var> *VarList, int Symmetry);
  void Sync(MyList<Patch> *PatL, MyList<var> *VarList, int Symmetry);
+
+  // Async Sync: overlap MPI communication with computation
+  struct TransferState
+  {
+    MPI_Request *reqs;
+    MPI_Status *stats;
+    int req_no;
+    double **send_data;
+    double **rec_data;
+    int cpusize;
+    MyList<gridseg> **transfer_src;
+    MyList<gridseg> **transfer_dst;
+    MyList<gridseg> **src;
+    MyList<gridseg> *dst;
+    MyList<var> *VarList1;
+    MyList<var> *VarList2;
+    int Symmetry;
+    bool owns_gsl; // true if this state owns and should free the GSLs
+  };
+  struct SyncHandle
+  {
+    TransferState *states;
+    int num_states;
+  };
+  SyncHandle *SyncBegin(Patch *Pat, MyList<var> *VarList, int Symmetry);
+  SyncHandle *SyncBegin(MyList<Patch> *PatL, MyList<var> *VarList, int Symmetry);
+  void SyncEnd(SyncHandle *handle);
+
+  // Cached GSL plan: pre-build grid segment lists once, reuse across multiple Sync calls
+  struct SyncPlanEntry
+  {
+    int cpusize;
+    MyList<gridseg> **transfer_src;
+    MyList<gridseg> **transfer_dst;
+    MyList<gridseg> **src;
+    MyList<gridseg> *dst;
+  };
+  struct SyncPlan
+  {
+    SyncPlanEntry *entries;
+    int num_entries;
+    int Symmetry;
+  };
+  SyncPlan *SyncPreparePlan(MyList<Patch> *PatL, int Symmetry);
+  void SyncFreePlan(SyncPlan *plan);
+  SyncHandle *SyncBeginWithPlan(SyncPlan *plan, MyList<var> *VarList);
+  SyncHandle *SyncBeginWithPlan(SyncPlan *plan, MyList<var> *VarList1, MyList<var> *VarList2);
  void OutBdLow2Hi(Patch *Patc, Patch *Patf,
                   MyList<var> *VarList1 /* source */, MyList<var> *VarList2 /* target */,
                   int Symmetry);
--- a/AMSS_NCKU_source/TwoPunctures.C
+++ b/AMSS_NCKU_source/TwoPunctures.C
--- a/AMSS_NCKU_source/TwoPunctures.h
+++ b/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)
--- a/AMSS_NCKU_source/Z4c_class.C
+++ b/AMSS_NCKU_source/Z4c_class.C
@@ -186,6 +186,12 @@ void Z4c_class::Step(int lev, int YN)
  int ERROR = 0;

  MyList<ss_patch> *sPp;
+
+  // Pre-build grid segment lists once for this level's patches.
+  // These are reused across predictor + 3 corrector SyncBegin calls,
+  // avoiding O(cpusize * blocks^2) rebuild each time.
+  Parallel::SyncPlan *sync_plan = Parallel::SyncPreparePlan(GH->PatL[lev], Symmetry);
+
  // Predictor
  MyList<Patch> *Pp = GH->PatL[lev];
  while (Pp)
@@ -321,13 +327,17 @@ void Z4c_class::Step(int lev, int YN)
    }
    Pp = Pp->next;
  }
-  // check error information
+  // Start async ghost zone exchange - overlaps with error check and Shell computation
+  Parallel::SyncHandle *sync_pre = Parallel::SyncBeginWithPlan(sync_plan, SynchList_pre);
+
+  // check error information (overlaps with MPI transfer)
  {
    int erh = ERROR;
    MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
  }
  if (ERROR)
  {
+    Parallel::SyncEnd(sync_pre); sync_pre = 0;
    Parallel::Dump_Data(GH->PatL[lev], StateList, 0, PhysTime, dT_lev);
    if (myrank == 0)
    {
@@ -475,6 +485,7 @@ void Z4c_class::Step(int lev, int YN)
  }
  if (ERROR)
  {
+    Parallel::SyncEnd(sync_pre); sync_pre = 0;
    SH->Dump_Data(StateList, 0, PhysTime, dT_lev);
    if (myrank == 0)
    {
@@ -485,7 +496,8 @@ void Z4c_class::Step(int lev, int YN)
  }
 #endif

-  Parallel::Sync(GH->PatL[lev], SynchList_pre, Symmetry);
+  // Complete async ghost zone exchange
+  if (sync_pre) Parallel::SyncEnd(sync_pre);

 #ifdef WithShell
  if (lev == 0)
@@ -693,13 +705,17 @@ void Z4c_class::Step(int lev, int YN)
      Pp = Pp->next;
    }

-    // check error information
+    // Start async ghost zone exchange - overlaps with error check and Shell computation
+    Parallel::SyncHandle *sync_cor = Parallel::SyncBeginWithPlan(sync_plan, SynchList_cor);
+
+    // check error information (overlaps with MPI transfer)
    {
      int erh = ERROR;
      MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
    }
    if (ERROR)
    {
+      Parallel::SyncEnd(sync_cor); sync_cor = 0;
      Parallel::Dump_Data(GH->PatL[lev], SynchList_pre, 0, PhysTime, dT_lev);
      if (myrank == 0)
      {
@@ -857,6 +873,7 @@ void Z4c_class::Step(int lev, int YN)
    }
    if (ERROR)
    {
+      Parallel::SyncEnd(sync_cor); sync_cor = 0;
      SH->Dump_Data(SynchList_pre, 0, PhysTime, dT_lev);
      if (myrank == 0)
      {
@@ -868,7 +885,8 @@ void Z4c_class::Step(int lev, int YN)
    }
 #endif

-    Parallel::Sync(GH->PatL[lev], SynchList_cor, Symmetry);
+    // Complete async ghost zone exchange
+    if (sync_cor) Parallel::SyncEnd(sync_cor);

 #ifdef WithShell
    if (lev == 0)
@@ -1042,6 +1060,8 @@ void Z4c_class::Step(int lev, int YN)
      Porg0[ithBH][2] = Porg1[ithBH][2];
    }
  }
+
+  Parallel::SyncFreePlan(sync_plan);
 }
 #else
 // for constraint preserving boundary (CPBC)
@@ -1075,6 +1095,10 @@ void Z4c_class::Step(int lev, int YN)
  int ERROR = 0;

  MyList<ss_patch> *sPp;
+
+  // Pre-build grid segment lists once for this level's patches.
+  Parallel::SyncPlan *sync_plan = Parallel::SyncPreparePlan(GH->PatL[lev], Symmetry);
+
  // Predictor
  MyList<Patch> *Pp = GH->PatL[lev];
  while (Pp)
@@ -1542,13 +1566,17 @@ void Z4c_class::Step(int lev, int YN)
  }
 #endif
  }
-  // check error information
+  // Start async ghost zone exchange - overlaps with error check
+  Parallel::SyncHandle *sync_pre = Parallel::SyncBeginWithPlan(sync_plan, SynchList_pre);
+
+  // check error information (overlaps with MPI transfer)
  {
    int erh = ERROR;
    MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
  }
  if (ERROR)
  {
+    Parallel::SyncEnd(sync_pre); sync_pre = 0;
    SH->Dump_Data(StateList, 0, PhysTime, dT_lev);
    if (myrank == 0)
    {
@@ -1558,7 +1586,8 @@ void Z4c_class::Step(int lev, int YN)
    }
  }

-  Parallel::Sync(GH->PatL[lev], SynchList_pre, Symmetry);
+  // Complete async ghost zone exchange
+  if (sync_pre) Parallel::SyncEnd(sync_pre);

  if (lev == 0)
  {
@@ -2103,13 +2132,17 @@ void Z4c_class::Step(int lev, int YN)
        sPp = sPp->next;
      }
    }
-    // check error information
+    // Start async ghost zone exchange - overlaps with error check
+    Parallel::SyncHandle *sync_cor = Parallel::SyncBeginWithPlan(sync_plan, SynchList_cor);
+
+    // check error information (overlaps with MPI transfer)
    {
      int erh = ERROR;
      MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
    }
    if (ERROR)
    {
+      Parallel::SyncEnd(sync_cor); sync_cor = 0;
      SH->Dump_Data(SynchList_pre, 0, PhysTime, dT_lev);
      if (myrank == 0)
      {
@@ -2120,7 +2153,8 @@ void Z4c_class::Step(int lev, int YN)
      }
    }

-    Parallel::Sync(GH->PatL[lev], SynchList_cor, Symmetry);
+    // Complete async ghost zone exchange
+    if (sync_cor) Parallel::SyncEnd(sync_cor);

    if (lev == 0)
    {
@@ -2346,6 +2380,8 @@ void Z4c_class::Step(int lev, int YN)
 	  DG_List->clearList();
 	}
 #endif
+
+  Parallel::SyncFreePlan(sync_plan);
 }
 #endif
 #undef MRBD
--- a/AMSS_NCKU_source/bssn_class.C
+++ b/AMSS_NCKU_source/bssn_class.C
@@ -3035,6 +3035,12 @@ void bssn_class::Step(int lev, int YN)
  int ERROR = 0;

  MyList<ss_patch> *sPp;
+
+  // Pre-build grid segment lists once for this level's patches.
+  // These are reused across predictor + 3 corrector SyncBegin calls,
+  // avoiding O(cpusize * blocks^2) rebuild each time.
+  Parallel::SyncPlan *sync_plan = Parallel::SyncPreparePlan(GH->PatL[lev], Symmetry);
+
  // Predictor
  MyList<Patch> *Pp = GH->PatL[lev];
  while (Pp)
@@ -3158,13 +3164,18 @@ void bssn_class::Step(int lev, int YN)
    }
    Pp = Pp->next;
  }
-  // check error information
+
+  // Start async ghost zone exchange - overlaps with error check and Shell computation
+  Parallel::SyncHandle *sync_pre = Parallel::SyncBeginWithPlan(sync_plan, SynchList_pre);
+
+  // check error information (overlaps with MPI transfer)
  {
    int erh = ERROR;
    MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
  }
  if (ERROR)
  {
+    Parallel::SyncEnd(sync_pre); sync_pre = 0;
    Parallel::Dump_Data(GH->PatL[lev], StateList, 0, PhysTime, dT_lev);
    if (myrank == 0)
    {
@@ -3324,6 +3335,7 @@ void bssn_class::Step(int lev, int YN)

  if (ERROR)
  {
+    Parallel::SyncEnd(sync_pre); sync_pre = 0;
    SH->Dump_Data(StateList, 0, PhysTime, dT_lev);
    if (myrank == 0)
    {
@@ -3334,7 +3346,8 @@ void bssn_class::Step(int lev, int YN)
  }
 #endif

-  Parallel::Sync(GH->PatL[lev], SynchList_pre, Symmetry);
+  // Complete async ghost zone exchange
+  if (sync_pre) Parallel::SyncEnd(sync_pre);

 #ifdef WithShell
  if (lev == 0)
@@ -3528,7 +3541,10 @@ void bssn_class::Step(int lev, int YN)
      Pp = Pp->next;
    }

-    // check error information
+    // Start async ghost zone exchange - overlaps with error check and Shell computation
+    Parallel::SyncHandle *sync_cor = Parallel::SyncBeginWithPlan(sync_plan, SynchList_cor);
+
+    // check error information (overlaps with MPI transfer)
    {
      int erh = ERROR;
      MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
@@ -3536,6 +3552,7 @@ void bssn_class::Step(int lev, int YN)

    if (ERROR)
    {
+      Parallel::SyncEnd(sync_cor); sync_cor = 0;
      Parallel::Dump_Data(GH->PatL[lev], SynchList_pre, 0, PhysTime, dT_lev);
      if (myrank == 0)
      {
@@ -3692,6 +3709,7 @@ void bssn_class::Step(int lev, int YN)
    }
    if (ERROR)
    {
+      Parallel::SyncEnd(sync_cor); sync_cor = 0;
      SH->Dump_Data(SynchList_pre, 0, PhysTime, dT_lev);
      if (myrank == 0)
      {
@@ -3704,7 +3722,8 @@ void bssn_class::Step(int lev, int YN)
    }
 #endif

-    Parallel::Sync(GH->PatL[lev], SynchList_cor, Symmetry);
+    // Complete async ghost zone exchange
+    if (sync_cor) Parallel::SyncEnd(sync_cor);

 #ifdef WithShell
    if (lev == 0)
@@ -3895,6 +3914,8 @@ void bssn_class::Step(int lev, int YN)
      Porg0[ithBH][2] = Porg1[ithBH][2];
    }
  }
+
+  Parallel::SyncFreePlan(sync_plan);
 }

 //================================================================================================
@@ -4817,6 +4838,12 @@ void bssn_class::Step(int lev, int YN)
  int ERROR = 0;

  MyList<ss_patch> *sPp;
+
+  // Pre-build grid segment lists once for this level's patches.
+  // These are reused across predictor + 3 corrector SyncBegin calls,
+  // avoiding O(cpusize * blocks^2) rebuild each time.
+  Parallel::SyncPlan *sync_plan = Parallel::SyncPreparePlan(GH->PatL[lev], Symmetry);
+
  // Predictor
  MyList<Patch> *Pp = GH->PatL[lev];
  while (Pp)
@@ -4943,13 +4970,17 @@ void bssn_class::Step(int lev, int YN)

  //   misc::tillherecheck(GH->Commlev[lev],GH->start_rank[lev],"after Predictor rhs calculation");

-  // check error information
+  // Start async ghost zone exchange - overlaps with error check and BH position
+  Parallel::SyncHandle *sync_pre = Parallel::SyncBeginWithPlan(sync_plan, SynchList_pre);
+
+  // check error information (overlaps with MPI transfer)
  {
    int erh = ERROR;
    MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, GH->Commlev[lev]);
  }
  if (ERROR)
  {
+    Parallel::SyncEnd(sync_pre); sync_pre = 0;
    Parallel::Dump_Data(GH->PatL[lev], StateList, 0, PhysTime, dT_lev);
    if (myrank == 0)
    {
@@ -4961,7 +4992,8 @@ 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);
+  // Complete async ghost zone exchange
+  if (sync_pre) Parallel::SyncEnd(sync_pre);

 #if (MAPBH == 0)
  // for black hole position
@@ -5140,13 +5172,17 @@ void bssn_class::Step(int lev, int YN)

    //   misc::tillherecheck(GH->Commlev[lev],GH->start_rank[lev],"before Corrector error check");

-    // check error information
+    // Start async ghost zone exchange - overlaps with error check and BH position
+    Parallel::SyncHandle *sync_cor = Parallel::SyncBeginWithPlan(sync_plan, SynchList_cor);
+
+    // check error information (overlaps with MPI transfer)
    {
      int erh = ERROR;
      MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, GH->Commlev[lev]);
    }
    if (ERROR)
    {
+      Parallel::SyncEnd(sync_cor); sync_cor = 0;
      Parallel::Dump_Data(GH->PatL[lev], SynchList_pre, 0, PhysTime, dT_lev);
      if (myrank == 0)
      {
@@ -5160,7 +5196,8 @@ 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);
+    // Complete async ghost zone exchange
+    if (sync_cor) Parallel::SyncEnd(sync_cor);

    //    misc::tillherecheck(GH->Commlev[lev],GH->start_rank[lev],"after Corrector sync");

@@ -5276,6 +5313,8 @@ void bssn_class::Step(int lev, int YN)

  //     if(myrank==GH->start_rank[lev]) cout<<GH->mylev<<endl;
  //     misc::tillherecheck(GH->Commlev[lev],GH->start_rank[lev],"complet GH Step");
+
+  Parallel::SyncFreePlan(sync_plan);
 }

 //================================================================================================
--- a/AMSS_NCKU_source/bssn_rhs.f90
+++ b/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
--- a/AMSS_NCKU_source/diff_new.f90
+++ b/AMSS_NCKU_source/diff_new.f90
@@ -1939,309 +1939,6 @@
  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 +2077,6 @@

  end subroutine fderivs
 !-----------------------------------------------------------------------------
-! batch first derivatives (4 fields), same symmetry setup
-!-----------------------------------------------------------------------------
-!-----------------------------------------------------------------------------
 !
 ! single derivatives dx
 !
--- a/AMSS_NCKU_source/enforce_algebra.f90
+++ b/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

--- a/AMSS_NCKU_source/fmisc.f90
+++ b/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)
@@ -886,7 +883,6 @@ 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+1,1:extc(2),1:extc(3))*SoA(1)
@@ -912,7 +908,6 @@ 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 +936,6 @@ 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)
@@ -1119,25 +1113,16 @@ end subroutine d2dump
 !------------------------------------------------------------------------------

  subroutine polint(xa, ya, x, y, dy, ordn)
-
  implicit none

-!~~~~~~> Input Parameter:
  integer, intent(in) :: ordn
  real*8, dimension(ordn), intent(in) :: xa, ya
  real*8, intent(in) :: x
  real*8, intent(out) :: y, dy

-!~~~~~~> Other parameter:
-
-  integer :: m,n,ns
-  real*8, dimension(ordn) :: c,d,den,ho
-  real*8 :: dif,dift
-
-!~~~~~~>
-
-  n=ordn
-  m=ordn
+  integer :: i, m, ns, n_m
+  real*8, dimension(ordn) :: c, d, ho
+  real*8 :: dif, dift, hp, h, den_val

  c = ya
  d = ya
@@ -1145,27 +1130,38 @@ end subroutine d2dump

  ns = 1
  dif = abs(x - xa(1))
-  do m=1,n
-   dift=abs(x-xa(m))
+
+  do i = 2, ordn
+    dift = abs(x - xa(i))
    if (dift < dif) then
-    ns=m
+      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
+
+  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
      end if
-    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
+
+      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)
@@ -1175,7 +1171,6 @@ end subroutine d2dump
  end do

  return
-
  end subroutine polint
 !------------------------------------------------------------------------------
 !
@@ -1183,35 +1178,37 @@ 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 +1216,15 @@ 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 +1233,33 @@ 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
--- a/AMSS_NCKU_source/lopsidediff.f90
+++ b/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
--- a/AMSS_NCKU_source/makefile
+++ b/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
--- a/AMSS_NCKU_source/makefile.inc
+++ b/AMSS_NCKU_source/makefile.inc
@@ -15,11 +15,10 @@ LDLIBS  = -L${MKLROOT}/lib -lmkl_intel_lp64 -lmkl_sequential -lmkl_core -lifcore
 ## -xHost: Optimize for the host CPU architecture (Intel/AMD compatible)
 ## -fp-model fast=2: Aggressive floating-point optimizations
 ## -fma: Enable fused multiply-add instructions
-## Note: OpenMP has been disabled (-qopenmp removed) due to performance issues
-CXXAPPFLAGS  = -O3 -xHost -fp-model fast=2 -fma \
+CXXAPPFLAGS  = -O3 -xHost -fp-model fast=2 -fma -ipo \
               -Dfortran3 -Dnewc -I${MKLROOT}/include
-f90appflags  = -O3 -xHost -fp-model fast=2 -fma \
-               -fpp -I${MKLROOT}/include
+f90appflags  = -O3 -xHost -fp-model fast=2 -fma -ipo \
+               -align array64byte -fpp -I${MKLROOT}/include
 f90          = ifx
 f77          = ifx
 CXX          = icpx
--- a/generate_macrodef.py
+++ b/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 )
--- a/inputfile_example/AMSS_NCKU_Input.py
+++ b/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)

 #################################################

--- a/makefile_and_run.py
+++ b/makefile_and_run.py
@@ -10,18 +10,17 @@

 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"

 ## 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 = 104


 ##################################################################
@@ -153,7 +152,7 @@ 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(                                                          )
@@ -180,7 +179,9 @@ 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

 ##################################################################
--- a/parallel_plot_helper.py
+++ b/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)
--- a/plot_GW_strain_amplitude_xiaoqu.py
+++ b/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

--- a/plot_binary_data.py
+++ b/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])
+
--- a/plot_xiaoqu.py
+++ b/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 " )
Author	SHA1	Message	Date
ianchb	8b68b5d782	fixup! Fix load explosion: use subprocess for binary data plots to avoid thread conflict * Seems we don't have to set so many variables, `OMP_NUM_THREADS` is enough. Test: Annotate the code for setting other environment variables. It runs normally.	2026-02-09 23:00:17 +08:00
ianchb	dd2443c926	Fix load explosion: use subprocess for binary data plots to avoid thread conflict Co-authored-by: copilot-swe-agent[bot] <198982749+copilot@users.noreply.github.com>	2026-02-09 21:40:27 +08:00
ianchb	2d7ba5c60c	[2/2] Implement multiprocessing-based parallel plotting	2026-02-09 21:36:45 +08:00
ianchb	4777cad4ed	[1/2] Implement multiprocessing-based parallel plotting	2026-02-09 15:13:18 +08:00
ianchb	afd4006da2	Cache GSL in SyncPlan and apply async Sync to Z4c_class Major optimization: Pre-build grid segment lists (GSLs) once per Step() call via SyncPreparePlan(), then reuse them across all 4 RK4 substep SyncBegin calls via SyncBeginWithPlan(). This eliminates the O(cpusize * blocks^2) GSL rebuild cost that was incurred on every ghost zone exchange. Applied async SyncBegin/SyncEnd overlap pattern to Z4c_class.C (ABEtype==2, the default configuration), which was still using blocking Parallel::Sync. Both the regular and CPBC variants of Z4c Step() are now optimized. Co-authored-by: copilot-swe-agent[bot] <198982749+copilot@users.noreply.github.com>	2026-02-08 16:46:44 +08:00
copilot-swe-agent[bot]	a918dc103e	Add SyncBegin/SyncEnd to Parallel for MPI communication-computation overlap Split the blocking Parallel::Sync into async SyncBegin (initiates local copy + MPI_Isend/Irecv) and SyncEnd (MPI_Waitall + unpack). This allows overlapping MPI ghost zone exchange with error checking and Shell patch computation. Modified Step() in bssn_class.C for both PSTR==0 and PSTR==1/2/3 versions to start Sync before error checks, overlapping the MPI_Allreduce with the ongoing ghost zone transfers. Co-authored-by: copilot-swe-agent[bot] <198982749+copilot@users.noreply.github.com>	2026-02-08 16:19:13 +08:00
copilot-swe-agent[bot]	38c2c30186	Merge lopsided advection + kodis dissipation to share symmetry_bd buffer Add lopsided_kodis subroutine in lopsidediff.f90 that combines upwind advection (lopsided) and Kreiss-Oliger dissipation (kodis) into one function sharing a single fh buffer from symmetry_bd. This eliminates 27 redundant full-grid copies per RHS evaluation (108 per timestep). For gxx/gyy/gzz variables: kodis stencil coefficients sum to zero (1-6+15-20+15-6+1=0), so using gxx(=dxx+1) instead of dxx for the dissipation buffer is mathematically exact. Update bssn_rhs.f90 to use the merged lopsided_kodis calls. Co-authored-by: ianchb <45872450+ianchb@users.noreply.github.com>	2026-02-08 15:42:44 +08:00
ianchb	b8e41b2b39	Only enable OpenMP for TwoPunctures	2026-02-08 13:00:37 +08:00
ianchb	133e4f13a2	Use OpenMP's parallel for with schedule(dynamic,1)	2026-02-07 19:48:24 +08:00
ianchb	914c4f4791	Optimize memory allocation in JFD_times_dv This should reduce the pressure on the memory allocator, indirectly improving caching behavior. Co-authored-by: copilot-swe-agent[bot] <198982749+copilot@users.noreply.github.com>	2026-02-07 15:55:45 +08:00
ianchb	f345b0e520	Performance optimization for the TwoPunctures module * Re-enabled OpenMP. 1. Batch spectral derivatives (Chebyshev & Fourier) via precomputed matrices: Chebyshev/Fourier transforms and derivatives are precomputed as explicit physical-space operator matrices. Batch DGEMM now applies to entire tensor fields, mathematically identical to original per-line transforms but vastly faster. 2. Gauss-Seidel relaxation & tridiagonal solver workspace reuse: Per-thread reusable workspaces replace per-call heap new/delete in all tridiagonal and relaxation routines. 3. Efficient OpenMP multithreading throughout relaxation/deriv: relax_omp and friends parallelize over grouped lines/planes, maximizing threading efficiency and memory independence. Co-authored-by: copilot-swe-agent[bot] <198982749+copilot@users.noreply.github.com>	2026-02-07 14:48:47 +08:00
ianchb	f5ed23d687	Revert "Eliminate hot-path heap allocations in TwoPunctures spectral solver" This reverts commit `09ffdb553d`.	2026-02-07 14:45:25 +08:00
ianchb	03d501db04	Display the runtime of TwoPunctures	2026-02-07 14:45:16 +08:00
CGH0S7	09ffdb553d	Eliminate hot-path heap allocations in TwoPunctures spectral solver Pre-allocate workspace buffers as class members to remove ~8M malloc/free pairs per Newton iteration from LineRelax, ThomasAlgorithm, JFD_times_dv, J_times_dv, chebft_Zeros, fourft, Derivatives_AB3, and F_of_v. Rewrite ThomasAlgorithm to operate in-place on input arrays. Results are bit-identical; no algorithmic changes. Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>	2026-02-06 21:20:35 +08:00
CGH0S7	699e443c7a	Optimize polint/polin2/polin3 interpolation for cache locality Changes: - polint: Rewrite Neville algorithm from array-slice operations to scalar loop. Mathematically identical, avoids temporary array allocations for den(1:n-m) slices. (Credit: yx-fmisc branch) - polin2: Swap interpolation order so inner loop accesses ya(:,j) (contiguous in Fortran column-major) instead of ya(i,:) (strided). Tensor product interpolation is commutative; all call sites pass identical coordinate arrays for all dimensions. - polin3: Swap interpolation order to process contiguous first dimension first: ya(:,j,k) -> yatmp(:,k) -> ymtmp(:). Same commutativity argument as polin2. Compile-time safety switch: -DPOLINT_LEGACY_ORDER restores original dimension ordering Default (no flag): uses optimized contiguous-memory ordering Usage: # Production (optimized order): make clean && make -j ABE # Fallback if results differ (original order): Add -DPOLINT_LEGACY_ORDER to f90appflags in makefile.inc Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>	2026-02-06 19:00:35 +08:00
CGH0S7	24bfa44911	Disable NaN sanity check in bssn_rhs.f90 for production builds Wrap the NaN sanity check (21 sum() full-array traversals per RHS call) with #ifdef DEBUG so it is compiled out in production builds. This eliminates 84 redundant full-array scans per timestep (21 per RHS call × 4 RK4 substages) that serve no purpose when input data is valid. Usage: - Production build (default): NaN check is disabled, no changes needed. - Debug build: add -DDEBUG to f90appflags in makefile.inc, e.g. f90appflags = -O3 ... -DDEBUG -fpp ... to re-enable the NaN sanity check. Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>	2026-02-06 18:36:29 +08:00
CGH0S7	6738854a9d	Compiler-level and hot-path optimizations for GW150914 - makefile.inc: add -ipo (interprocedural optimization) and -align array64byte (64-byte array alignment for vectorization) - fmisc.f90: remove redundant funcc=0.d0 zeroing from symmetry_bd, symmetry_tbd, symmetry_stbd (~328+ full-array memsets eliminated per timestep) - enforce_algebra.f90: rewrite enforce_ag and enforce_ga as point-wise loops, replacing 12 stack-allocated 3D temporary arrays with scalar locals for better cache locality All changes are mathematically equivalent — no algorithmic modifications. Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>	2026-02-06 17:13:39 +08:00
CGH0S7	223ec17a54	input updated	2026-02-06 13:57:48 +08:00