Optimize bssn_rhs.f90: Fuse loops for metric inversion and Christoffel symbols to improve cache locality

优化 compute_rhs_bssn 热点路径并加入 NaN 检查开关
- 用 DEBUG_NAN_CHECK 宏按需启用 NaN 检查，并在输入/宏生成器中新增 Debug_NaN_Check 配置 - 逆度量改为先求行列式再乘法展开，减少除法；并在 Gam^i/Christoffel 处提取公共子表达式 - 预置批量 fderivs 辅助例程，便于后续矢量化/合并导数计算 - 将默认 MPI_processes 调整为 8 变更涉及： - AMSS_NCKU_source/bssn_rhs.f90 - generate_macrodef.py - AMSS_NCKU_Input.py - AMSS_NCKU_Input_Mini.py - inputfile_example/AMSS_NCKU_Input.py - AMSS_NCKU_source/diff_new.f90 TODO: fmisc.f90 polint()
2026-01-21 11:22:33 +08:00 · 2026-01-20 19:37:26 +08:00 · 2026-01-20 00:31:40 +08:00
25 changed files with 2422 additions and 3079 deletions
--- a/.gitignore
+++ b/.gitignore
@@ -1,6 +1,7 @@
 __pycache__
 GW150914
 GW150914-origin
+GW150914-mini
 docs
 *.tmp

--- a/AMSS_NCKU_Input.py
+++ b/AMSS_NCKU_Input.py
@@ -16,12 +16,14 @@ 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    = 64                             ## number of mpi processes used in the simulation
+MPI_processes    = 8                             ## 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
@@ -0,0 +1,233 @@
+
+#################################################
+##
+## 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
@@ -0,0 +1,224 @@
+##################################################################
+##
+## 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
@@ -8,14 +8,6 @@
 ##
 ##################################################################

-## 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)
-

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

@@ -432,31 +424,26 @@ 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_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_xiaoqu.generate_puncture_orbit_plot(   binary_results_directory, figure_directory )
+plot_xiaoqu.generate_puncture_orbit_plot3D( binary_results_directory, figure_directory )

 ## Plot black hole separation vs. time
-plot_tasks.append( ( plot_xiaoqu.generate_puncture_distence_plot, (binary_results_directory, figure_directory) ) )
+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_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_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 ADM mass evolution
 for i in range(input_data.Detector_Number):
-    plot_tasks.append( ( plot_xiaoqu.generate_ADMmass_plot, (binary_results_directory, figure_directory, i) ) )
+    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_tasks.append( ( plot_xiaoqu.generate_constraint_check_plot, (binary_results_directory, figure_directory, i) ) )
-
-run_plot_tasks_parallel(plot_tasks)
+    plot_xiaoqu.generate_constraint_check_plot( binary_results_directory, figure_directory, i )

 ## 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
@@ -3756,358 +3756,6 @@ 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
@@ -81,53 +81,6 @@ 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
@@ -1,8 +1,7 @@
+
 #ifndef TWO_PUNCTURES_H
 #define TWO_PUNCTURES_H

-#include <omp.h>
-
 #define StencilSize 19
 #define N_PlaneRelax 1
 #define NRELAX 200
@@ -43,18 +42,6 @@ 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;
@@ -71,28 +58,6 @@ 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);
@@ -151,11 +116,23 @@ 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
@@ -186,12 +186,6 @@ 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)
@@ -327,17 +321,13 @@ void Z4c_class::Step(int lev, int YN)
    }
    Pp = Pp->next;
  }
-  // 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)
+  // check error information
  {
    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)
    {
@@ -485,7 +475,6 @@ 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)
    {
@@ -496,8 +485,7 @@ void Z4c_class::Step(int lev, int YN)
  }
 #endif

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

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

-    // 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)
+    // check error information
    {
      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)
      {
@@ -873,7 +857,6 @@ 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)
      {
@@ -885,8 +868,7 @@ void Z4c_class::Step(int lev, int YN)
    }
 #endif

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

 #ifdef WithShell
    if (lev == 0)
@@ -1060,8 +1042,6 @@ void Z4c_class::Step(int lev, int YN)
      Porg0[ithBH][2] = Porg1[ithBH][2];
    }
  }
-
-  Parallel::SyncFreePlan(sync_plan);
 }
 #else
 // for constraint preserving boundary (CPBC)
@@ -1095,10 +1075,6 @@ 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)
@@ -1566,17 +1542,13 @@ void Z4c_class::Step(int lev, int YN)
  }
 #endif
  }
-  // 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)
+  // check error information
  {
    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)
    {
@@ -1586,8 +1558,7 @@ void Z4c_class::Step(int lev, int YN)
    }
  }

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

  if (lev == 0)
  {
@@ -2132,17 +2103,13 @@ void Z4c_class::Step(int lev, int YN)
        sPp = sPp->next;
      }
    }
-    // 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)
+    // check error information
    {
      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)
      {
@@ -2153,8 +2120,7 @@ void Z4c_class::Step(int lev, int YN)
      }
    }

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

    if (lev == 0)
    {
@@ -2380,8 +2346,6 @@ 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
@@ -3035,12 +3035,6 @@ 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)
@@ -3164,18 +3158,13 @@ void bssn_class::Step(int lev, int YN)
    }
    Pp = Pp->next;
  }
-
-  // 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)
+  // check error information
  {
    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)
    {
@@ -3335,7 +3324,6 @@ 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)
    {
@@ -3346,8 +3334,7 @@ void bssn_class::Step(int lev, int YN)
  }
 #endif

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

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

-    // 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)
+    // check error information
    {
      int erh = ERROR;
      MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
@@ -3552,7 +3536,6 @@ 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)
      {
@@ -3709,7 +3692,6 @@ 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)
      {
@@ -3722,8 +3704,7 @@ void bssn_class::Step(int lev, int YN)
    }
 #endif

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

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

 //================================================================================================
@@ -4838,12 +4817,6 @@ 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)
@@ -4970,17 +4943,13 @@ void bssn_class::Step(int lev, int YN)

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

-  // 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)
+  // check error information
  {
    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)
    {
@@ -4992,8 +4961,7 @@ void bssn_class::Step(int lev, int YN)

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

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

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

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

-    // 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)
+    // check error information
    {
      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)
      {
@@ -5196,8 +5160,7 @@ void bssn_class::Step(int lev, int YN)

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

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

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

@@ -5313,8 +5276,6 @@ 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
@@ -61,7 +61,9 @@
  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
+  integer::gont,i,j,k
+  real*8 :: val1, val2
+  real*8 :: det, t_gupxx, t_gupxy, t_gupxz, t_gupyy, t_gupyz, t_gupzz

 !~~~~~~> Other variables:

@@ -84,7 +86,10 @@
  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            :: dX, dY, dZ, PI
+  real*8            :: PI
+#if (DEBUG_NAN_CHECK)
+  real*8            :: dX
+#endif
  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
@@ -106,8 +111,8 @@
  call getpbh(BHN,Porg,Mass)
 #endif

-!!! sanity check (disabled in production builds for performance)
-#ifdef DEBUG
+#if (DEBUG_NAN_CHECK)
+!!! sanity check
  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)                                           &
@@ -141,10 +146,6 @@

  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
@@ -158,15 +159,15 @@
  div_beta = betaxx + betayy + betazz

  call fderivs(ex,chi,chix,chiy,chiz,X,Y,Z,SYM,SYM,SYM,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,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,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)
+
+  chi_rhs = F2o3 *chin1*( alpn1 * trK - div_beta ) !rhs for chi

  gxx_rhs = - TWO * alpn1 * Axx    -  F2o3 * gxx * div_beta          + &
              TWO *(  gxx * betaxx +   gxy * betayx +   gxz * betazx)
@@ -192,71 +193,99 @@
                                       gyz * betayx +   gzz * betazx   &
                                                    -   gxz * betayy     !rhs for gij

-! 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
+! 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

     if(co == 0)then
-! 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))
+        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)))
     endif

-! 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 ))
+     ! 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 )

-  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 * 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 )

-  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 = 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) )

-  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 ) )
+     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

-  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 + &
@@ -287,30 +316,40 @@
  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/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            ) + &
+        -F3o2 * ONE/chin1 * Gamxa - &
+              gupxx * fxx - &
+              gupxy * fxy - &
+              gupxz * fxz + &
                        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/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            ) + &
+        -F3o2 * ONE/chin1 * Gamya - &
+              gupxy * fxx - &
+              gupyy * fxy - &
+              gupyz * fxz + &
                        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/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            ) + &
+        -F3o2 * ONE/chin1 * Gamza - &
+              gupxz * fxx - &
+              gupyz * fxy - &
+              gupzz * fxz + &
                        Gamzxx * Rxx + Gamzyy * Ryy + Gamzzz * Rzz   + &
                TWO * ( Gamzxy * Rxy + Gamzxz * Rxz + Gamzyz * Ryz ) )

@@ -612,47 +651,47 @@
  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/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 ) 
+  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 ) 
 ! Add chi part to Ricci tensor:

-  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
+  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

 ! 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)/chin1
-  gxxy = (gupxy * chix + gupyy * chiy + gupyz * chiz)/chin1
-  gxxz = (gupxz * chix + gupyz * chiy + gupzz * chiz)/chin1
+  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
 ! now get physical second kind of connection
-  Gamxxx = Gamxxx - ( (chix + chix)/chin1 - gxx * gxxx )*HALF
+  Gamxxx = Gamxxx - ( TWO * chix * ONE/chin1 - gxx * gxxx )*HALF
  Gamyxx = Gamyxx - (                     - gxx * gxxy )*HALF
  Gamzxx = Gamzxx - (                     - gxx * gxxz )*HALF
  Gamxyy = Gamxyy - (                     - gyy * gxxx )*HALF
-  Gamyyy = Gamyyy - ( (chiy + chiy)/chin1 - gyy * gxxy )*HALF
+  Gamyyy = Gamyyy - ( TWO * chiy * ONE/chin1 - gyy * gxxy )*HALF
  Gamzyy = Gamzyy - (                     - gyy * gxxz )*HALF
  Gamxzz = Gamxzz - (                     - gzz * gxxx )*HALF
  Gamyzz = Gamyzz - (                     - gzz * gxxy )*HALF
-  Gamzzz = Gamzzz - ( (chiz + chiz)/chin1 - gzz * gxxz )*HALF
-  Gamxxy = Gamxxy - (  chiy        /chin1 - gxy * gxxx )*HALF
-  Gamyxy = Gamyxy - (         chix /chin1 - gxy * 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
  Gamzxy = Gamzxy - (                     - gxy * gxxz )*HALF
-  Gamxxz = Gamxxz - (  chiz        /chin1 - gxz * gxxx )*HALF
+  Gamxxz = Gamxxz - (  chiz * ONE/chin1 - gxz * gxxx )*HALF
  Gamyxz = Gamyxz - (                     - gxz * gxxy )*HALF
-  Gamzxz = Gamzxz - (         chix /chin1 - gxz * gxxz )*HALF
+  Gamzxz = Gamzxz - (  chix * ONE/chin1 - gxz * gxxz )*HALF
  Gamxyz = Gamxyz - (                     - gyz * gxxx )*HALF
-  Gamyyz = Gamyyz - (  chiz        /chin1 - gyz * gxxy )*HALF
-  Gamzyz = Gamzyz - (         chiy /chin1 - gyz * gxxz )*HALF
+  Gamyyz = Gamyyz - (  chiz * ONE/chin1 - gyz * gxxy )*HALF
+  Gamzyz = Gamzyz - (  chiy * ONE/chin1 - gyz * gxxz )*HALF

  fxx = fxx - Gamxxx*Lapx - Gamyxx*Lapy - Gamzxx*Lapz
  fyy = fyy - Gamxyy*Lapx - Gamyyy*Lapy - Gamzyy*Lapz
@@ -695,7 +734,7 @@
       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/chin1*f)
+        TWO* ( gupxy * fxy + gupxz * fxz + gupyz * fyz ) + alpn1 * ONE/chin1 * f)
  
  fxx = alpn1 * (Rxx - EIGHT * PI * Sxx) - fxx
  fxy = alpn1 * (Rxy - EIGHT * PI * Sxy) - fxy
@@ -815,7 +854,8 @@
  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/chin1)/(1-dsqrt(chin1))**2
+  fxx = dsqrt(chin1)
+  reta = 1.31d0/2*dsqrt(reta*ONE/chin1)/(ONE-fxx)**2
  dtSfx_rhs = Gamx_rhs - reta*dtSfx
  dtSfy_rhs = Gamy_rhs - reta*dtSfy
  dtSfz_rhs = Gamz_rhs - reta*dtSfz
@@ -827,7 +867,7 @@
  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/chin1)/(1-chin1)**2
+  reta = 1.31d0/2*dsqrt(reta*ONE/chin1)/(ONE-chin1)**2
  dtSfx_rhs = Gamx_rhs - reta*dtSfx
  dtSfy_rhs = Gamy_rhs - reta*dtSfy
  dtSfz_rhs = Gamz_rhs - reta*dtSfz
@@ -835,7 +875,8 @@
  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/chin1)/(1-dsqrt(chin1))**2
+  fxx = dsqrt(chin1)
+  reta = 1.31d0/2*dsqrt(reta*ONE/chin1)/(ONE-fxx)**2
  betax_rhs = FF*Gamx - reta*betax
  betay_rhs = FF*Gamy - reta*betay
  betaz_rhs = FF*Gamz - reta*betaz
@@ -847,7 +888,7 @@
  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/chin1)/(1-chin1)**2
+  reta = 1.31d0/2*dsqrt(reta*ONE/chin1)/(ONE-chin1)**2
  betax_rhs = FF*Gamx - reta*betax
  betay_rhs = FF*Gamy - reta*betay
  betaz_rhs = FF*Gamz - reta*betaz
@@ -945,60 +986,103 @@
  SSA(2)=SYM
  SSA(3)=ANTI

-!!!!!!!!!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.
+!!!!!!!!!advection term part

-  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,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,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,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,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,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,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,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(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
@@ -1036,48 +1120,48 @@
 ! 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/chin1
+                 + Gamxxx * Axx + Gamyxx * Axy + Gamzxx * Axz) - chix*Axx*ONE/chin1
  gxyx = gxyx - (  Gamxxy * Axx + Gamyxy * Axy + Gamzxy * Axz &
-                 + Gamxxx * Axy + Gamyxx * Ayy + Gamzxx * Ayz) - chix*Axy/chin1
+                 + Gamxxx * Axy + Gamyxx * Ayy + Gamzxx * Ayz) - chix*Axy*ONE/chin1
  gxzx = gxzx - (  Gamxxz * Axx + Gamyxz * Axy + Gamzxz * Axz &
-                 + Gamxxx * Axz + Gamyxx * Ayz + Gamzxx * Azz) - chix*Axz/chin1
+                 + Gamxxx * Axz + Gamyxx * Ayz + Gamzxx * Azz) - chix*Axz*ONE/chin1
  gyyx = gyyx - (  Gamxxy * Axy + Gamyxy * Ayy + Gamzxy * Ayz &
-                 + Gamxxy * Axy + Gamyxy * Ayy + Gamzxy * Ayz) - chix*Ayy/chin1
+                 + Gamxxy * Axy + Gamyxy * Ayy + Gamzxy * Ayz) - chix*Ayy*ONE/chin1
  gyzx = gyzx - (  Gamxxz * Axy + Gamyxz * Ayy + Gamzxz * Ayz &
-                 + Gamxxy * Axz + Gamyxy * Ayz + Gamzxy * Azz) - chix*Ayz/chin1
+                 + Gamxxy * Axz + Gamyxy * Ayz + Gamzxy * Azz) - chix*Ayz*ONE/chin1
  gzzx = gzzx - (  Gamxxz * Axz + Gamyxz * Ayz + Gamzxz * Azz &
-                 + Gamxxz * Axz + Gamyxz * Ayz + Gamzxz * Azz) - chix*Azz/chin1
+                 + Gamxxz * Axz + Gamyxz * Ayz + Gamzxz * Azz) - chix*Azz*ONE/chin1
  gxxy = gxxy - (  Gamxxy * Axx + Gamyxy * Axy + Gamzxy * Axz &
-                 + Gamxxy * Axx + Gamyxy * Axy + Gamzxy * Axz) - chiy*Axx/chin1
+                 + Gamxxy * Axx + Gamyxy * Axy + Gamzxy * Axz) - chiy*Axx*ONE/chin1
  gxyy = gxyy - (  Gamxyy * Axx + Gamyyy * Axy + Gamzyy * Axz &
-                 + Gamxxy * Axy + Gamyxy * Ayy + Gamzxy * Ayz) - chiy*Axy/chin1
+                 + Gamxxy * Axy + Gamyxy * Ayy + Gamzxy * Ayz) - chiy*Axy*ONE/chin1
  gxzy = gxzy - (  Gamxyz * Axx + Gamyyz * Axy + Gamzyz * Axz &
-                 + Gamxxy * Axz + Gamyxy * Ayz + Gamzxy * Azz) - chiy*Axz/chin1
+                 + Gamxxy * Axz + Gamyxy * Ayz + Gamzxy * Azz) - chiy*Axz*ONE/chin1
  gyyy = gyyy - (  Gamxyy * Axy + Gamyyy * Ayy + Gamzyy * Ayz &
-                 + Gamxyy * Axy + Gamyyy * Ayy + Gamzyy * Ayz) - chiy*Ayy/chin1
+                 + Gamxyy * Axy + Gamyyy * Ayy + Gamzyy * Ayz) - chiy*Ayy*ONE/chin1
  gyzy = gyzy - (  Gamxyz * Axy + Gamyyz * Ayy + Gamzyz * Ayz &
-                 + Gamxyy * Axz + Gamyyy * Ayz + Gamzyy * Azz) - chiy*Ayz/chin1
+                 + Gamxyy * Axz + Gamyyy * Ayz + Gamzyy * Azz) - chiy*Ayz*ONE/chin1
  gzzy = gzzy - (  Gamxyz * Axz + Gamyyz * Ayz + Gamzyz * Azz &
-                 + Gamxyz * Axz + Gamyyz * Ayz + Gamzyz * Azz) - chiy*Azz/chin1
+                 + Gamxyz * Axz + Gamyyz * Ayz + Gamzyz * Azz) - chiy*Azz*ONE/chin1
  gxxz = gxxz - (  Gamxxz * Axx + Gamyxz * Axy + Gamzxz * Axz &
-                 + Gamxxz * Axx + Gamyxz * Axy + Gamzxz * Axz) - chiz*Axx/chin1
+                 + Gamxxz * Axx + Gamyxz * Axy + Gamzxz * Axz) - chiz*Axx*ONE/chin1
  gxyz = gxyz - (  Gamxyz * Axx + Gamyyz * Axy + Gamzyz * Axz &
-                 + Gamxxz * Axy + Gamyxz * Ayy + Gamzxz * Ayz) - chiz*Axy/chin1
+                 + Gamxxz * Axy + Gamyxz * Ayy + Gamzxz * Ayz) - chiz*Axy*ONE/chin1
  gxzz = gxzz - (  Gamxzz * Axx + Gamyzz * Axy + Gamzzz * Axz &
-                 + Gamxxz * Axz + Gamyxz * Ayz + Gamzxz * Azz) - chiz*Axz/chin1
+                 + Gamxxz * Axz + Gamyxz * Ayz + Gamzxz * Azz) - chiz*Axz*ONE/chin1
  gyyz = gyyz - (  Gamxyz * Axy + Gamyyz * Ayy + Gamzyz * Ayz &
-                 + Gamxyz * Axy + Gamyyz * Ayy + Gamzyz * Ayz) - chiz*Ayy/chin1
+                 + Gamxyz * Axy + Gamyyz * Ayy + Gamzyz * Ayz) - chiz*Ayy*ONE/chin1
  gyzz = gyzz - (  Gamxzz * Axy + Gamyzz * Ayy + Gamzzz * Ayz &
-                 + Gamxyz * Axz + Gamyyz * Ayz + Gamzyz * Azz) - chiz*Ayz/chin1
+                 + Gamxyz * Axz + Gamyyz * Ayz + Gamzyz * Azz) - chiz*Ayz*ONE/chin1
  gzzz = gzzz - (  Gamxzz * Axz + Gamyzz * Ayz + Gamzzz * Azz &
-                 + Gamxzz * Axz + Gamyzz * Ayz + Gamzzz * Azz) - chiz*Azz/chin1
+                 + Gamxzz * Axz + Gamyzz * Ayz + Gamzzz * Azz) - chiz*Azz*ONE/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
@@ -1939,6 +1939,309 @@
  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
@@ -2077,6 +2380,9 @@

  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
@@ -19,60 +19,48 @@

 !~~~~~~~> Local variable:
  
-  integer :: i,j,k
-  real*8 :: lgxx,lgyy,lgzz,ldetg
-  real*8 :: lgupxx,lgupxy,lgupxz,lgupyy,lgupyz,lgupzz
-  real*8 :: ltrA,lscale
+  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
  real*8, parameter :: F1o3 = 1.D0 / 3.D0, ONE = 1.D0, TWO = 2.D0

 !~~~~~~>

-  do k=1,ex(3)
-  do j=1,ex(2)
-  do i=1,ex(1)
+  gxx = dxx + ONE
+  gyy = dyy + ONE
+  gzz = dzz + ONE

-    lgxx = dxx(i,j,k) + ONE
-    lgyy = dyy(i,j,k) + ONE
-    lgzz = dzz(i,j,k) + ONE
+  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

-    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)
+  trA =         gupxx * Axx + gupyy * Ayy + gupzz * Azz &
+       + TWO * (gupxy * Axy + gupxz * Axz + gupyz * Ayz)

-    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
+  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

-    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))
+  detg = ONE / ( detg ** F1o3 ) 
  
-    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
+  gxx = gxx * detg
+  gxy = gxy * detg
+  gxz = gxz * detg
+  gyy = gyy * detg
+  gyz = gyz * detg
+  gzz = gzz * detg

-    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
+  dxx = gxx - ONE
+  dyy = gyy - ONE
+  dzz = gzz - ONE

  return

@@ -95,70 +83,50 @@

 !~~~~~~~> Local variable:
  
-  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, 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
  real*8, parameter :: F1o3 = 1.D0 / 3.D0, ONE = 1.D0, TWO = 2.D0

 !~~~~~~>

-  do k=1,ex(3)
-  do j=1,ex(2)
-  do i=1,ex(1)
+  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

-! 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)
+  gupzz = ONE / ( gupzz ** F1o3 ) 
  
-    lscale =  lgxx * lgyy * lgzz + lgxy * lgyz * lgxz &
-            + lgxz * lgxy * lgyz - lgxz * lgyy * lgxz &
-            - lgxy * lgxy * lgzz - lgxx * lgyz * lgyz
+  gxx = gxx * gupzz
+  gxy = gxy * gupzz
+  gxz = gxz * gupzz
+  gyy = gyy * gupzz
+  gyz = gyz * gupzz
+  gzz = gzz * gupzz

-    lscale = ONE / ( lscale ** F1o3 )
+  dxx = gxx - ONE
+  dyy = gyy - ONE
+  dzz = gzz - ONE
+! for A  

-    lgxx = lgxx * lscale
-    lgxy = lgxy * lscale
-    lgxz = lgxz * lscale
-    lgyy = lgyy * lscale
-    lgyz = lgyz * lscale
-    lgzz = lgzz * lscale
+  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 )

-    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
+  trA =         gupxx * Axx + gupyy * Ayy + gupzz * Azz &
+       + TWO * (gupxy * Axy + gupxz * Axz + gupyz * Ayz)

-! 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
+  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

  return

--- a/AMSS_NCKU_source/fmisc.f90
+++ b/AMSS_NCKU_source/fmisc.f90
@@ -324,6 +324,7 @@ 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)
@@ -349,6 +350,7 @@ 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)
@@ -377,6 +379,7 @@ 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)
@@ -883,6 +886,7 @@ 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)
@@ -908,6 +912,7 @@ 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)
@@ -936,6 +941,7 @@ 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)
@@ -1112,65 +1118,64 @@ end subroutine d2dump
 ! Lagrangian polynomial interpolation
 !------------------------------------------------------------------------------

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

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

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

-  c = ya
-  d = ya
-  ho = xa - x
+  integer :: m,n,ns
+  real*8, dimension(ordn) :: c,d,den,ho
+  real*8 :: dif,dift

-  ns = 1
-  dif = abs(x - xa(1))
+!~~~~~~>

-  do i = 2, ordn
-    dift = abs(x - xa(i))
-    if (dift < dif) then
-      ns = i
-      dif = dift
+  n=ordn
+  m=ordn
+
+  c=ya
+  d=ya
+  ho=xa-x
+
+  ns=1
+  dif=abs(x-xa(1))
+  do m=1,n
+   dift=abs(x-xa(m))
+   if(dift < dif) then
+    ns=m
+    dif=dift
   end if
  end do

-  y = ya(ns)
-  ns = ns - 1
-
-  do m = 1, ordn - 1
-    n_m = ordn - m
-    do i = 1, n_m
-      hp = ho(i)
-      h  = ho(i+m)
-      den_val = hp - h
-
-      if (den_val == 0.0d0) then
+  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
      write(*,*) 'failure in polint for point',x
      write(*,*) 'with input points: ',xa
      stop
-      end if
-
-      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)
+    endif
+    den(1:n-m)=(c(2:n-m+1)-d(1:n-m))/den(1:n-m)
+    d(1:n-m)=ho(1+m:n)*den(1:n-m)
+    c(1:n-m)=ho(1:n-m)*den(1:n-m)
+    if (2*ns < n-m) then
+      dy=c(ns+1)
    else
-      dy = d(ns)
-      ns = ns - 1
+      dy=d(ns)
+      ns=ns-1
    end if
-    y = y + dy
+    y=y+dy
  end do

  return
+
  end subroutine polint
 !------------------------------------------------------------------------------
 !
@@ -1178,37 +1183,35 @@ 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

-#ifdef POLINT_LEGACY_ORDER
+!~~~~~~> Other parameters:
+
  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
+
+  call polint(x1a,ymtmp,x1,y,dy,ordn)

  return
+
  end subroutine polin2
 !------------------------------------------------------------------------------
 !
@@ -1216,15 +1219,18 @@ 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

-#ifdef POLINT_LEGACY_ORDER
+!~~~~~~> Other parameters:
+
  integer  :: i,j,m,n
  real*8, dimension(ordn,ordn) :: yatmp
  real*8, dimension(ordn) :: ymtmp
@@ -1233,33 +1239,24 @@ 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
+
+  call polint(x1a,ymtmp,x1,y,dy,ordn)

  return
+
  end subroutine polin3
 !--------------------------------------------------------------------------------------
 ! calculate L2norm
--- a/AMSS_NCKU_source/lopsidediff.f90
+++ b/AMSS_NCKU_source/lopsidediff.f90
@@ -487,201 +487,6 @@ 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
@@ -16,12 +16,6 @@ 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\
@@ -102,7 +96,7 @@ ABEGPU: $(C++FILES_GPU) $(F90FILES) $(F77FILES) $(AHFDOBJS) $(CUDAFILES)
 	$(CLINKER) $(CXXAPPFLAGS) -o $@ $(C++FILES_GPU) $(F90FILES) $(F77FILES) $(AHFDOBJS) $(CUDAFILES) $(LDLIBS)

 TwoPunctureABE: $(TwoPunctureFILES)
-	$(CLINKER) $(CXXAPPFLAGS) -qopenmp -o $@ $(TwoPunctureFILES) $(LDLIBS)
+	$(CLINKER) $(CXXAPPFLAGS) -o $@ $(TwoPunctureFILES) $(LDLIBS)

 clean:
 	rm *.o ABE ABEGPU TwoPunctureABE make.log -f
--- a/AMSS_NCKU_source/makefile.inc
+++ b/AMSS_NCKU_source/makefile.inc
@@ -15,10 +15,11 @@ LDLIBS  = -L${MKLROOT}/lib -lmkl_intel_lp64 -lmkl_sequential -lmkl_core -lifcore
 ## -xHost: Optimize for the host CPU architecture (Intel/AMD compatible)
 ## -fp-model fast=2: Aggressive floating-point optimizations
 ## -fma: Enable fused multiply-add instructions
-CXXAPPFLAGS  = -O3 -xHost -fp-model fast=2 -fma -ipo \
+## Note: OpenMP has been disabled (-qopenmp removed) due to performance issues
+CXXAPPFLAGS  = -O3 -xHost -fp-model fast=2 -fma \
               -Dfortran3 -Dnewc -I${MKLROOT}/include
-f90appflags  = -O3 -xHost -fp-model fast=2 -fma -ipo \
-               -align array64byte -fpp -I${MKLROOT}/include
+f90appflags  = -O3 -xHost -fp-model fast=2 -fma \
+               -fpp -I${MKLROOT}/include
 f90          = ifx
 f77          = ifx
 CXX          = icpx
--- a/generate_macrodef.py
+++ b/generate_macrodef.py
@@ -392,6 +392,17 @@ 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

@@ -514,6 +525,9 @@ 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
@@ -36,6 +36,7 @@ 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
@@ -10,17 +10,18 @@

 import AMSS_NCKU_Input as input_data
 import subprocess
-import time
+
 ## CPU core binding configuration using taskset
 ## taskset ensures all child processes inherit the CPU affinity mask
 ## This forces make and all compiler processes to use only nohz_full cores (4-55, 60-111)
 ## Format: taskset -c 4-55,60-111 ensures processes only run on these cores
-NUMACTL_CPU_BIND = "taskset -c 0-111"
+#NUMACTL_CPU_BIND = "taskset -c 4-55,60-111"
+NUMACTL_CPU_BIND = ""

 ## 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 = 104
+BUILD_JOBS = 14


 ##################################################################
@@ -152,7 +153,7 @@ 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(                                                          )
@@ -179,9 +180,7 @@ 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
@@ -1,29 +0,0 @@
-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
@@ -11,8 +11,6 @@
 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
@@ -8,23 +8,16 @@
 ##
 #################################################

-## 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
+

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

@@ -199,19 +192,3 @@ 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
@@ -8,8 +8,6 @@
 #################################################

 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
@@ -17,9 +15,6 @@ 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

@@ -55,40 +50,10 @@ def generate_binary_data_plot( binary_outdir, figure_outdir ):
        file_list.append(x)
        print(x)

-    ## 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  = []
+    ## Plot each file in the list
    for filename in file_list:
        print(filename)
-        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 )
+        plot_binary_data.plot_binary_data(filename, binary_outdir, figure_outdir)

    print(                        )
    print( " Binary Data Plot Has been Finished " )