修改block划分，对负载高的rank所在block进行划分，添加到空rank，空rank是平移得到的

Rank [  20]: Calc 0.000003 s

Rank [  35]: Calc 0.000003 s

Rank [  10]: Calc 0.000010 s

Rank [  17]: Calc 0.000005 s

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

Rank [  27]: Calc 0.331978 s

Rank [  35]: Calc 0.242219 s

Rank [  28]: Calc 0.242132 s

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

AMSS_NCKU_ABEtest.py

@@ -1,447 +0,0 @@
-
-##################################################################
-##
-## AMSS-NCKU ABE Test Program (Skip TwoPuncture if data exists)
-## Modified from AMSS_NCKU_Program.py
-## Author: Xiaoqu
-## Modified: 2026/02/01
-##
-##################################################################
-
-
-##################################################################
-
-## Print program introduction
-
-import print_information
-
-print_information.print_program_introduction()
-
-##################################################################
-
-import AMSS_NCKU_Input as input_data
-
-##################################################################
-
-## Create directories to store program run data
-
-import os
-import shutil
-import sys
-import time
-
-## Set the output directory according to the input file
-File_directory = os.path.join(input_data.File_directory)
-
-## Check if output directory exists and if TwoPuncture data is available
-#skip_twopuncture = False
-skip_twopuncture = True
-output_directory = os.path.join(File_directory, "AMSS_NCKU_output")
-binary_results_directory = os.path.join(output_directory, input_data.Output_directory)
-
-if os.path.exists(File_directory):
-    print( " Output directory already exists." )
-    print()
-    '''
-    # Check if TwoPuncture initial data files exist
-    if (input_data.Initial_Data_Method == "Ansorg-TwoPuncture"):
-        twopuncture_output = os.path.join(output_directory, "TwoPunctureABE")
-        input_par = os.path.join(output_directory, "input.par")
-
-        if os.path.exists(twopuncture_output) and os.path.exists(input_par):
-            print( " Found existing TwoPuncture initial data." )
-            print( " Do you want to skip TwoPuncture phase and reuse existing data?" )
-            print( " Input 'skip' to skip TwoPuncture and start ABE directly" )
-            print( " Input 'regenerate' to regenerate everything from scratch" )
-            print()
-            
-            while True:
-                try:
-                    inputvalue = input()
-                    if ( inputvalue == "skip" ):
-                        print( " Skipping TwoPuncture phase, will reuse existing initial data." )
-                        print()
-                        skip_twopuncture = True
-                        break
-                    elif ( inputvalue == "regenerate" ):
-                        print( " Regenerating everything from scratch." )
-                        print()
-                        skip_twopuncture = False
-                        break
-                    else:
-                        print( " Please input 'skip' or 'regenerate'." )
-                except ValueError:
-                    print( " Please input 'skip' or 'regenerate'." )
-            
-        else:
-            print( " TwoPuncture initial data not found, will regenerate everything." )
-            print()
-'''
-    # If not skipping, remove and recreate directory
-    if not skip_twopuncture:
-        shutil.rmtree(File_directory, ignore_errors=True)
-        os.mkdir(File_directory)
-        os.mkdir(output_directory)
-        os.mkdir(binary_results_directory)
-        figure_directory = os.path.join(File_directory, "figure")
-        os.mkdir(figure_directory)
-        shutil.copy("AMSS_NCKU_Input.py", File_directory)
-        print( " Output directory has been regenerated." )
-        print()
-else:
-    # Create fresh directory structure
-    os.mkdir(File_directory)
-    shutil.copy("AMSS_NCKU_Input.py", File_directory)
-    os.mkdir(output_directory)
-    os.mkdir(binary_results_directory)
-    figure_directory = os.path.join(File_directory, "figure")
-    os.mkdir(figure_directory)
-    print( " Output directory has been generated." )
-    print()
-
-# Ensure figure directory exists
-figure_directory = os.path.join(File_directory, "figure")
-if not os.path.exists(figure_directory):
-    os.mkdir(figure_directory)
-
-##################################################################
-
-## Output related parameter information
-
-import setup
-
-## Print and save input parameter information
-setup.print_input_data( File_directory )
-
-if not skip_twopuncture:
-    setup.generate_AMSSNCKU_input()
-
-setup.print_puncture_information()
-
-
-##################################################################
-
-## Generate AMSS-NCKU program input files based on the configured parameters
-
-if not skip_twopuncture:
-    print()
-    print( " Generating the AMSS-NCKU input parfile for the ABE executable." )
-    print()
-
-    ## Generate cgh-related input files from the grid information
-
-    import numerical_grid
-
-    numerical_grid.append_AMSSNCKU_cgh_input()
-
-    print()
-    print( " The input parfile for AMSS-NCKU C++ executable file ABE has been generated." )
-    print( " However, the input relevant to TwoPuncture need to be appended later." )
-    print()
-
-
-##################################################################
-
-## Plot the initial grid configuration
-
-if not skip_twopuncture:
-    print()
-    print( " Schematically plot the numerical grid structure." )
-    print()
-
-    import numerical_grid
-    numerical_grid.plot_initial_grid()
-
-
-##################################################################
-
-## Generate AMSS-NCKU macro files according to the numerical scheme and parameters
-
-if not skip_twopuncture:
-    print()
-    print( " Automatically generating the macro file for AMSS-NCKU C++ executable file ABE " )
-    print( " (Based on the finite-difference numerical scheme) " )
-    print()
-
-    import generate_macrodef
-
-    generate_macrodef.generate_macrodef_h()
-    print( " AMSS-NCKU macro file macrodef.h has been generated. " )
-
-    generate_macrodef.generate_macrodef_fh()
-    print( " AMSS-NCKU macro file macrodef.fh has been generated. " )
-
-
-##################################################################
-
-# Compile the AMSS-NCKU program according to user requirements
-# NOTE: ABE compilation is always performed, even when skipping TwoPuncture
-
-print()
-print( " Preparing to compile and run the AMSS-NCKU code as requested " )
-print( " Compiling the AMSS-NCKU code based on the generated macro files " )
-print()
-
-AMSS_NCKU_source_path = "AMSS_NCKU_source"
-AMSS_NCKU_source_copy = os.path.join(File_directory, "AMSS_NCKU_source_copy")
-
-## If AMSS_NCKU source folder is missing, create it and prompt the user
-if not os.path.exists(AMSS_NCKU_source_path):
-    os.makedirs(AMSS_NCKU_source_path)
-    print( " The AMSS-NCKU source files are incomplete; copy all source files into ./AMSS_NCKU_source. " )
-    print( " Press Enter to continue. " )
-    inputvalue = input()
-
-# Copy AMSS-NCKU source files to prepare for compilation
-# If skipping TwoPuncture and source_copy already exists, remove it first
-if skip_twopuncture and os.path.exists(AMSS_NCKU_source_copy):
-    shutil.rmtree(AMSS_NCKU_source_copy)
-
-shutil.copytree(AMSS_NCKU_source_path, AMSS_NCKU_source_copy)
-
-# Copy the generated macro files into the AMSS_NCKU source folder
-if not skip_twopuncture:
-    macrodef_h_path  = os.path.join(File_directory, "macrodef.h")
-    macrodef_fh_path = os.path.join(File_directory, "macrodef.fh")
-else:
-    # When skipping TwoPuncture, use existing macro files from previous run
-    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)
-
-# Compile related programs
-import makefile_and_run
-
-## Change working directory to the target source copy
-os.chdir(AMSS_NCKU_source_copy)
-
-## Build the main AMSS-NCKU executable (ABE or ABEGPU)
-makefile_and_run.makefile_ABE()
-
-## If the initial-data method is Ansorg-TwoPuncture, build the TwoPunctureABE executable
-## Only build TwoPunctureABE if not skipping TwoPuncture phase
-if (input_data.Initial_Data_Method == "Ansorg-TwoPuncture" ) and not skip_twopuncture:
-    makefile_and_run.makefile_TwoPunctureABE()
-
-## Change current working directory back up two levels
-os.chdir('..')
-os.chdir('..')
-
-print()
-
-##################################################################
-
-## Copy the AMSS-NCKU executable (ABE/ABEGPU) to the run directory
-
-if (input_data.GPU_Calculation == "no"):
-    ABE_file = os.path.join(AMSS_NCKU_source_copy, "ABE")
-elif (input_data.GPU_Calculation == "yes"):
-    ABE_file = os.path.join(AMSS_NCKU_source_copy, "ABEGPU")
-
-if not os.path.exists( ABE_file ):
-    print()
-    print( " Lack of AMSS-NCKU executable file ABE/ABEGPU; recompile AMSS_NCKU_source manually. " )
-    print( " When recompilation is finished, press Enter to continue. " )
-    inputvalue = input()
-
-## Copy the executable ABE (or ABEGPU) into the run directory
-shutil.copy2(ABE_file, output_directory)
-
-## If the initial-data method is TwoPuncture, copy the TwoPunctureABE executable to the run directory
-## Only copy TwoPunctureABE if not skipping TwoPuncture phase
-if (input_data.Initial_Data_Method == "Ansorg-TwoPuncture" ) and not skip_twopuncture:
-    TwoPuncture_file = os.path.join(AMSS_NCKU_source_copy, "TwoPunctureABE")
-
-    if not os.path.exists( TwoPuncture_file ):
-        print()
-        print( " Lack of AMSS-NCKU executable file TwoPunctureABE; recompile TwoPunctureABE in AMSS_NCKU_source. " )
-        print( " When recompilation is finished, press Enter to continue. " )
-        inputvalue = input()
-
-    ## Copy the TwoPunctureABE executable into the run directory
-    shutil.copy2(TwoPuncture_file, output_directory)
-
-##################################################################
-
-## If the initial-data method is TwoPuncture, generate the TwoPuncture input files
-
-if (input_data.Initial_Data_Method == "Ansorg-TwoPuncture" ) and not skip_twopuncture:
-
-    print()
-    print( " Initial data is chosen as Ansorg-TwoPuncture" )
-    print()
-    
-    print()
-    print( " Automatically generating the input parfile for the TwoPunctureABE executable " )
-    print()
-    
-    import generate_TwoPuncture_input
-    
-    generate_TwoPuncture_input.generate_AMSSNCKU_TwoPuncture_input()
-    
-    print()
-    print( " The input parfile for the TwoPunctureABE executable has been generated. " )
-    print()
-    
-    ## Generated AMSS-NCKU TwoPuncture input filename
-    AMSS_NCKU_TwoPuncture_inputfile      = 'AMSS-NCKU-TwoPuncture.input'
-    AMSS_NCKU_TwoPuncture_inputfile_path = os.path.join( File_directory, AMSS_NCKU_TwoPuncture_inputfile )
- 
-    ## Copy and rename the file
-    shutil.copy2( AMSS_NCKU_TwoPuncture_inputfile_path, os.path.join(output_directory, 'TwoPunctureinput.par') )
-    
-    ## Run TwoPuncture to generate initial-data files
-    
-    start_time = time.time()  # Record start time
-
-    print()
-    print()
-    
-    ## Change to the output (run) directory
-    os.chdir(output_directory)
-
-    ## Run the TwoPuncture executable
-    import makefile_and_run
-    makefile_and_run.run_TwoPunctureABE()
-    
-    ## Change current working directory back up two levels
-    os.chdir('..')
-    os.chdir('..')
-
-elif (input_data.Initial_Data_Method == "Ansorg-TwoPuncture" ) and skip_twopuncture:
-    print()
-    print( " Skipping TwoPuncture execution, using existing initial data." )
-    print()
-    start_time = time.time()  # Record start time for ABE only
-else:
-    start_time = time.time()  # Record start time
-    
-##################################################################
-    
-## Update puncture data based on TwoPuncture run results
-
-if not skip_twopuncture:
-    import renew_puncture_parameter
-    renew_puncture_parameter.append_AMSSNCKU_BSSN_input(File_directory, output_directory)
-
-    ## Generated AMSS-NCKU input filename
-    AMSS_NCKU_inputfile      = 'AMSS-NCKU.input'
-    AMSS_NCKU_inputfile_path = os.path.join(File_directory, AMSS_NCKU_inputfile)
- 
-    ## Copy and rename the file
-    shutil.copy2( AMSS_NCKU_inputfile_path, os.path.join(output_directory, 'input.par') )
-
-    print()
-    print( " Successfully copy all AMSS-NCKU input parfile to target dictionary. " )
-    print()
-else:
-    print()
-    print( " Using existing input.par file from previous run." )
-    print()
-
-##################################################################
-
-## Launch the AMSS-NCKU program
-
-print()
-print()
-
-## Change to the run directory
-os.chdir( output_directory )
-
-import makefile_and_run
-makefile_and_run.run_ABE()
-
-## Change current working directory back up two levels
-os.chdir('..')
-os.chdir('..')
-
-end_time = time.time()
-elapsed_time = end_time - start_time
-
-##################################################################
-
-## Copy some basic input and log files out to facilitate debugging
-
-## Path to the file that stores calculation settings
-AMSS_NCKU_error_file_path = os.path.join(binary_results_directory, "setting.par")
-## Copy and rename the file for easier inspection
-shutil.copy( AMSS_NCKU_error_file_path, os.path.join(output_directory, "AMSSNCKU_setting_parameter") )
-
-## Path to the error log file
-AMSS_NCKU_error_file_path = os.path.join(binary_results_directory, "Error.log")
-## Copy and rename the error log
-shutil.copy( AMSS_NCKU_error_file_path, os.path.join(output_directory, "Error.log") )
-
-## Primary program outputs
-AMSS_NCKU_BH_data         = os.path.join(binary_results_directory, "bssn_BH.dat"        )
-AMSS_NCKU_ADM_data        = os.path.join(binary_results_directory, "bssn_ADMQs.dat"     )
-AMSS_NCKU_psi4_data       = os.path.join(binary_results_directory, "bssn_psi4.dat"      )
-AMSS_NCKU_constraint_data = os.path.join(binary_results_directory, "bssn_constraint.dat")
-## copy and rename the file
-shutil.copy( AMSS_NCKU_BH_data,         os.path.join(output_directory, "bssn_BH.dat"        ) )
-shutil.copy( AMSS_NCKU_ADM_data,        os.path.join(output_directory, "bssn_ADMQs.dat"     ) )
-shutil.copy( AMSS_NCKU_psi4_data,       os.path.join(output_directory, "bssn_psi4.dat"      ) )
-shutil.copy( AMSS_NCKU_constraint_data, os.path.join(output_directory, "bssn_constraint.dat") )
-
-## Additional program outputs
-if (input_data.Equation_Class == "BSSN-EM"):
-    AMSS_NCKU_phi1_data = os.path.join(binary_results_directory, "bssn_phi1.dat" )
-    AMSS_NCKU_phi2_data = os.path.join(binary_results_directory, "bssn_phi2.dat" )
-    shutil.copy( AMSS_NCKU_phi1_data, os.path.join(output_directory, "bssn_phi1.dat" ) )
-    shutil.copy( AMSS_NCKU_phi2_data, os.path.join(output_directory, "bssn_phi2.dat" ) )
-elif (input_data.Equation_Class == "BSSN-EScalar"):
-    AMSS_NCKU_maxs_data = os.path.join(binary_results_directory, "bssn_maxs.dat" )
-    shutil.copy( AMSS_NCKU_maxs_data, os.path.join(output_directory, "bssn_maxs.dat" ) )
-
-##################################################################
-
-## Plot the AMSS-NCKU program results
-
-print()
-print( " Plotting the txt and binary results data from the AMSS-NCKU simulation " )
-print()
-
-
-import plot_xiaoqu
-import plot_GW_strain_amplitude_xiaoqu
-
-## Plot black hole trajectory
-plot_xiaoqu.generate_puncture_orbit_plot(   binary_results_directory, figure_directory )
-plot_xiaoqu.generate_puncture_orbit_plot3D( binary_results_directory, figure_directory )
-
-## Plot black hole separation vs. time
-plot_xiaoqu.generate_puncture_distence_plot( binary_results_directory, figure_directory )
-
-## Plot gravitational waveforms (psi4 and strain amplitude)
-for i in range(input_data.Detector_Number):
-    plot_xiaoqu.generate_gravitational_wave_psi4_plot( binary_results_directory, figure_directory, i )
-    plot_GW_strain_amplitude_xiaoqu.generate_gravitational_wave_amplitude_plot( binary_results_directory, figure_directory, i )
-
-## Plot ADM mass evolution
-for i in range(input_data.Detector_Number):
-    plot_xiaoqu.generate_ADMmass_plot( binary_results_directory, figure_directory, i )
-
-## Plot Hamiltonian constraint violation over time
-for i in range(input_data.grid_level):
-    plot_xiaoqu.generate_constraint_check_plot( binary_results_directory, figure_directory, i )
-
-## Plot stored binary data
-plot_xiaoqu.generate_binary_data_plot( binary_results_directory, figure_directory )
-
-print()
-print( f" This Program Cost = {elapsed_time} Seconds " )
-print()
-
-
-##################################################################
-
-print()
-print( " The AMSS-NCKU-Python simulation is successfully finished, thanks for using !!! " )
-print()
-
-##################################################################
-
-

AMSS_NCKU_Program.py

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

AMSS_NCKU_source/MPatch.h

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

AMSS_NCKU_source/NullShellPatch.h

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

AMSS_NCKU_source/Parallel.h

@@ -1,167 +1,235 @@
-
+
-#ifndef PARALLEL_H
+#ifndef PARALLEL_H
-#define PARALLEL_H
+#define PARALLEL_H
-
+
-#include <iostream>
+#include <iostream>
-#include <iomanip>
+#include <iomanip>
-#include <fstream>
+#include <fstream>
-#include <cstdlib>
+#include <cstdlib>
-#include <cstdio>
+#include <cstdio>
-#include <string>
+#include <string>
-#include <cmath>
+#include <cmath>
-#include <new>
+#include <new>
-using namespace std;
+using namespace std;
-
+#include <memory.h>
-#include "Parallel_bam.h"
+#include "Parallel_bam.h"
-#include "var.h"
+#include "var.h"
-#include "MPatch.h"
+#include "MPatch.h"
-#include "Block.h"
+#include "Block.h"
-#include "MyList.h"
+#include "MyList.h"
-#include "macrodef.h" //need dim; ghost_width; CONTRACT
+#include "macrodef.h" //need dim; ghost_width; CONTRACT
-namespace Parallel
+namespace Parallel
-{
+{
-  struct gridseg
+  struct gridseg
-  {
+  {
-    double llb[dim];
+    double llb[dim];
-    double uub[dim];
+    double uub[dim];
-    int shape[dim];
+    int shape[dim];
-    double illb[dim], iuub[dim]; // only use for OutBdLow2Hi
+    double illb[dim], iuub[dim]; // only use for OutBdLow2Hi
-    Block *Bg;
+    Block *Bg;
-  };
+  };
-  int partition1(int &nx, int split_size, int min_width, int cpusize, int shape);    // special for 1 diemnsion
+  int partition1(int &nx, int split_size, int min_width, int cpusize, int shape);    // special for 1 diemnsion
-  int partition2(int *nxy, int split_size, int *min_width, int cpusize, int *shape); // special for 2 diemnsions
+  int partition2(int *nxy, int split_size, int *min_width, int cpusize, int *shape); // special for 2 diemnsions
-  int partition3(int *nxyz, int split_size, int *min_width, int cpusize, int *shape);
+  int partition3(int *nxyz, int split_size, int *min_width, int cpusize, int *shape);
-  MyList<Block> *distribute(MyList<Patch> *PatchLIST, int cpusize, int ingfsi, int fngfs, bool periodic, int nodes = 0); // produce corresponding Blocks
+  MyList<Block> *distribute(MyList<Patch> *PatchLIST, int cpusize, int ingfsi, int fngfs, bool periodic, int nodes = 0); // produce corresponding Blocks
-  void KillBlocks(MyList<Patch> *PatchLIST);
+  MyList<Block> *distribute_hard(MyList<Patch> *PatchLIST, int cpusize, int ingfsi, int fngfs, bool periodic, int nodes = 0); // produce corresponding Blocks
-
+  Block* splitHotspotBlock(MyList<Block>* &BlL, int _dim, 
-  void setfunction(MyList<Block> *BlL, var *vn, double func(double x, double y, double z));
+                                 int ib0_orig, int ib3_orig, 
-  void setfunction(int rank, MyList<Block> *BlL, var *vn, double func(double x, double y, double z));
+                                 int jb1_orig, int jb4_orig, 
-  void writefile(double time, int nx, int ny, int nz, double xmin, double xmax, double ymin, double ymax,
+                                 int kb2_orig, int kb5_orig, 
-                 double zmin, double zmax, char *filename, double *data_out);
+                                 Patch* PP, int r_left, int r_right, 
-  void writefile(double time, int nx, int ny, double xmin, double xmax, double ymin, double ymax,
+                                 int ingfsi, int fngfsi, bool periodic,
-                 char *filename, double *datain);
+                                 Block* &split_first_block, Block* &split_last_block);
-  void getarrayindex(int DIM, int *shape, int *index, int n);
+  Block* createMappedBlock(MyList<Block>* &BlL, int _dim, int* shape, double* bbox,
-  int getarraylocation(int DIM, int *shape, int *index);
+                        int block_id, int ingfsi, int fngfsi, int lev); 
-  void copy(int DIM, double *llbout, double *uubout, int *Dshape, double *DD, double *llbin, double *uubin,
+  void KillBlocks(MyList<Patch> *PatchLIST);
-            int *shape, double *datain, double *llb, double *uub);
+
-  void Dump_CPU_Data(MyList<Block> *BlL, MyList<var> *DumpList, char *tag, double time, double dT);
+  void setfunction(MyList<Block> *BlL, var *vn, double func(double x, double y, double z));
-  void Dump_Data(MyList<Patch> *PL, MyList<var> *DumpList, char *tag, double time, double dT);
+  void setfunction(int rank, MyList<Block> *BlL, var *vn, double func(double x, double y, double z));
-  void Dump_Data(Patch *PP, MyList<var> *DumpList, char *tag, double time, double dT, int grd);
+  void writefile(double time, int nx, int ny, int nz, double xmin, double xmax, double ymin, double ymax,
-  double *Collect_Data(Patch *PP, var *VP);
+                 double zmin, double zmax, char *filename, double *data_out);
-  void d2Dump_Data(MyList<Patch> *PL, MyList<var> *DumpList, char *tag, double time, double dT);
+  void writefile(double time, int nx, int ny, double xmin, double xmax, double ymin, double ymax,
-  void d2Dump_Data(Patch *PP, MyList<var> *DumpList, char *tag, double time, double dT, int grd);
+                 char *filename, double *datain);
-  void Dump_Data0(Patch *PP, MyList<var> *DumpList, char *tag, double time, double dT);
+  void getarrayindex(int DIM, int *shape, int *index, int n);
-  double global_interp(int DIM, int *ext, double **CoX, double *datain,
+  int getarraylocation(int DIM, int *shape, int *index);
-                       double *poX, int ordn, double *SoA, int Symmetry);
+  void copy(int DIM, double *llbout, double *uubout, int *Dshape, double *DD, double *llbin, double *uubin,
-  double global_interp(int DIM, int *ext, double **CoX, double *datain,
+            int *shape, double *datain, double *llb, double *uub);
-                       double *poX, int ordn);
+  void Dump_CPU_Data(MyList<Block> *BlL, MyList<var> *DumpList, char *tag, double time, double dT);
-  double Lagrangian_Int(double x, int npts, double *xpts, double *funcvals);
+  void Dump_Data(MyList<Patch> *PL, MyList<var> *DumpList, char *tag, double time, double dT);
-  double LagrangePoly(double x, int pt, int npts, double *xpts);
+  void Dump_Data(Patch *PP, MyList<var> *DumpList, char *tag, double time, double dT, int grd);
-  MyList<gridseg> *build_complete_gsl(Patch *Pat);
+  double *Collect_Data(Patch *PP, var *VP);
-  MyList<gridseg> *build_complete_gsl(MyList<Patch> *PatL);
+  void d2Dump_Data(MyList<Patch> *PL, MyList<var> *DumpList, char *tag, double time, double dT);
-  MyList<gridseg> *build_complete_gsl_virtual(MyList<Patch> *PatL);
+  void d2Dump_Data(Patch *PP, MyList<var> *DumpList, char *tag, double time, double dT, int grd);
-  MyList<gridseg> *build_complete_gsl_virtual2(MyList<Patch> *PatL);        // - buffer
+  void Dump_Data0(Patch *PP, MyList<var> *DumpList, char *tag, double time, double dT);
-  MyList<gridseg> *build_owned_gsl0(Patch *Pat, int rank_in);               // - ghost without extension, special for Sync usage
+  double global_interp(int DIM, int *ext, double **CoX, double *datain,
-  MyList<gridseg> *build_owned_gsl1(Patch *Pat, int rank_in);               // - ghost, similar to build_owned_gsl0 but extend one point on left side for vertex grid
+                       double *poX, int ordn, double *SoA, int Symmetry);
-  MyList<gridseg> *build_owned_gsl2(Patch *Pat, int rank_in);               // - buffer - ghost
+  double global_interp(int DIM, int *ext, double **CoX, double *datain,
-  MyList<gridseg> *build_owned_gsl3(Patch *Pat, int rank_in, int Symmetry); // - ghost - BD ghost
+                       double *poX, int ordn);
-  MyList<gridseg> *build_owned_gsl4(Patch *Pat, int rank_in, int Symmetry); // - buffer - ghost - BD ghost
+  double Lagrangian_Int(double x, int npts, double *xpts, double *funcvals);
-  MyList<gridseg> *build_owned_gsl5(Patch *Pat, int rank_in);               // similar to build_owned_gsl2 but no extension
+  double LagrangePoly(double x, int pt, int npts, double *xpts);
-  MyList<gridseg> *build_owned_gsl(MyList<Patch> *PatL, int rank_in, int type, int Symmetry);
+  MyList<gridseg> *build_complete_gsl(Patch *Pat);
-  void build_gstl(MyList<gridseg> *srci, MyList<gridseg> *dsti, MyList<gridseg> **out_src, MyList<gridseg> **out_dst);
+  MyList<gridseg> *build_complete_gsl(MyList<Patch> *PatL);
-  int data_packer(double *data, MyList<gridseg> *src, MyList<gridseg> *dst, int rank_in, int dir,
+  MyList<gridseg> *build_complete_gsl_virtual(MyList<Patch> *PatL);
-                  MyList<var> *VarLists, MyList<var> *VarListd, int Symmetry);
+  MyList<gridseg> *build_complete_gsl_virtual2(MyList<Patch> *PatL);        // - buffer
-  void transfer(MyList<gridseg> **src, MyList<gridseg> **dst,
+  MyList<gridseg> *build_owned_gsl0(Patch *Pat, int rank_in);               // - ghost without extension, special for Sync usage
-                MyList<var> *VarList1 /* source */, MyList<var> *VarList2 /*target */,
+  MyList<gridseg> *build_owned_gsl1(Patch *Pat, int rank_in);               // - ghost, similar to build_owned_gsl0 but extend one point on left side for vertex grid
-                int Symmetry);
+  MyList<gridseg> *build_owned_gsl2(Patch *Pat, int rank_in);               // - buffer - ghost
-  int data_packermix(double *data, MyList<gridseg> *src, MyList<gridseg> *dst, int rank_in, int dir,
+  MyList<gridseg> *build_owned_gsl3(Patch *Pat, int rank_in, int Symmetry); // - ghost - BD ghost
-                     MyList<var> *VarLists, MyList<var> *VarListd, int Symmetry);
+  MyList<gridseg> *build_owned_gsl4(Patch *Pat, int rank_in, int Symmetry); // - buffer - ghost - BD ghost
-  void transfermix(MyList<gridseg> **src, MyList<gridseg> **dst,
+  MyList<gridseg> *build_owned_gsl5(Patch *Pat, int rank_in);               // similar to build_owned_gsl2 but no extension
-                   MyList<var> *VarList1 /* source */, MyList<var> *VarList2 /*target */,
+  MyList<gridseg> *build_owned_gsl(MyList<Patch> *PatL, int rank_in, int type, int Symmetry);
-                   int Symmetry);
+  void build_gstl(MyList<gridseg> *srci, MyList<gridseg> *dsti, MyList<gridseg> **out_src, MyList<gridseg> **out_dst);
-  void Sync(Patch *Pat, MyList<var> *VarList, int Symmetry);
+  int data_packer(double *data, MyList<gridseg> *src, MyList<gridseg> *dst, int rank_in, int dir,
-  void Sync(MyList<Patch> *PatL, MyList<var> *VarList, int Symmetry);
+                  MyList<var> *VarLists, MyList<var> *VarListd, int Symmetry);
-  void OutBdLow2Hi(Patch *Patc, Patch *Patf,
+  void transfer(MyList<gridseg> **src, MyList<gridseg> **dst,
-                   MyList<var> *VarList1 /* source */, MyList<var> *VarList2 /* target */,
+                MyList<var> *VarList1 /* source */, MyList<var> *VarList2 /*target */,
-                   int Symmetry);
+                int Symmetry);
-  void OutBdLow2Hi(MyList<Patch> *PatcL, MyList<Patch> *PatfL,
+  int data_packermix(double *data, MyList<gridseg> *src, MyList<gridseg> *dst, int rank_in, int dir,
-                   MyList<var> *VarList1 /* source */, MyList<var> *VarList2 /* target */,
+                     MyList<var> *VarLists, MyList<var> *VarListd, int Symmetry);
-                   int Symmetry);
+  void transfermix(MyList<gridseg> **src, MyList<gridseg> **dst,
-  void OutBdLow2Himix(Patch *Patc, Patch *Patf,
+                   MyList<var> *VarList1 /* source */, MyList<var> *VarList2 /*target */,
-                      MyList<var> *VarList1 /* source */, MyList<var> *VarList2 /* target */,
+                   int Symmetry);
-                      int Symmetry);
+  void Sync(Patch *Pat, MyList<var> *VarList, int Symmetry);
-  void OutBdLow2Himix(MyList<Patch> *PatcL, MyList<Patch> *PatfL,
+  void Sync(MyList<Patch> *PatL, MyList<var> *VarList, int Symmetry);
-                      MyList<var> *VarList1 /* source */, MyList<var> *VarList2 /* target */,
+  void Sync_merged(MyList<Patch> *PatL, MyList<var> *VarList, int Symmetry);
-                      int Symmetry);
+
-  void Prolong(Patch *Patc, Patch *Patf,
+  struct SyncCache {
-               MyList<var> *VarList1 /* source */, MyList<var> *VarList2 /* target */,
+    bool valid;
-               int Symmetry);
+    int cpusize;
-  void Prolongint(Patch *Patc, Patch *Patf,
+    MyList<gridseg> **combined_src;
-                  MyList<var> *VarList1 /* source */, MyList<var> *VarList2 /* target */,
+    MyList<gridseg> **combined_dst;
-                  int Symmetry);
+    int *send_lengths;
-  void Restrict(MyList<Patch> *PatcL, MyList<Patch> *PatfL,
+    int *recv_lengths;
-                MyList<var> *VarList1 /* source */, MyList<var> *VarList2 /* target */,
+    double **send_bufs;
-                int Symmetry);
+    double **recv_bufs;
-  void Restrict_after(MyList<Patch> *PatcL, MyList<Patch> *PatfL,
+    int *send_buf_caps;
-                      MyList<var> *VarList1 /* source */, MyList<var> *VarList2 /* target */,
+    int *recv_buf_caps;
-                      int Symmetry); // for -ghost - BDghost
+    MPI_Request *reqs;
-  MyList<Parallel::gridseg> *build_PhysBD_gsl(Patch *Pat);
+    MPI_Status *stats;
-  MyList<Parallel::gridseg> *build_ghost_gsl(MyList<Patch> *PatL);
+    int max_reqs;
-  MyList<Parallel::gridseg> *build_ghost_gsl(Patch *Pat);
+    bool lengths_valid;
-  MyList<Parallel::gridseg> *build_buffer_gsl(Patch *Pat);
+    SyncCache();
-  MyList<Parallel::gridseg> *build_buffer_gsl(MyList<Patch> *PatL);
+    void invalidate();
-  MyList<Parallel::gridseg> *gsl_subtract(MyList<Parallel::gridseg> *A, MyList<Parallel::gridseg> *B);
+    void destroy();
-  MyList<Parallel::gridseg> *gs_subtract(MyList<Parallel::gridseg> *A, MyList<Parallel::gridseg> *B);
+  };
-  MyList<Parallel::gridseg> *gsl_and(MyList<Parallel::gridseg> *A, MyList<Parallel::gridseg> *B);
+
-  MyList<Parallel::gridseg> *gs_and(MyList<Parallel::gridseg> *A, MyList<Parallel::gridseg> *B);
+  void Sync_cached(MyList<Patch> *PatL, MyList<var> *VarList, int Symmetry, SyncCache &cache);
-  MyList<Parallel::gridseg> *clone_gsl(MyList<Parallel::gridseg> *p, bool first_only);
+  void transfer_cached(MyList<gridseg> **src, MyList<gridseg> **dst,
-  MyList<Parallel::gridseg> *build_bulk_gsl(Patch *Pat); // similar to build_owned_gsl0 but does not care rank issue
+                       MyList<var> *VarList1, MyList<var> *VarList2,
-  MyList<Parallel::gridseg> *build_bulk_gsl(Block *bp, Patch *Pat);
+                       int Symmetry, SyncCache &cache);
-  void build_PhysBD_gstl(Patch *Pat, MyList<Parallel::gridseg> *srci, MyList<Parallel::gridseg> *dsti,
+
-                         MyList<Parallel::gridseg> **out_src, MyList<Parallel::gridseg> **out_dst);
+  struct AsyncSyncState {
-  void PeriodicBD(Patch *Pat, MyList<var> *VarList, int Symmetry);
+    int req_no;
-  double L2Norm(Patch *Pat, var *vf);
+    bool active;
-  void checkgsl(MyList<Parallel::gridseg> *pp, bool first_only);
+    AsyncSyncState() : req_no(0), active(false) {}
-  void checkvarl(MyList<var> *pp, bool first_only);
+  };
-  MyList<Parallel::gridseg> *divide_gsl(MyList<Parallel::gridseg> *p, Patch *Pat);
+
-  MyList<Parallel::gridseg> *divide_gs(MyList<Parallel::gridseg> *p, Patch *Pat);
+  void Sync_start(MyList<Patch> *PatL, MyList<var> *VarList, int Symmetry,
-  void prepare_inter_time_level(Patch *Pat,
+                  SyncCache &cache, AsyncSyncState &state);
-                                MyList<var> *VarList1 /* source (t+dt) */, MyList<var> *VarList2 /* source (t) */,
+  void Sync_finish(SyncCache &cache, AsyncSyncState &state,
-                                MyList<var> *VarList3 /* target (t+a*dt) */, int tindex);
+                   MyList<var> *VarList, int Symmetry);
-  void prepare_inter_time_level(Patch *Pat,
+  void OutBdLow2Hi(Patch *Patc, Patch *Patf,
-                                MyList<var> *VarList1 /* source (t+dt) */, MyList<var> *VarList2 /* source (t) */,
+                   MyList<var> *VarList1 /* source */, MyList<var> *VarList2 /* target */,
-                                MyList<var> *VarList3 /* source (t-dt) */, MyList<var> *VarList4 /* target (t+a*dt) */, int tindex);
+                   int Symmetry);
-  void prepare_inter_time_level(MyList<Patch> *PatL,
+  void OutBdLow2Hi(MyList<Patch> *PatcL, MyList<Patch> *PatfL,
-                                MyList<var> *VarList1 /* source (t+dt) */, MyList<var> *VarList2 /* source (t) */,
+                   MyList<var> *VarList1 /* source */, MyList<var> *VarList2 /* target */,
-                                MyList<var> *VarList3 /* target (t+a*dt) */, int tindex);
+                   int Symmetry);
-  void prepare_inter_time_level(MyList<Patch> *Pat,
+  void OutBdLow2Himix(Patch *Patc, Patch *Patf,
-                                MyList<var> *VarList1 /* source (t+dt) */, MyList<var> *VarList2 /* source (t) */,
+                      MyList<var> *VarList1 /* source */, MyList<var> *VarList2 /* target */,
-                                MyList<var> *VarList3 /* source (t-dt) */, MyList<var> *VarList4 /* target (t+a*dt) */, int tindex);
+                      int Symmetry);
-  void merge_gsl(MyList<gridseg> *&A, const double ratio);
+  void OutBdLow2Himix(MyList<Patch> *PatcL, MyList<Patch> *PatfL,
-  bool merge_gs(MyList<gridseg> *D, MyList<gridseg> *B, MyList<gridseg> *&C, const double ratio);
+                      MyList<var> *VarList1 /* source */, MyList<var> *VarList2 /* target */,
-  // Add ghost region to tangent plane
+                      int Symmetry);
-  // we assume the grids have the same resolution
+  void Restrict_cached(MyList<Patch> *PatcL, MyList<Patch> *PatfL,
-  void add_ghost_touch(MyList<gridseg> *&A);
+                       MyList<var> *VarList1, MyList<var> *VarList2,
-  void cut_gsl(MyList<gridseg> *&A);
+                       int Symmetry, SyncCache &cache);
-  bool cut_gs(MyList<gridseg> *D, MyList<gridseg> *B, MyList<gridseg> *&C);
+  void OutBdLow2Hi_cached(MyList<Patch> *PatcL, MyList<Patch> *PatfL,
-  MyList<Parallel::gridseg> *gs_subtract_virtual(MyList<Parallel::gridseg> *A, MyList<Parallel::gridseg> *B);
+                          MyList<var> *VarList1, MyList<var> *VarList2,
-  void fill_level_data(MyList<Patch> *PatLd, MyList<Patch> *PatLs, MyList<Patch> *PatcL,
+                          int Symmetry, SyncCache &cache);
-                       MyList<var> *OldList, MyList<var> *StateList, MyList<var> *FutureList,
+  void OutBdLow2Himix_cached(MyList<Patch> *PatcL, MyList<Patch> *PatfL,
-                       MyList<var> *tmList, int Symmetry, bool BB, bool CC);
+                             MyList<var> *VarList1, MyList<var> *VarList2,
-  bool PatList_Interp_Points(MyList<Patch> *PatL, MyList<var> *VarList,
+                             int Symmetry, SyncCache &cache);
-                             int NN, double **XX,
+  void Prolong(Patch *Patc, Patch *Patf,
-                             double *Shellf, int Symmetry);
+               MyList<var> *VarList1 /* source */, MyList<var> *VarList2 /* target */,
-  void aligncheck(double *bbox0, double *bboxl, int lev, double *DH0, int *shape);
+               int Symmetry);
-  bool point_locat_gsl(double *pox, MyList<Parallel::gridseg> *gsl);
+  void Prolongint(Patch *Patc, Patch *Patf,
-  void checkpatchlist(MyList<Patch> *PatL, bool buflog);
+                  MyList<var> *VarList1 /* source */, MyList<var> *VarList2 /* target */,
-
+                  int Symmetry);
-  double L2Norm(Patch *Pat, var *vf, MPI_Comm Comm_here);
+  void Restrict(MyList<Patch> *PatcL, MyList<Patch> *PatfL,
-  bool PatList_Interp_Points(MyList<Patch> *PatL, MyList<var> *VarList,
+                MyList<var> *VarList1 /* source */, MyList<var> *VarList2 /* target */,
-                             int NN, double **XX,
+                int Symmetry);
-                             double *Shellf, int Symmetry, MPI_Comm Comm_here);
+  void Restrict_after(MyList<Patch> *PatcL, MyList<Patch> *PatfL,
-#if (PSTR == 1 || PSTR == 2 || PSTR == 3)
+                      MyList<var> *VarList1 /* source */, MyList<var> *VarList2 /* target */,
-  MyList<Block> *distribute(MyList<Patch> *PatchLIST, int cpusize, int ingfsi, int fngfsi,
+                      int Symmetry); // for -ghost - BDghost
-                            bool periodic, int start_rank, int end_rank, int nodes = 0);
+  MyList<Parallel::gridseg> *build_PhysBD_gsl(Patch *Pat);
-#endif
+  MyList<Parallel::gridseg> *build_ghost_gsl(MyList<Patch> *PatL);
-}
+  MyList<Parallel::gridseg> *build_ghost_gsl(Patch *Pat);
-#endif /*PARALLEL_H */
+  MyList<Parallel::gridseg> *build_buffer_gsl(Patch *Pat);
+  MyList<Parallel::gridseg> *build_buffer_gsl(MyList<Patch> *PatL);
+  MyList<Parallel::gridseg> *gsl_subtract(MyList<Parallel::gridseg> *A, MyList<Parallel::gridseg> *B);
+  MyList<Parallel::gridseg> *gs_subtract(MyList<Parallel::gridseg> *A, MyList<Parallel::gridseg> *B);
+  MyList<Parallel::gridseg> *gsl_and(MyList<Parallel::gridseg> *A, MyList<Parallel::gridseg> *B);
+  MyList<Parallel::gridseg> *gs_and(MyList<Parallel::gridseg> *A, MyList<Parallel::gridseg> *B);
+  MyList<Parallel::gridseg> *clone_gsl(MyList<Parallel::gridseg> *p, bool first_only);
+  MyList<Parallel::gridseg> *build_bulk_gsl(Patch *Pat); // similar to build_owned_gsl0 but does not care rank issue
+  MyList<Parallel::gridseg> *build_bulk_gsl(Block *bp, Patch *Pat);
+  void build_PhysBD_gstl(Patch *Pat, MyList<Parallel::gridseg> *srci, MyList<Parallel::gridseg> *dsti,
+                         MyList<Parallel::gridseg> **out_src, MyList<Parallel::gridseg> **out_dst);
+  void PeriodicBD(Patch *Pat, MyList<var> *VarList, int Symmetry);
+  double L2Norm(Patch *Pat, var *vf);
+  void checkgsl(MyList<Parallel::gridseg> *pp, bool first_only);
+  void checkvarl(MyList<var> *pp, bool first_only);
+  MyList<Parallel::gridseg> *divide_gsl(MyList<Parallel::gridseg> *p, Patch *Pat);
+  MyList<Parallel::gridseg> *divide_gs(MyList<Parallel::gridseg> *p, Patch *Pat);
+  void prepare_inter_time_level(Patch *Pat,
+                                MyList<var> *VarList1 /* source (t+dt) */, MyList<var> *VarList2 /* source (t) */,
+                                MyList<var> *VarList3 /* target (t+a*dt) */, int tindex);
+  void prepare_inter_time_level(Patch *Pat,
+                                MyList<var> *VarList1 /* source (t+dt) */, MyList<var> *VarList2 /* source (t) */,
+                                MyList<var> *VarList3 /* source (t-dt) */, MyList<var> *VarList4 /* target (t+a*dt) */, int tindex);
+  void prepare_inter_time_level(MyList<Patch> *PatL,
+                                MyList<var> *VarList1 /* source (t+dt) */, MyList<var> *VarList2 /* source (t) */,
+                                MyList<var> *VarList3 /* target (t+a*dt) */, int tindex);
+  void prepare_inter_time_level(MyList<Patch> *Pat,
+                                MyList<var> *VarList1 /* source (t+dt) */, MyList<var> *VarList2 /* source (t) */,
+                                MyList<var> *VarList3 /* source (t-dt) */, MyList<var> *VarList4 /* target (t+a*dt) */, int tindex);
+  void merge_gsl(MyList<gridseg> *&A, const double ratio);
+  bool merge_gs(MyList<gridseg> *D, MyList<gridseg> *B, MyList<gridseg> *&C, const double ratio);
+  // Add ghost region to tangent plane
+  // we assume the grids have the same resolution
+  void add_ghost_touch(MyList<gridseg> *&A);
+  void cut_gsl(MyList<gridseg> *&A);
+  bool cut_gs(MyList<gridseg> *D, MyList<gridseg> *B, MyList<gridseg> *&C);
+  MyList<Parallel::gridseg> *gs_subtract_virtual(MyList<Parallel::gridseg> *A, MyList<Parallel::gridseg> *B);
+  void fill_level_data(MyList<Patch> *PatLd, MyList<Patch> *PatLs, MyList<Patch> *PatcL,
+                       MyList<var> *OldList, MyList<var> *StateList, MyList<var> *FutureList,
+                       MyList<var> *tmList, int Symmetry, bool BB, bool CC);
+  bool PatList_Interp_Points(MyList<Patch> *PatL, MyList<var> *VarList,
+                             int NN, double **XX,
+                             double *Shellf, int Symmetry);
+  void aligncheck(double *bbox0, double *bboxl, int lev, double *DH0, int *shape);
+  bool point_locat_gsl(double *pox, MyList<Parallel::gridseg> *gsl);
+  void checkpatchlist(MyList<Patch> *PatL, bool buflog);
+
+  double L2Norm(Patch *Pat, var *vf, MPI_Comm Comm_here);
+  bool PatList_Interp_Points(MyList<Patch> *PatL, MyList<var> *VarList,
+                             int NN, double **XX,
+                             double *Shellf, int Symmetry, MPI_Comm Comm_here);
+#if (PSTR == 1 || PSTR == 2 || PSTR == 3)
+  MyList<Block> *distribute(MyList<Patch> *PatchLIST, int cpusize, int ingfsi, int fngfsi,
+                            bool periodic, int start_rank, int end_rank, int nodes = 0);
+  
+  // Redistribute blocks with time statistics for load balancing
+  MyList<Block> *distribute(MyList<Patch> *PatchLIST, MyList<Block> *OldBlockL,
+                            int cpusize, int ingfsi, int fngfsi,
+                            bool periodic, int start_rank, int end_rank, int nodes = 0);
+#endif
+
+  // Dynamic load balancing: split blocks for heavy ranks
+    void split_heavy_blocks(MyList<Patch> *PatL, int *heavy_ranks, int num_heavy,
+                            int split_factor, int cpusize, int ingfsi, int fngfsi);
+
+    // Check if load balancing is needed based on interpolation times
+    bool check_load_balance_need(double *rank_times, int nprocs, int &num_heavy, int *heavy_ranks);
+  }
+  #endif /*PARALLEL_H */

AMSS_NCKU_source/bssn_class.C

@@ -730,6 +730,12 @@ void bssn_class::Initialize()
     PhysTime = StartTime;
     Setup_Black_Hole_position();
   }
+
+  // Initialize sync caches (per-level, for predictor and corrector)
+  sync_cache_pre = new Parallel::SyncCache[GH->levels];
+  sync_cache_cor = new Parallel::SyncCache[GH->levels];
+  sync_cache_rp_coarse = new Parallel::SyncCache[GH->levels];
+  sync_cache_rp_fine = new Parallel::SyncCache[GH->levels];
 }
 
 //================================================================================================
@@ -981,6 +987,32 @@ bssn_class::~bssn_class()
   delete Azzz;
 #endif
 
+  // Destroy sync caches before GH
+  if (sync_cache_pre)
+  {
+    for (int i = 0; i < GH->levels; i++)
+      sync_cache_pre[i].destroy();
+    delete[] sync_cache_pre;
+  }
+  if (sync_cache_cor)
+  {
+    for (int i = 0; i < GH->levels; i++)
+      sync_cache_cor[i].destroy();
+    delete[] sync_cache_cor;
+  }
+  if (sync_cache_rp_coarse)
+  {
+    for (int i = 0; i < GH->levels; i++)
+      sync_cache_rp_coarse[i].destroy();
+    delete[] sync_cache_rp_coarse;
+  }
+  if (sync_cache_rp_fine)
+  {
+    for (int i = 0; i < GH->levels; i++)
+      sync_cache_rp_fine[i].destroy();
+    delete[] sync_cache_rp_fine;
+  }
+
   delete GH;
 #ifdef WithShell
   delete SH;
@@ -2181,6 +2213,7 @@ void bssn_class::Evolve(int Steps)
     GH->Regrid(Symmetry, BH_num, Porgbr, Porg0,
                SynchList_cor, OldStateList, StateList, SynchList_pre,
                fgt(PhysTime - dT_mon, StartTime, dT_mon / 2), ErrorMonitor);
+    for (int il = 0; il < GH->levels; il++) { sync_cache_pre[il].invalidate(); sync_cache_cor[il].invalidate(); sync_cache_rp_coarse[il].invalidate(); sync_cache_rp_fine[il].invalidate(); }
 #endif
 
 #if (REGLEV == 0 && (PSTR == 1 || PSTR == 2))
@@ -2396,6 +2429,7 @@ void bssn_class::RecursiveStep(int lev)
   GH->Regrid_Onelevel(lev, Symmetry, BH_num, Porgbr, Porg0,
                       SynchList_cor, OldStateList, StateList, SynchList_pre,
                       fgt(PhysTime - dT_lev, StartTime, dT_lev / 2), ErrorMonitor);
+  for (int il = 0; il < GH->levels; il++) { sync_cache_pre[il].invalidate(); sync_cache_cor[il].invalidate(); sync_cache_rp_coarse[il].invalidate(); sync_cache_rp_fine[il].invalidate(); }
 #endif
 }
 
@@ -2574,6 +2608,7 @@ void bssn_class::ParallelStep()
   GH->Regrid_Onelevel(GH->mylev, Symmetry, BH_num, Porgbr, Porg0,
                       SynchList_cor, OldStateList, StateList, SynchList_pre,
                       fgt(PhysTime - dT_lev, StartTime, dT_lev / 2), ErrorMonitor);
+  for (int il = 0; il < GH->levels; il++) { sync_cache_pre[il].invalidate(); sync_cache_cor[il].invalidate(); sync_cache_rp_coarse[il].invalidate(); sync_cache_rp_fine[il].invalidate(); }
 #endif
 }
 
@@ -2740,6 +2775,7 @@ void bssn_class::ParallelStep()
         GH->Regrid_Onelevel(lev + 1, Symmetry, BH_num, Porgbr, Porg0,
                             SynchList_cor, OldStateList, StateList, SynchList_pre,
                             fgt(PhysTime - dT_levp1, StartTime, dT_levp1 / 2), ErrorMonitor);
+        for (int il = 0; il < GH->levels; il++) { sync_cache_pre[il].invalidate(); sync_cache_cor[il].invalidate(); sync_cache_rp_coarse[il].invalidate(); sync_cache_rp_fine[il].invalidate(); }
 
         //               a_stream.clear();
         //               a_stream.str("");
@@ -2754,6 +2790,7 @@ void bssn_class::ParallelStep()
       GH->Regrid_Onelevel(lev, Symmetry, BH_num, Porgbr, Porg0,
                           SynchList_cor, OldStateList, StateList, SynchList_pre,
                           fgt(PhysTime - dT_lev, StartTime, dT_lev / 2), ErrorMonitor);
+      for (int il = 0; il < GH->levels; il++) { sync_cache_pre[il].invalidate(); sync_cache_cor[il].invalidate(); sync_cache_rp_coarse[il].invalidate(); sync_cache_rp_fine[il].invalidate(); }
 
       //               a_stream.clear();
       //               a_stream.str("");
@@ -2772,6 +2809,7 @@ void bssn_class::ParallelStep()
           GH->Regrid_Onelevel(lev - 1, Symmetry, BH_num, Porgbr, Porg0,
                               SynchList_cor, OldStateList, StateList, SynchList_pre,
                               fgt(PhysTime - dT_lev, StartTime, dT_levm1 / 2), ErrorMonitor);
+          for (int il = 0; il < GH->levels; il++) { sync_cache_pre[il].invalidate(); sync_cache_cor[il].invalidate(); sync_cache_rp_coarse[il].invalidate(); sync_cache_rp_fine[il].invalidate(); }
 
           //               a_stream.clear();
           //               a_stream.str("");
@@ -2787,6 +2825,7 @@ void bssn_class::ParallelStep()
           GH->Regrid_Onelevel(lev - 1, Symmetry, BH_num, Porgbr, Porg0,
                               SynchList_cor, OldStateList, StateList, SynchList_pre,
                               fgt(PhysTime - dT_lev, StartTime, dT_levm1 / 2), ErrorMonitor);
+          for (int il = 0; il < GH->levels; il++) { sync_cache_pre[il].invalidate(); sync_cache_cor[il].invalidate(); sync_cache_rp_coarse[il].invalidate(); sync_cache_rp_fine[il].invalidate(); }
 
           //               a_stream.clear();
           //               a_stream.str("");
@@ -3158,21 +3197,7 @@ void bssn_class::Step(int lev, int YN)
     }
     Pp = Pp->next;
   }
-  // check error information
+  // NOTE: error check deferred to after Shell Patch computation to reduce MPI_Allreduce calls
-  {
-    int erh = ERROR;
-    MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
-  }
-  if (ERROR)
-  {
-    Parallel::Dump_Data(GH->PatL[lev], StateList, 0, PhysTime, dT_lev);
-    if (myrank == 0)
-    {
-      if (ErrorMonitor->outfile)
-        ErrorMonitor->outfile << "find NaN in state variables at t = " << PhysTime << ", lev = " << lev << endl;
-      MPI_Abort(MPI_COMM_WORLD, 1);
-    }
-  }
 
 #ifdef WithShell
   // evolve Shell Patches
@@ -3190,9 +3215,9 @@ void bssn_class::Step(int lev, int YN)
         {
 #if (AGM == 0)
           f_enforce_ga(cg->shape,
-                       cg->fgfs[gxx0->sgfn], cg->fgfs[gxy0->sgfn], cg->fgfs[gxz0->sgfn], 
+                       cg->fgfs[gxx0->sgfn], cg->fgfs[gxy0->sgfn], cg->fgfs[gxz0->sgfn],
                        cg->fgfs[gyy0->sgfn], cg->fgfs[gyz0->sgfn], cg->fgfs[gzz0->sgfn],
-                       cg->fgfs[Axx0->sgfn], cg->fgfs[Axy0->sgfn], cg->fgfs[Axz0->sgfn], 
+                       cg->fgfs[Axx0->sgfn], cg->fgfs[Axy0->sgfn], cg->fgfs[Axz0->sgfn],
                        cg->fgfs[Ayy0->sgfn], cg->fgfs[Ayz0->sgfn], cg->fgfs[Azz0->sgfn]);
 #endif
 
@@ -3316,25 +3341,16 @@ void bssn_class::Step(int lev, int YN)
 #endif
   }
 
-  // check error information
+  // Non-blocking error reduction overlapped with Sync to hide Allreduce latency
+  MPI_Request err_req;
   {
     int erh = ERROR;
-    MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
+    MPI_Iallreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD, &err_req);
-  }
-
-  if (ERROR)
-  {
-    SH->Dump_Data(StateList, 0, PhysTime, dT_lev);
-    if (myrank == 0)
-    {
-      if (ErrorMonitor->outfile)
-        ErrorMonitor->outfile << "find NaN in state variables on Shell Patches at t = " << PhysTime << endl;
-      MPI_Abort(MPI_COMM_WORLD, 1);
-    }
   }
 #endif
 
-  Parallel::Sync(GH->PatL[lev], SynchList_pre, Symmetry);
+  Parallel::AsyncSyncState async_pre;
+  Parallel::Sync_start(GH->PatL[lev], SynchList_pre, Symmetry, sync_cache_pre[lev], async_pre);
 
 #ifdef WithShell
   if (lev == 0)
@@ -3347,12 +3363,29 @@ void bssn_class::Step(int lev, int YN)
     {
       prev_clock = curr_clock;
       curr_clock = clock();
-      cout << " Shell stuff synchronization used " 
+      cout << " Shell stuff synchronization used "
-           << (double)(curr_clock - prev_clock) / ((double)CLOCKS_PER_SEC) 
+           << (double)(curr_clock - prev_clock) / ((double)CLOCKS_PER_SEC)
            << " seconds! " << endl;
     }
   }
 #endif
+  Parallel::Sync_finish(sync_cache_pre[lev], async_pre, SynchList_pre, Symmetry);
+
+#ifdef WithShell
+  // Complete non-blocking error reduction and check
+  MPI_Wait(&err_req, MPI_STATUS_IGNORE);
+  if (ERROR)
+  {
+    Parallel::Dump_Data(GH->PatL[lev], StateList, 0, PhysTime, dT_lev);
+    SH->Dump_Data(StateList, 0, PhysTime, dT_lev);
+    if (myrank == 0)
+    {
+      if (ErrorMonitor->outfile)
+        ErrorMonitor->outfile << "find NaN in state variables at t = " << PhysTime << ", lev = " << lev << endl;
+      MPI_Abort(MPI_COMM_WORLD, 1);
+    }
+  }
+#endif
 
 #if (MAPBH == 0)
   // for black hole position
@@ -3528,24 +3561,7 @@ void bssn_class::Step(int lev, int YN)
       Pp = Pp->next;
     }
 
-    // check error information
+    // NOTE: error check deferred to after Shell Patch computation to reduce MPI_Allreduce calls
-    {
-      int erh = ERROR;
-      MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
-    }
-
-    if (ERROR)
-    {
-      Parallel::Dump_Data(GH->PatL[lev], SynchList_pre, 0, PhysTime, dT_lev);
-      if (myrank == 0)
-      {
-        if (ErrorMonitor->outfile)
-          ErrorMonitor->outfile << "find NaN in RK4 substep#" << iter_count 
-                                << " variables at t = " << PhysTime 
-                                << ", lev = " << lev << endl;
-        MPI_Abort(MPI_COMM_WORLD, 1);
-      }
-    }
 
 #ifdef WithShell
     // evolve Shell Patches
@@ -3563,9 +3579,9 @@ void bssn_class::Step(int lev, int YN)
           {
 #if (AGM == 0)
             f_enforce_ga(cg->shape,
-                         cg->fgfs[gxx->sgfn], cg->fgfs[gxy->sgfn], cg->fgfs[gxz->sgfn], 
+                         cg->fgfs[gxx->sgfn], cg->fgfs[gxy->sgfn], cg->fgfs[gxz->sgfn],
                          cg->fgfs[gyy->sgfn], cg->fgfs[gyz->sgfn], cg->fgfs[gzz->sgfn],
-                         cg->fgfs[Axx->sgfn], cg->fgfs[Axy->sgfn], cg->fgfs[Axz->sgfn], 
+                         cg->fgfs[Axx->sgfn], cg->fgfs[Axy->sgfn], cg->fgfs[Axz->sgfn],
                          cg->fgfs[Ayy->sgfn], cg->fgfs[Ayz->sgfn], cg->fgfs[Azz->sgfn]);
 #elif (AGM == 1)
             if (iter_count == 3)
@@ -3685,26 +3701,16 @@ void bssn_class::Step(int lev, int YN)
         sPp = sPp->next;
       }
     }
-    // check error information
+    // Non-blocking error reduction overlapped with Sync to hide Allreduce latency
+    MPI_Request err_req_cor;
     {
       int erh = ERROR;
-      MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
+      MPI_Iallreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD, &err_req_cor);
-    }
-    if (ERROR)
-    {
-      SH->Dump_Data(SynchList_pre, 0, PhysTime, dT_lev);
-      if (myrank == 0)
-      {
-        if (ErrorMonitor->outfile)
-          ErrorMonitor->outfile << "find NaN on Shell Patches in RK4 substep#" 
-                                << iter_count << " variables at t = " 
-                                << PhysTime << endl;
-        MPI_Abort(MPI_COMM_WORLD, 1);
-      }
     }
 #endif
 
-    Parallel::Sync(GH->PatL[lev], SynchList_cor, Symmetry);
+    Parallel::AsyncSyncState async_cor;
+    Parallel::Sync_start(GH->PatL[lev], SynchList_cor, Symmetry, sync_cache_cor[lev], async_cor);
 
 #ifdef WithShell
     if (lev == 0)
@@ -3717,12 +3723,31 @@ void bssn_class::Step(int lev, int YN)
       {
         prev_clock = curr_clock;
         curr_clock = clock();
-        cout << " Shell stuff synchronization used " 
+        cout << " Shell stuff synchronization used "
-             << (double)(curr_clock - prev_clock) / ((double)CLOCKS_PER_SEC) 
+             << (double)(curr_clock - prev_clock) / ((double)CLOCKS_PER_SEC)
              << " seconds! " << endl;
       }
     }
 #endif
+    Parallel::Sync_finish(sync_cache_cor[lev], async_cor, SynchList_cor, Symmetry);
+
+#ifdef WithShell
+    // Complete non-blocking error reduction and check
+    MPI_Wait(&err_req_cor, MPI_STATUS_IGNORE);
+    if (ERROR)
+    {
+      Parallel::Dump_Data(GH->PatL[lev], SynchList_pre, 0, PhysTime, dT_lev);
+      SH->Dump_Data(SynchList_pre, 0, PhysTime, dT_lev);
+      if (myrank == 0)
+      {
+        if (ErrorMonitor->outfile)
+          ErrorMonitor->outfile << "find NaN in RK4 substep#" << iter_count
+                                << " variables at t = " << PhysTime
+                                << ", lev = " << lev << endl;
+        MPI_Abort(MPI_COMM_WORLD, 1);
+      }
+    }
+#endif
 
 #if (MAPBH == 0)
     // for black hole position
@@ -4034,22 +4059,7 @@ void bssn_class::Step(int lev, int YN)
     }
     Pp = Pp->next;
   }
-  // check error information
+  // NOTE: error check deferred to after Shell Patch computation to reduce MPI_Allreduce calls
-  {
-    int erh = ERROR;
-    MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
-  }
-  if (ERROR)
-  {
-    Parallel::Dump_Data(GH->PatL[lev], StateList, 0, PhysTime, dT_lev);
-    if (myrank == 0)
-    {
-      if (ErrorMonitor->outfile)
-        ErrorMonitor->outfile << "find NaN in state variables at t = " << PhysTime 
-                              << ", lev = " << lev << endl;
-      MPI_Abort(MPI_COMM_WORLD, 1);
-    }
-  }
 
 #ifdef WithShell
   // evolve Shell Patches
@@ -4067,15 +4077,15 @@ void bssn_class::Step(int lev, int YN)
         {
 #if (AGM == 0)
           f_enforce_ga(cg->shape,
-                       cg->fgfs[gxx0->sgfn], cg->fgfs[gxy0->sgfn], cg->fgfs[gxz0->sgfn], 
+                       cg->fgfs[gxx0->sgfn], cg->fgfs[gxy0->sgfn], cg->fgfs[gxz0->sgfn],
                        cg->fgfs[gyy0->sgfn], cg->fgfs[gyz0->sgfn], cg->fgfs[gzz0->sgfn],
-                       cg->fgfs[Axx0->sgfn], cg->fgfs[Axy0->sgfn], cg->fgfs[Axz0->sgfn], 
+                       cg->fgfs[Axx0->sgfn], cg->fgfs[Axy0->sgfn], cg->fgfs[Axz0->sgfn],
                        cg->fgfs[Ayy0->sgfn], cg->fgfs[Ayz0->sgfn], cg->fgfs[Azz0->sgfn]);
 #endif
 
           if (f_compute_rhs_bssn_ss(cg->shape, TRK4, cg->X[0], cg->X[1], cg->X[2],
-                                    cg->fgfs[fngfs + ShellPatch::gx], 
+                                    cg->fgfs[fngfs + ShellPatch::gx],
-                                    cg->fgfs[fngfs + ShellPatch::gy], 
+                                    cg->fgfs[fngfs + ShellPatch::gy],
                                     cg->fgfs[fngfs + ShellPatch::gz],
                                     cg->fgfs[fngfs + ShellPatch::drhodx], 
                                     cg->fgfs[fngfs + ShellPatch::drhody], 
@@ -4190,25 +4200,16 @@ void bssn_class::Step(int lev, int YN)
   }
 #endif
   }
-  // check error information
+  // Non-blocking error reduction overlapped with Sync to hide Allreduce latency
+  MPI_Request err_req;
   {
     int erh = ERROR;
-    MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
+    MPI_Iallreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD, &err_req);
-  }
-  if (ERROR)
-  {
-    SH->Dump_Data(StateList, 0, PhysTime, dT_lev);
-    if (myrank == 0)
-    {
-      if (ErrorMonitor->outfile)
-        ErrorMonitor->outfile << "find NaN in state variables on Shell Patches at t = " 
-                              << PhysTime << endl;
-      MPI_Abort(MPI_COMM_WORLD, 1);
-    }
   }
 #endif
 
-  Parallel::Sync(GH->PatL[lev], SynchList_pre, Symmetry);
+  Parallel::AsyncSyncState async_pre;
+  Parallel::Sync_start(GH->PatL[lev], SynchList_pre, Symmetry, sync_cache_pre[lev], async_pre);
 
 #ifdef WithShell
   if (lev == 0)
@@ -4221,9 +4222,27 @@ void bssn_class::Step(int lev, int YN)
     {
       prev_clock = curr_clock;
       curr_clock = clock();
-      cout << " Shell stuff synchronization used " 
+      cout << " Shell stuff synchronization used "
-      << (double)(curr_clock - prev_clock) / ((double)CLOCKS_PER_SEC) 
+           << (double)(curr_clock - prev_clock) / ((double)CLOCKS_PER_SEC)
-      << " seconds! " << endl;
+           << " seconds! " << endl;
+    }
+  }
+#endif
+  Parallel::Sync_finish(sync_cache_pre[lev], async_pre, SynchList_pre, Symmetry);
+
+#ifdef WithShell
+  // Complete non-blocking error reduction and check
+  MPI_Wait(&err_req, MPI_STATUS_IGNORE);
+  if (ERROR)
+  {
+    Parallel::Dump_Data(GH->PatL[lev], StateList, 0, PhysTime, dT_lev);
+    SH->Dump_Data(StateList, 0, PhysTime, dT_lev);
+    if (myrank == 0)
+    {
+      if (ErrorMonitor->outfile)
+        ErrorMonitor->outfile << "find NaN in state variables at t = " << PhysTime
+                              << ", lev = " << lev << endl;
+      MPI_Abort(MPI_COMM_WORLD, 1);
     }
   }
 #endif
@@ -4386,23 +4405,7 @@ void bssn_class::Step(int lev, int YN)
       Pp = Pp->next;
     }
 
-    // check error information
+    // NOTE: error check deferred to after Shell Patch computation to reduce MPI_Allreduce calls
-    {
-      int erh = ERROR;
-      MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
-    }
-    if (ERROR)
-    {
-      Parallel::Dump_Data(GH->PatL[lev], SynchList_pre, 0, PhysTime, dT_lev);
-      if (myrank == 0)
-      {
-        if (ErrorMonitor->outfile)
-          ErrorMonitor->outfile << "find NaN in RK4 substep#" << iter_count 
-                                << " variables at t = " << PhysTime 
-                                << ", lev = " << lev << endl;
-        MPI_Abort(MPI_COMM_WORLD, 1);
-      }
-    }
 
 #ifdef WithShell
     // evolve Shell Patches
@@ -4420,9 +4423,9 @@ void bssn_class::Step(int lev, int YN)
           {
 #if (AGM == 0)
             f_enforce_ga(cg->shape,
-                         cg->fgfs[gxx->sgfn], cg->fgfs[gxy->sgfn], cg->fgfs[gxz->sgfn], 
+                         cg->fgfs[gxx->sgfn], cg->fgfs[gxy->sgfn], cg->fgfs[gxz->sgfn],
                          cg->fgfs[gyy->sgfn], cg->fgfs[gyz->sgfn], cg->fgfs[gzz->sgfn],
-                         cg->fgfs[Axx->sgfn], cg->fgfs[Axy->sgfn], cg->fgfs[Axz->sgfn], 
+                         cg->fgfs[Axx->sgfn], cg->fgfs[Axy->sgfn], cg->fgfs[Axz->sgfn],
                          cg->fgfs[Ayy->sgfn], cg->fgfs[Ayz->sgfn], cg->fgfs[Azz->sgfn]);
 #elif (AGM == 1)
             if (iter_count == 3)
@@ -4542,25 +4545,16 @@ void bssn_class::Step(int lev, int YN)
         sPp = sPp->next;
       }
     }
-    // check error information
+    // Non-blocking error reduction overlapped with Sync to hide Allreduce latency
+    MPI_Request err_req_cor;
     {
       int erh = ERROR;
-      MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
+      MPI_Iallreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD, &err_req_cor);
-    }
-    if (ERROR)
-    {
-      SH->Dump_Data(SynchList_pre, 0, PhysTime, dT_lev);
-      if (myrank == 0)
-      {
-        if (ErrorMonitor->outfile)
-          ErrorMonitor->outfile << "find NaN on Shell Patches in RK4 substep#" << iter_count 
-                                << " variables at t = " << PhysTime << endl;
-        MPI_Abort(MPI_COMM_WORLD, 1);
-      }
     }
 #endif
 
-    Parallel::Sync(GH->PatL[lev], SynchList_cor, Symmetry);
+    Parallel::AsyncSyncState async_cor;
+    Parallel::Sync_start(GH->PatL[lev], SynchList_cor, Symmetry, sync_cache_cor[lev], async_cor);
 
 #ifdef WithShell
     if (lev == 0)
@@ -4573,11 +4567,30 @@ void bssn_class::Step(int lev, int YN)
       {
         prev_clock = curr_clock;
         curr_clock = clock();
-        cout << " Shell stuff synchronization used " 
+        cout << " Shell stuff synchronization used "
-             << (double)(curr_clock - prev_clock) / ((double)CLOCKS_PER_SEC) 
+             << (double)(curr_clock - prev_clock) / ((double)CLOCKS_PER_SEC)
              << " seconds! " << endl;
       }
     }
+#endif
+    Parallel::Sync_finish(sync_cache_cor[lev], async_cor, SynchList_cor, Symmetry);
+
+#ifdef WithShell
+    // Complete non-blocking error reduction and check
+    MPI_Wait(&err_req_cor, MPI_STATUS_IGNORE);
+    if (ERROR)
+    {
+      Parallel::Dump_Data(GH->PatL[lev], SynchList_pre, 0, PhysTime, dT_lev);
+      SH->Dump_Data(SynchList_pre, 0, PhysTime, dT_lev);
+      if (myrank == 0)
+      {
+        if (ErrorMonitor->outfile)
+          ErrorMonitor->outfile << "find NaN in RK4 substep#" << iter_count
+                                << " variables at t = " << PhysTime
+                                << ", lev = " << lev << endl;
+        MPI_Abort(MPI_COMM_WORLD, 1);
+      }
+    }
 #endif
     // for black hole position
     if (BH_num > 0 && lev == GH->levels - 1)
@@ -4943,11 +4956,19 @@ void bssn_class::Step(int lev, int YN)
 
   //   misc::tillherecheck(GH->Commlev[lev],GH->start_rank[lev],"after Predictor rhs calculation");
 
-  // check error information
+  // Non-blocking error reduction overlapped with Sync to hide Allreduce latency
+  MPI_Request err_req;
   {
     int erh = ERROR;
-    MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, GH->Commlev[lev]);
+    MPI_Iallreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, GH->Commlev[lev], &err_req);
   }
+
+  //   misc::tillherecheck(GH->Commlev[lev],GH->start_rank[lev],"before Predictor sync");
+
+  Parallel::Sync_cached(GH->PatL[lev], SynchList_pre, Symmetry, sync_cache_pre[lev]);
+
+  // Complete non-blocking error reduction and check
+  MPI_Wait(&err_req, MPI_STATUS_IGNORE);
   if (ERROR)
   {
     Parallel::Dump_Data(GH->PatL[lev], StateList, 0, PhysTime, dT_lev);
@@ -4959,10 +4980,6 @@ void bssn_class::Step(int lev, int YN)
     }
   }
 
-  //   misc::tillherecheck(GH->Commlev[lev],GH->start_rank[lev],"before Predictor sync");
-
-  Parallel::Sync(GH->PatL[lev], SynchList_pre, Symmetry);
-
 #if (MAPBH == 0)
   // for black hole position
   if (BH_num > 0 && lev == GH->levels - 1)
@@ -5140,30 +5157,34 @@ void bssn_class::Step(int lev, int YN)
 
     //   misc::tillherecheck(GH->Commlev[lev],GH->start_rank[lev],"before Corrector error check");
 
-    // check error information
+    // Non-blocking error reduction overlapped with Sync to hide Allreduce latency
+    MPI_Request err_req_cor;
     {
       int erh = ERROR;
-      MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, GH->Commlev[lev]);
+      MPI_Iallreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, GH->Commlev[lev], &err_req_cor);
     }
+
+    //    misc::tillherecheck(GH->Commlev[lev],GH->start_rank[lev],"before Corrector sync");
+
+    Parallel::Sync_cached(GH->PatL[lev], SynchList_cor, Symmetry, sync_cache_cor[lev]);
+
+    //    misc::tillherecheck(GH->Commlev[lev],GH->start_rank[lev],"after Corrector sync");
+
+    // Complete non-blocking error reduction and check
+    MPI_Wait(&err_req_cor, MPI_STATUS_IGNORE);
     if (ERROR)
     {
       Parallel::Dump_Data(GH->PatL[lev], SynchList_pre, 0, PhysTime, dT_lev);
       if (myrank == 0)
       {
         if (ErrorMonitor->outfile)
-          ErrorMonitor->outfile << "find NaN in RK4 substep#" << iter_count 
+          ErrorMonitor->outfile << "find NaN in RK4 substep#" << iter_count
-                                << " variables at t = " << PhysTime 
+                                << " variables at t = " << PhysTime
                                 << ", lev = " << lev << endl;
         MPI_Abort(MPI_COMM_WORLD, 1);
       }
     }
 
-    //    misc::tillherecheck(GH->Commlev[lev],GH->start_rank[lev],"before Corrector sync");
-
-    Parallel::Sync(GH->PatL[lev], SynchList_cor, Symmetry);
-
-    //    misc::tillherecheck(GH->Commlev[lev],GH->start_rank[lev],"after Corrector sync");
-
 #if (MAPBH == 0)
     // for black hole position
     if (BH_num > 0 && lev == GH->levels - 1)
@@ -5447,21 +5468,11 @@ void bssn_class::SHStep()
 #if (PSTR == 1 || PSTR == 2)
 //   misc::tillherecheck(GH->Commlev[lev],GH->start_rank[lev],"before Predictor's error check");
 #endif
-  // check error information
+  // Non-blocking error reduction overlapped with Synch to hide Allreduce latency
+  MPI_Request err_req;
   {
     int erh = ERROR;
-    MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
+    MPI_Iallreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD, &err_req);
-  }
-
-  if (ERROR)
-  {
-    SH->Dump_Data(StateList, 0, PhysTime, dT_lev);
-    if (myrank == 0)
-    {
-      if (ErrorMonitor->outfile)
-        ErrorMonitor->outfile << "find NaN in state variables on Shell Patches at t = " << PhysTime << endl;
-      MPI_Abort(MPI_COMM_WORLD, 1);
-    }
   }
 
   {
@@ -5473,12 +5484,25 @@ void bssn_class::SHStep()
     {
       prev_clock = curr_clock;
       curr_clock = clock();
-      cout << " Shell stuff synchronization used " 
+      cout << " Shell stuff synchronization used "
-           << (double)(curr_clock - prev_clock) / ((double)CLOCKS_PER_SEC) 
+           << (double)(curr_clock - prev_clock) / ((double)CLOCKS_PER_SEC)
            << " seconds! " << endl;
     }
   }
 
+  // Complete non-blocking error reduction and check
+  MPI_Wait(&err_req, MPI_STATUS_IGNORE);
+  if (ERROR)
+  {
+    SH->Dump_Data(StateList, 0, PhysTime, dT_lev);
+    if (myrank == 0)
+    {
+      if (ErrorMonitor->outfile)
+        ErrorMonitor->outfile << "find NaN in state variables on Shell Patches at t = " << PhysTime << endl;
+      MPI_Abort(MPI_COMM_WORLD, 1);
+    }
+  }
+
   // corrector
   for (iter_count = 1; iter_count < 4; iter_count++)
   {
@@ -5621,21 +5645,11 @@ void bssn_class::SHStep()
         sPp = sPp->next;
       }
     }
-    // check error information
+    // Non-blocking error reduction overlapped with Synch to hide Allreduce latency
+    MPI_Request err_req_cor;
     {
       int erh = ERROR;
-      MPI_Allreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
+      MPI_Iallreduce(&erh, &ERROR, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD, &err_req_cor);
-    }
-    if (ERROR)
-    {
-      SH->Dump_Data(SynchList_pre, 0, PhysTime, dT_lev);
-      if (myrank == 0)
-      {
-        if (ErrorMonitor->outfile)
-          ErrorMonitor->outfile << "find NaN on Shell Patches in RK4 substep#" << iter_count 
-                                << " variables at t = " << PhysTime << endl;
-        MPI_Abort(MPI_COMM_WORLD, 1);
-      }
     }
 
     {
@@ -5647,12 +5661,26 @@ void bssn_class::SHStep()
       {
         prev_clock = curr_clock;
         curr_clock = clock();
-        cout << " Shell stuff synchronization used " 
+        cout << " Shell stuff synchronization used "
-             << (double)(curr_clock - prev_clock) / ((double)CLOCKS_PER_SEC) 
+             << (double)(curr_clock - prev_clock) / ((double)CLOCKS_PER_SEC)
              << " seconds! " << endl;
       }
     }
 
+    // Complete non-blocking error reduction and check
+    MPI_Wait(&err_req_cor, MPI_STATUS_IGNORE);
+    if (ERROR)
+    {
+      SH->Dump_Data(SynchList_pre, 0, PhysTime, dT_lev);
+      if (myrank == 0)
+      {
+        if (ErrorMonitor->outfile)
+          ErrorMonitor->outfile << "find NaN on Shell Patches in RK4 substep#" << iter_count
+                                << " variables at t = " << PhysTime << endl;
+        MPI_Abort(MPI_COMM_WORLD, 1);
+      }
+    }
+
     sPp = SH->PatL;
     while (sPp)
     {
@@ -5781,7 +5809,7 @@ void bssn_class::RestrictProlong(int lev, int YN, bool BB,
 //       misc::tillherecheck(GH->Commlev[GH->mylev],GH->start_rank[GH->mylev],a_stream.str());
 #endif
 
-      Parallel::Sync(GH->PatL[lev - 1], SynchList_pre, Symmetry);
+      Parallel::Sync_cached(GH->PatL[lev - 1], SynchList_pre, Symmetry, sync_cache_rp_coarse[lev]);
 
 #if (PSTR == 1 || PSTR == 2)
 //       a_stream.clear();
@@ -5791,21 +5819,11 @@ void bssn_class::RestrictProlong(int lev, int YN, bool BB,
 #endif
 
 #if (RPB == 0)
-      Ppc = GH->PatL[lev - 1];
-      while (Ppc)
-      {
-        Pp = GH->PatL[lev];
-        while (Pp)
-        {
 #if (MIXOUTB == 0)
-          Parallel::OutBdLow2Hi(Ppc->data, Pp->data, SynchList_pre, SL, Symmetry);
+      Parallel::OutBdLow2Hi(GH->PatL[lev - 1], GH->PatL[lev], SynchList_pre, SL, Symmetry);
 #elif (MIXOUTB == 1)
-          Parallel::OutBdLow2Himix(Ppc->data, Pp->data, SynchList_pre, SL, Symmetry);
+      Parallel::OutBdLow2Himix(GH->PatL[lev - 1], GH->PatL[lev], SynchList_pre, SL, Symmetry);
 #endif
-          Pp = Pp->next;
-        }
-        Ppc = Ppc->next;
-      }
 #elif (RPB == 1)
       //       Parallel::OutBdLow2Hi_bam(GH->PatL[lev-1],GH->PatL[lev],SynchList_pre,SL,Symmetry);
       Parallel::OutBdLow2Hi_bam(GH->PatL[lev - 1], GH->PatL[lev], SynchList_pre, SL, GH->bdsul[lev], Symmetry);
@@ -5842,7 +5860,7 @@ void bssn_class::RestrictProlong(int lev, int YN, bool BB,
 //       misc::tillherecheck(GH->Commlev[GH->mylev],GH->start_rank[GH->mylev],a_stream.str());
 #endif
 
-      Parallel::Sync(GH->PatL[lev - 1], SL, Symmetry);
+      Parallel::Sync_cached(GH->PatL[lev - 1], SL, Symmetry, sync_cache_rp_coarse[lev]);
 
 #if (PSTR == 1 || PSTR == 2)
 //       a_stream.clear();
@@ -5852,21 +5870,11 @@ void bssn_class::RestrictProlong(int lev, int YN, bool BB,
 #endif
 
 #if (RPB == 0)
-      Ppc = GH->PatL[lev - 1];
-      while (Ppc)
-      {
-        Pp = GH->PatL[lev];
-        while (Pp)
-        {
 #if (MIXOUTB == 0)
-          Parallel::OutBdLow2Hi(Ppc->data, Pp->data, SL, SL, Symmetry);
+      Parallel::OutBdLow2Hi(GH->PatL[lev - 1], GH->PatL[lev], SL, SL, Symmetry);
 #elif (MIXOUTB == 1)
-          Parallel::OutBdLow2Himix(Ppc->data, Pp->data, SL, SL, Symmetry);
+      Parallel::OutBdLow2Himix(GH->PatL[lev - 1], GH->PatL[lev], SL, SL, Symmetry);
 #endif
-          Pp = Pp->next;
-        }
-        Ppc = Ppc->next;
-      }
 #elif (RPB == 1)
       //       Parallel::OutBdLow2Hi_bam(GH->PatL[lev-1],GH->PatL[lev],SL,SL,Symmetry);
       Parallel::OutBdLow2Hi_bam(GH->PatL[lev - 1], GH->PatL[lev], SL, SL, GH->bdsul[lev], Symmetry);
@@ -5880,7 +5888,7 @@ void bssn_class::RestrictProlong(int lev, int YN, bool BB,
 #endif
     }
 
-    Parallel::Sync(GH->PatL[lev], SL, Symmetry);
+    Parallel::Sync_cached(GH->PatL[lev], SL, Symmetry, sync_cache_rp_fine[lev]);
 
 #if (PSTR == 1 || PSTR == 2)
 //    a_stream.clear();
@@ -5938,24 +5946,14 @@ void bssn_class::RestrictProlong_aux(int lev, int YN, bool BB,
       Parallel::Restrict_bam(GH->PatL[lev - 1], GH->PatL[lev], SL, SynchList_pre, GH->rsul[lev], Symmetry);
 #endif
 
-      Parallel::Sync(GH->PatL[lev - 1], SynchList_pre, Symmetry);
+      Parallel::Sync_cached(GH->PatL[lev - 1], SynchList_pre, Symmetry, sync_cache_rp_coarse[lev]);
 
 #if (RPB == 0)
-      Ppc = GH->PatL[lev - 1];
-      while (Ppc)
-      {
-        Pp = GH->PatL[lev];
-        while (Pp)
-        {
 #if (MIXOUTB == 0)
-          Parallel::OutBdLow2Hi(Ppc->data, Pp->data, SynchList_pre, SL, Symmetry);
+      Parallel::OutBdLow2Hi(GH->PatL[lev - 1], GH->PatL[lev], SynchList_pre, SL, Symmetry);
 #elif (MIXOUTB == 1)
-          Parallel::OutBdLow2Himix(Ppc->data, Pp->data, SynchList_pre, SL, Symmetry);
+      Parallel::OutBdLow2Himix(GH->PatL[lev - 1], GH->PatL[lev], SynchList_pre, SL, Symmetry);
 #endif
-          Pp = Pp->next;
-        }
-        Ppc = Ppc->next;
-      }
 #elif (RPB == 1)
       //       Parallel::OutBdLow2Hi_bam(GH->PatL[lev-1],GH->PatL[lev],SynchList_pre,SL,Symmetry);
       Parallel::OutBdLow2Hi_bam(GH->PatL[lev - 1], GH->PatL[lev], SynchList_pre, SL, GH->bdsul[lev], Symmetry);
@@ -5970,31 +5968,21 @@ void bssn_class::RestrictProlong_aux(int lev, int YN, bool BB,
       Parallel::Restrict_bam(GH->PatL[lev - 1], GH->PatL[lev], SL, SL, GH->rsul[lev], Symmetry);
 #endif
 
-      Parallel::Sync(GH->PatL[lev - 1], SL, Symmetry);
+      Parallel::Sync_cached(GH->PatL[lev - 1], SL, Symmetry, sync_cache_rp_coarse[lev]);
 
 #if (RPB == 0)
-      Ppc = GH->PatL[lev - 1];
-      while (Ppc)
-      {
-        Pp = GH->PatL[lev];
-        while (Pp)
-        {
 #if (MIXOUTB == 0)
-          Parallel::OutBdLow2Hi(Ppc->data, Pp->data, SL, SL, Symmetry);
+      Parallel::OutBdLow2Hi(GH->PatL[lev - 1], GH->PatL[lev], SL, SL, Symmetry);
 #elif (MIXOUTB == 1)
-          Parallel::OutBdLow2Himix(Ppc->data, Pp->data, SL, SL, Symmetry);
+      Parallel::OutBdLow2Himix(GH->PatL[lev - 1], GH->PatL[lev], SL, SL, Symmetry);
 #endif
-          Pp = Pp->next;
-        }
-        Ppc = Ppc->next;
-      }
 #elif (RPB == 1)
       //       Parallel::OutBdLow2Hi_bam(GH->PatL[lev-1],GH->PatL[lev],SL,SL,Symmetry);
       Parallel::OutBdLow2Hi_bam(GH->PatL[lev - 1], GH->PatL[lev], SL, SL, GH->bdsul[lev], Symmetry);
 #endif
     }
 
-    Parallel::Sync(GH->PatL[lev], SL, Symmetry);
+    Parallel::Sync_cached(GH->PatL[lev], SL, Symmetry, sync_cache_rp_fine[lev]);
   }
 }
 
@@ -6045,24 +6033,14 @@ void bssn_class::RestrictProlong(int lev, int YN, bool BB)
       Parallel::Restrict_bam(GH->PatL[lev - 1], GH->PatL[lev], SynchList_cor, SynchList_pre, GH->rsul[lev], Symmetry);
 #endif
 
-      Parallel::Sync(GH->PatL[lev - 1], SynchList_pre, Symmetry);
+      Parallel::Sync_cached(GH->PatL[lev - 1], SynchList_pre, Symmetry, sync_cache_rp_coarse[lev]);
 
 #if (RPB == 0)
-      Ppc = GH->PatL[lev - 1];
-      while (Ppc)
-      {
-        Pp = GH->PatL[lev];
-        while (Pp)
-        {
 #if (MIXOUTB == 0)
-          Parallel::OutBdLow2Hi(Ppc->data, Pp->data, SynchList_pre, SynchList_cor, Symmetry);
+      Parallel::OutBdLow2Hi(GH->PatL[lev - 1], GH->PatL[lev], SynchList_pre, SynchList_cor, Symmetry);
 #elif (MIXOUTB == 1)
-          Parallel::OutBdLow2Himix(Ppc->data, Pp->data, SynchList_pre, SynchList_cor, Symmetry);
+      Parallel::OutBdLow2Himix(GH->PatL[lev - 1], GH->PatL[lev], SynchList_pre, SynchList_cor, Symmetry);
 #endif
-          Pp = Pp->next;
-        }
-        Ppc = Ppc->next;
-      }
 #elif (RPB == 1)
       //       Parallel::OutBdLow2Hi_bam(GH->PatL[lev-1],GH->PatL[lev],SynchList_pre,SynchList_cor,Symmetry);
       Parallel::OutBdLow2Hi_bam(GH->PatL[lev - 1], GH->PatL[lev], SynchList_pre, SynchList_cor, GH->bdsul[lev], Symmetry);
@@ -6079,31 +6057,21 @@ void bssn_class::RestrictProlong(int lev, int YN, bool BB)
       Parallel::Restrict_bam(GH->PatL[lev - 1], GH->PatL[lev], SynchList_cor, StateList, GH->rsul[lev], Symmetry);
 #endif
 
-      Parallel::Sync(GH->PatL[lev - 1], StateList, Symmetry);
+      Parallel::Sync_cached(GH->PatL[lev - 1], StateList, Symmetry, sync_cache_rp_coarse[lev]);
 
 #if (RPB == 0)
-      Ppc = GH->PatL[lev - 1];
-      while (Ppc)
-      {
-        Pp = GH->PatL[lev];
-        while (Pp)
-        {
 #if (MIXOUTB == 0)
-          Parallel::OutBdLow2Hi(Ppc->data, Pp->data, StateList, SynchList_cor, Symmetry);
+      Parallel::OutBdLow2Hi(GH->PatL[lev - 1], GH->PatL[lev], StateList, SynchList_cor, Symmetry);
 #elif (MIXOUTB == 1)
-          Parallel::OutBdLow2Himix(Ppc->data, Pp->data, StateList, SynchList_cor, Symmetry);
+      Parallel::OutBdLow2Himix(GH->PatL[lev - 1], GH->PatL[lev], StateList, SynchList_cor, Symmetry);
 #endif
-          Pp = Pp->next;
-        }
-        Ppc = Ppc->next;
-      }
 #elif (RPB == 1)
       //       Parallel::OutBdLow2Hi_bam(GH->PatL[lev-1],GH->PatL[lev],StateList,SynchList_cor,Symmetry);
       Parallel::OutBdLow2Hi_bam(GH->PatL[lev - 1], GH->PatL[lev], StateList, SynchList_cor, GH->bdsul[lev], Symmetry);
 #endif
     }
 
-    Parallel::Sync(GH->PatL[lev], SynchList_cor, Symmetry);
+    Parallel::Sync_cached(GH->PatL[lev], SynchList_cor, Symmetry, sync_cache_rp_fine[lev]);
   }
 }
 
@@ -6133,21 +6101,11 @@ void bssn_class::ProlongRestrict(int lev, int YN, bool BB)
       }
 
 #if (RPB == 0)
-      Ppc = GH->PatL[lev - 1];
-      while (Ppc)
-      {
-        Pp = GH->PatL[lev];
-        while (Pp)
-        {
 #if (MIXOUTB == 0)
-          Parallel::OutBdLow2Hi(Ppc->data, Pp->data, SynchList_pre, SynchList_cor, Symmetry);
+      Parallel::OutBdLow2Hi(GH->PatL[lev - 1], GH->PatL[lev], SynchList_pre, SynchList_cor, Symmetry);
 #elif (MIXOUTB == 1)
-          Parallel::OutBdLow2Himix(Ppc->data, Pp->data, SynchList_pre, SynchList_cor, Symmetry);
+      Parallel::OutBdLow2Himix(GH->PatL[lev - 1], GH->PatL[lev], SynchList_pre, SynchList_cor, Symmetry);
 #endif
-          Pp = Pp->next;
-        }
-        Ppc = Ppc->next;
-      }
 #elif (RPB == 1)
       //       Parallel::OutBdLow2Hi_bam(GH->PatL[lev-1],GH->PatL[lev],SynchList_pre,SynchList_cor,Symmetry);
       Parallel::OutBdLow2Hi_bam(GH->PatL[lev - 1], GH->PatL[lev], SynchList_pre, SynchList_cor, GH->bdsul[lev], Symmetry);
@@ -6156,21 +6114,11 @@ void bssn_class::ProlongRestrict(int lev, int YN, bool BB)
     else // no time refinement levels and for all same time levels
     {
 #if (RPB == 0)
-      Ppc = GH->PatL[lev - 1];
-      while (Ppc)
-      {
-        Pp = GH->PatL[lev];
-        while (Pp)
-        {
 #if (MIXOUTB == 0)
-          Parallel::OutBdLow2Hi(Ppc->data, Pp->data, StateList, SynchList_cor, Symmetry);
+      Parallel::OutBdLow2Hi(GH->PatL[lev - 1], GH->PatL[lev], StateList, SynchList_cor, Symmetry);
 #elif (MIXOUTB == 1)
-          Parallel::OutBdLow2Himix(Ppc->data, Pp->data, StateList, SynchList_cor, Symmetry);
+      Parallel::OutBdLow2Himix(GH->PatL[lev - 1], GH->PatL[lev], StateList, SynchList_cor, Symmetry);
 #endif
-          Pp = Pp->next;
-        }
-        Ppc = Ppc->next;
-      }
 #elif (RPB == 1)
       //       Parallel::OutBdLow2Hi_bam(GH->PatL[lev-1],GH->PatL[lev],StateList,SynchList_cor,Symmetry);
       Parallel::OutBdLow2Hi_bam(GH->PatL[lev - 1], GH->PatL[lev], StateList, SynchList_cor, GH->bdsul[lev], Symmetry);
@@ -6186,10 +6134,10 @@ void bssn_class::ProlongRestrict(int lev, int YN, bool BB)
 #else
       Parallel::Restrict_after(GH->PatL[lev - 1], GH->PatL[lev], SynchList_cor, StateList, Symmetry);
 #endif
-      Parallel::Sync(GH->PatL[lev - 1], StateList, Symmetry);
+      Parallel::Sync_cached(GH->PatL[lev - 1], StateList, Symmetry, sync_cache_rp_coarse[lev]);
     }
 
-    Parallel::Sync(GH->PatL[lev], SynchList_cor, Symmetry);
+    Parallel::Sync_cached(GH->PatL[lev], SynchList_cor, Symmetry, sync_cache_rp_fine[lev]);
   }
 }
 #undef MIXOUTB

AMSS_NCKU_source/bssn_class.h

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

AMSS_NCKU_source/bssn_rhs.f90

@@ -161,8 +161,36 @@
 
   chi_rhs = F2o3 *chin1*( alpn1 * trK - div_beta ) !rhs for chi
 
+  call fderivs(ex,dxx,gxxx,gxxy,gxxz,X,Y,Z,SYM ,SYM ,SYM ,Symmetry,Lev)
+  call fderivs(ex,gxy,gxyx,gxyy,gxyz,X,Y,Z,ANTI,ANTI,SYM ,Symmetry,Lev)
+  call fderivs(ex,gxz,gxzx,gxzy,gxzz,X,Y,Z,ANTI,SYM ,ANTI,Symmetry,Lev)
+  call fderivs(ex,dyy,gyyx,gyyy,gyyz,X,Y,Z,SYM ,SYM ,SYM ,Symmetry,Lev)
+  call fderivs(ex,gyz,gyzx,gyzy,gyzz,X,Y,Z,SYM ,ANTI,ANTI,Symmetry,Lev)
+  call fderivs(ex,dzz,gzzx,gzzy,gzzz,X,Y,Z,SYM ,SYM ,SYM ,Symmetry,Lev)
 
+  gxx_rhs = - TWO * alpn1 * Axx    -  F2o3 * gxx * div_beta          + &
+              TWO *(  gxx * betaxx +   gxy * betayx +   gxz * betazx)
 
+  gyy_rhs = - TWO * alpn1 * Ayy    -  F2o3 * gyy * div_beta          + &
+              TWO *(  gxy * betaxy +   gyy * betayy +   gyz * betazy)
+
+  gzz_rhs = - TWO * alpn1 * Azz    -  F2o3 * gzz * div_beta          + &
+              TWO *(  gxz * betaxz +   gyz * betayz +   gzz * betazz)
+
+  gxy_rhs = - TWO * alpn1 * Axy    +  F1o3 * gxy    * div_beta       + &
+                      gxx * betaxy                  +   gxz * betazy + &
+                                       gyy * betayx +   gyz * betazx   &
+                                                    -   gxy * betazz
+
+  gyz_rhs = - TWO * alpn1 * Ayz    +  F1o3 * gyz    * div_beta       + &
+                      gxy * betaxz +   gyy * betayz                  + &
+                      gxz * betaxy                  +   gzz * betazy   &
+                                                    -   gyz * betaxx
+ 
+  gxz_rhs = - TWO * alpn1 * Axz    +  F1o3 * gxz    * div_beta       + &
+                      gxx * betaxz +   gxy * betayz                  + &
+                                       gyz * betayx +   gzz * betazx   &
+                                                    -   gxz * betayy     !rhs for gij
 
 ! invert tilted metric
   gupzz =  gxx * gyy * gzz + gxy * gyz * gxz + gxz * gxy * gyz - &
@@ -173,12 +201,7 @@
   gupyy =   ( gxx * gzz - gxz * gxz ) / gupzz
   gupyz = - ( gxx * gyz - gxy * gxz ) / gupzz
   gupzz =   ( gxx * gyy - gxy * gxy ) / gupzz
-  call fderivs(ex,dxx,gxxx,gxxy,gxxz,X,Y,Z,SYM ,SYM ,SYM ,Symmetry,Lev)
+
-  call fderivs(ex,gxy,gxyx,gxyy,gxyz,X,Y,Z,ANTI,ANTI,SYM ,Symmetry,Lev)
-  call fderivs(ex,gxz,gxzx,gxzy,gxzz,X,Y,Z,ANTI,SYM ,ANTI,Symmetry,Lev)
-  call fderivs(ex,dyy,gyyx,gyyy,gyyz,X,Y,Z,SYM ,SYM ,SYM ,Symmetry,Lev)
-  call fderivs(ex,gyz,gyzx,gyzy,gyzz,X,Y,Z,SYM ,ANTI,ANTI,Symmetry,Lev)
-  call fderivs(ex,dzz,gzzx,gzzy,gzzz,X,Y,Z,SYM ,SYM ,SYM ,Symmetry,Lev)
   if(co == 0)then
 ! Gam^i_Res = Gam^i + gup^ij_,j
   Gmx_Res = Gamx - (gupxx*(gupxx*gxxx+gupxy*gxyx+gupxz*gxzx)&
@@ -922,103 +945,60 @@
   SSA(2)=SYM
   SSA(3)=ANTI
 
-!!!!!!!!!advection term part
+!!!!!!!!!advection term + Kreiss-Oliger dissipation (merged for cache efficiency)
+! lopsided_kodis shares the symmetry_bd buffer between advection and
+! dissipation, eliminating redundant full-grid copies. For metric variables
+! gxx/gyy/gzz (=dxx/dyy/dzz+1): kodis stencil coefficients sum to zero,
+! so the constant offset has no effect on dissipation.
 
-  gxx_rhs = - TWO * alpn1 * Axx    -  F2o3 * gxx * div_beta          + &
+  call lopsided_kodis(ex,X,Y,Z,gxx,gxx_rhs,betax,betay,betaz,Symmetry,SSS,eps)
-              TWO *(  gxx * betaxx +   gxy * betayx +   gxz * betazx)
+  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)
 
-  gyy_rhs = - TWO * alpn1 * Ayy    -  F2o3 * gyy * div_beta          + &
+  call lopsided_kodis(ex,X,Y,Z,Axx,Axx_rhs,betax,betay,betaz,Symmetry,SSS,eps)
-              TWO *(  gxy * betaxy +   gyy * betayy +   gyz * betazy)
+  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)
 
-  gzz_rhs = - TWO * alpn1 * Azz    -  F2o3 * gzz * div_beta          + &
+  call lopsided_kodis(ex,X,Y,Z,chi,chi_rhs,betax,betay,betaz,Symmetry,SSS,eps)
-              TWO *(  gxz * betaxz +   gyz * betayz +   gzz * betazz)
+  call lopsided_kodis(ex,X,Y,Z,trK,trK_rhs,betax,betay,betaz,Symmetry,SSS,eps)
 
-  gxy_rhs = - TWO * alpn1 * Axy    +  F1o3 * gxy    * div_beta       + &
+  call lopsided_kodis(ex,X,Y,Z,Gamx,Gamx_rhs,betax,betay,betaz,Symmetry,ASS,eps)
-                      gxx * betaxy                  +   gxz * betazy + &
+  call lopsided_kodis(ex,X,Y,Z,Gamy,Gamy_rhs,betax,betay,betaz,Symmetry,SAS,eps)
-                                        gyy * betayx +   gyz * betazx   &
+  call lopsided_kodis(ex,X,Y,Z,Gamz,Gamz_rhs,betax,betay,betaz,Symmetry,SSA,eps)
-                                                    -   gxy * betazz
 
-  gyz_rhs = - TWO * alpn1 * Ayz    +  F1o3 * gyz    * div_beta       + &
+#if 1 
-                      gxy * betaxz +   gyy * betayz                  + &
+!! bam does not apply dissipation on gauge variables
-                      gxz * betaxy                  +   gzz * betazy   &
+  call lopsided_kodis(ex,X,Y,Z,Lap,Lap_rhs,betax,betay,betaz,Symmetry,SSS,eps)
-                                                    -   gyz * betaxx
+#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)
-  gxz_rhs = - TWO * alpn1 * Axz    +  F1o3 * gxz    * div_beta       + &
+  call lopsided_kodis(ex,X,Y,Z,betay,betay_rhs,betax,betay,betaz,Symmetry,SAS,eps)
-                      gxx * betaxz +   gxy * betayz                  + &
+  call lopsided_kodis(ex,X,Y,Z,betaz,betaz_rhs,betax,betay,betaz,Symmetry,SSA,eps)
-                                        gyz * betayx +   gzz * betazx   &
+#endif
-                                                    -   gxz * betayy     !rhs for gij
+#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
-  if(eps>0)then 
+! No dissipation on gauge variables (advection only)
-! usual Kreiss-Oliger dissipation     
-  call merge_lopsided_kodis(ex,X,Y,Z,chi,chi_rhs,betax,betay,betaz,Symmetry,SSS,eps) 
-  call merge_lopsided_kodis(ex,X,Y,Z,gxx,gxx_rhs,betax,betay,betaz,Symmetry,SSS,eps)
-  call merge_lopsided_kodis(ex,X,Y,Z,gxy,gxy_rhs,betax,betay,betaz,Symmetry,AAS,eps)
-  call merge_lopsided_kodis(ex,X,Y,Z,gxz,gxz_rhs,betax,betay,betaz,Symmetry,ASA,eps)
-  call merge_lopsided_kodis(ex,X,Y,Z,gyy,gyy_rhs,betax,betay,betaz,Symmetry,SSS,eps)
-  call merge_lopsided_kodis(ex,X,Y,Z,gyz,gyz_rhs,betax,betay,betaz,Symmetry,SAA,eps)
-  call merge_lopsided_kodis(ex,X,Y,Z,gzz,gzz_rhs,betax,betay,betaz,Symmetry,SSS,eps)
-  call merge_lopsided_kodis(ex,X,Y,Z,Axx,Axx_rhs,betax,betay,betaz,Symmetry,SSS,eps)
-  call merge_lopsided_kodis(ex,X,Y,Z,Axy,Axy_rhs,betax,betay,betaz,Symmetry,AAS,eps)
-  call merge_lopsided_kodis(ex,X,Y,Z,Axz,Axz_rhs,betax,betay,betaz,Symmetry,ASA,eps)
-  call merge_lopsided_kodis(ex,X,Y,Z,Ayy,Ayy_rhs,betax,betay,betaz,Symmetry,SSS,eps)
-  call merge_lopsided_kodis(ex,X,Y,Z,Ayz,Ayz_rhs,betax,betay,betaz,Symmetry,SAA,eps)
-  call merge_lopsided_kodis(ex,X,Y,Z,Azz,Azz_rhs,betax,betay,betaz,Symmetry,SSS,eps)
-  call merge_lopsided_kodis(ex,X,Y,Z,chi,chi_rhs,betax,betay,betaz,Symmetry,SSS,eps)
-  call merge_lopsided_kodis(ex,X,Y,Z,trK,trK_rhs,betax,betay,betaz,Symmetry,SSS,eps)
-  call merge_lopsided_kodis(ex,X,Y,Z,Gamx,Gamx_rhs,betax,betay,betaz,Symmetry,ASS,eps)
-  call merge_lopsided_kodis(ex,X,Y,Z,Gamy,Gamy_rhs,betax,betay,betaz,Symmetry,SAS,eps)
-  call merge_lopsided_kodis(ex,X,Y,Z,Gamz,Gamz_rhs,betax,betay,betaz,Symmetry,SSA,eps)
-  call merge_lopsided_kodis(ex,X,Y,Z,Lap,Lap_rhs,betax,betay,betaz,Symmetry,SSS,eps)
-  call merge_lopsided_kodis(ex,X,Y,Z,betax,betax_rhs,betax,betay,betaz,Symmetry,ASS,eps)
-  call merge_lopsided_kodis(ex,X,Y,Z,betay,betay_rhs,betax,betay,betaz,Symmetry,SAS,eps)
-  call merge_lopsided_kodis(ex,X,Y,Z,betaz,betaz_rhs,betax,betay,betaz,Symmetry,SSA,eps)
-  call merge_lopsided_kodis(ex,X,Y,Z,dtSfx,dtSfx_rhs,betax,betay,betaz,Symmetry,ASS,eps)
-  call merge_lopsided_kodis(ex,X,Y,Z,dtSfy,dtSfy_rhs,betax,betay,betaz,Symmetry,SAS,eps)
-  call merge_lopsided_kodis(ex,X,Y,Z,dtSfz,dtSfz_rhs,betax,betay,betaz,Symmetry,SSA,eps)
-
-
-
-
-
-
-
-
-
-
-
-
-  else 
-  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(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(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(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
-
+#endif
-  endif
 
   if(co == 0)then
 ! ham_Res = trR + 2/3 * K^2 - A_ij * A^ij - 16 * PI * rho
@@ -1163,265 +1143,3 @@ endif
   return
 
   end function compute_rhs_bssn
-
-
-
-
-  subroutine merge_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
-
-  !~~~~~~> 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_lopsided,jmin_lopsided,kmin_lopsided,imin_kodis,jmin_kodis,kmin_kodis,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
-    real*8, parameter :: SIX=6.d0,FIT=1.5d1,TWT=2.d1
-    real*8,parameter::cof=6.4d1   ! 2^6
-    real*8,intent(in) :: eps
-    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_lopsided = 1
-    jmin_lopsided = 1
-    kmin_lopsided = 1
-    if(Symmetry > NO_SYMM .and. dabs(Z(1)) < dZ) kmin_lopsided = -2
-    if(Symmetry > EQ_SYMM .and. dabs(X(1)) < dX) imin_lopsided = -2
-    if(Symmetry > EQ_SYMM .and. dabs(Y(1)) < dY) jmin_lopsided = -2
-
-    imin_kodis = 1
-    jmin_kodis = 1
-    kmin_kodis = 1
-
-    if(Symmetry > NO_SYMM .and. dabs(Z(1)) < dZ) kmin_kodis = -2
-    if(Symmetry == OCTANT .and. dabs(X(1)) < dX) imin_kodis = -2
-    if(Symmetry == OCTANT .and. dabs(Y(1)) < dY) jmin_kodis = -2
-
-
-    call symmetry_bd(3,ex,f,fh,SoA)
-
-  ! 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
-
-  !! new code, 2012dec27, based on bam
-  ! x direction   
-      if(Sfx(i,j,k) > ZEO)then
-        if(i+3 <= imax)then
-  !         v
-  ! D f = ------[ - 3f    - 10f  + 18f    - 6f     + f     ]
-  !  i     12dx       i-v      i      i+v     i+2v    i+3v
-      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(i-2) - 8 f(i-1) + 8 f(i+1) - f(i+2)
-  !  fx(i) = ---------------------------------------------
-  !                             12 dx
-      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
-  !         v
-  ! D f = ------[   3f    + 10f  - 18f    + 6f     - f     ]
-  !  i     12dx       i+v      i      i-v     i-2v    i-3v
-      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))
-  ! set imax and imin_lopsided 0
-      endif
-    elseif(Sfx(i,j,k) < ZEO)then
-        if(i-3 >= imin_lopsided)then
-  !         v
-  ! D f = ------[ - 3f    - 10f  + 18f    - 6f     + f     ]
-  !  i     12dx       i-v      i      i+v     i+2v    i+3v
-      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_lopsided)then
-  !
-  !              f(i-2) - 8 f(i-1) + 8 f(i+1) - f(i+2)
-  !  fx(i) = ---------------------------------------------
-  !                             12 dx
-      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_lopsided)then
-  !         v
-  ! D f = ------[   3f    + 10f  - 18f    + 6f     - f     ]
-  !  i     12dx       i+v      i      i-v     i-2v    i-3v
-      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))
-  ! set imax and imin_lopsided 0
-      endif
-    endif
-
-  ! y direction   
-      if(Sfy(i,j,k) > ZEO)then
-        if(j+3 <= jmax)then
-  !         v
-  ! D f = ------[ - 3f    - 10f  + 18f    - 6f     + f     ]
-  !  i     12dx       i-v      i      i+v     i+2v    i+3v
-      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(i-2) - 8 f(i-1) + 8 f(i+1) - f(i+2)
-  !  fx(i) = ---------------------------------------------
-  !                             12 dx
-      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
-  !         v
-  ! D f = ------[   3f    + 10f  - 18f    + 6f     - f     ]
-  !  i     12dx       i+v      i      i-v     i-2v    i-3v
-      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))
-  ! set imax and imin_lopsided 0
-      endif
-    elseif(Sfy(i,j,k) < ZEO)then
-        if(j-3 >= jmin_lopsided)then
-  !         v
-  ! D f = ------[ - 3f    - 10f  + 18f    - 6f     + f     ]
-  !  i     12dx       i-v      i      i+v     i+2v    i+3v
-      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_lopsided)then
-  !
-  !              f(i-2) - 8 f(i-1) + 8 f(i+1) - f(i+2)
-  !  fx(i) = ---------------------------------------------
-  !                             12 dx
-      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_lopsided)then
-  !         v
-  ! D f = ------[   3f    + 10f  - 18f    + 6f     - f     ]
-  !  i     12dx       i+v      i      i-v     i-2v    i-3v
-      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))
-  ! set jmax and jmin_lopsided 0
-      endif
-    endif
-
-  ! z direction   
-      if(Sfz(i,j,k) > ZEO)then
-        if(k+3 <= kmax)then
-  !         v
-  ! D f = ------[ - 3f    - 10f  + 18f    - 6f     + f     ]
-  !  i     12dx       i-v      i      i+v     i+2v    i+3v
-      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(i-2) - 8 f(i-1) + 8 f(i+1) - f(i+2)
-  !  fx(i) = ---------------------------------------------
-  !                             12 dx
-      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
-  !         v
-  ! D f = ------[   3f    + 10f  - 18f    + 6f     - f     ]
-  !  i     12dx       i+v      i      i-v     i-2v    i-3v
-      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))
-  ! set imax and imin_lopsided 0
-      endif
-    elseif(Sfz(i,j,k) < ZEO)then
-        if(k-3 >= kmin_lopsided)then
-  !         v
-  ! D f = ------[ - 3f    - 10f  + 18f    - 6f     + f     ]
-  !  i     12dx       i-v      i      i+v     i+2v    i+3v
-      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_lopsided)then
-  !
-  !              f(i-2) - 8 f(i-1) + 8 f(i+1) - f(i+2)
-  !  fx(i) = ---------------------------------------------
-  !                             12 dx
-      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_lopsided)then
-  !         v
-  ! D f = ------[   3f    + 10f  - 18f    + 6f     - f     ]
-  !  i     12dx       i+v      i      i-v     i-2v    i-3v
-      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))
-  ! set kmax and kmin_lopsided 0
-      endif
-    endif
-
-
-    if(i-3 >= imin_kodis .and. i+3 <= imax .and. &
-      j-3 >= jmin_kodis .and. j+3 <= jmax .and. &
-      k-3 >= kmin_kodis .and. k+3 <= kmax) then
-
-  ! calculation order if important ?
-    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
-
-    return
-
-
-
-  end subroutine merge_lopsided_kodis

AMSS_NCKU_source/cgh.h

@@ -1,92 +1,107 @@
-
+
-#ifndef CGH_H
+#ifndef CGH_H
-#define CGH_H
+#define CGH_H
-
+
-#include <mpi.h>
+#include <mpi.h>
-#include "MyList.h"
+#include "MyList.h"
-#include "MPatch.h"
+#include "MPatch.h"
-#include "macrodef.h"
+#include "macrodef.h"
-#include "monitor.h"
+#include "monitor.h"
-#include "Parallel.h"
+#include "Parallel.h"
-
+
-class cgh
+class cgh
-{
+{
-
+
-public:
+public:
-   int levels, movls, BH_num_in;
+   int levels, movls, BH_num_in;
-   // information of boxes
+   // information of boxes
-   int *grids;
+   int *grids;
-   double ***bbox;
+   double ***bbox;
-   int ***shape;
+   int ***shape;
-   double ***handle;
+   double ***handle;
-   double ***Porgls;
+   double ***Porgls;
-   double *Lt;
+   double *Lt;
-
+
-   // information of Patch list
+   // information of Patch list
-   MyList<Patch> **PatL;
+   MyList<Patch> **PatL;
-
+
-// information of OutBdLow2Hi point list and Restrict point list
+// information of OutBdLow2Hi point list and Restrict point list
-#if (RPB == 1)
+#if (RPB == 1)
-   MyList<Parallel::pointstru_bam> **bdsul, **rsul;
+   MyList<Parallel::pointstru_bam> **bdsul, **rsul;
-#endif
+#endif
-
+
-#if (PSTR == 1 || PSTR == 2 || PSTR == 3)
+#if (PSTR == 1 || PSTR == 2 || PSTR == 3)
-   int mylev;
+   int mylev;
-   int *start_rank, *end_rank;
+   int *start_rank, *end_rank;
-   MPI_Comm *Commlev;
+   MPI_Comm *Commlev;
-#endif
+#endif
-
+
-protected:
+protected:
-   int ingfs, fngfs;
+   int ingfs, fngfs;
-   static constexpr double ratio = 0.75;
+   static constexpr double ratio = 0.75;
-   int trfls;
+   int trfls;
-
+
-public:
+public:
-   cgh(int ingfsi, int fngfsi, int Symmetry, char *filename, int checkrun, monitor *ErrorMonitor);
+   cgh(int ingfsi, int fngfsi, int Symmetry, char *filename, int checkrun, monitor *ErrorMonitor);
-
+
-   ~cgh();
+   ~cgh();
-
+
-   void compose_cgh(int nprocs);
+   void compose_cgh(int nprocs);
-   void sethandle(monitor *ErrorMonitor);
+   void sethandle(monitor *ErrorMonitor);
-   void checkPatchList(MyList<Patch> *PatL, bool buflog);
+   void checkPatchList(MyList<Patch> *PatL, bool buflog);
-   void Regrid(int Symmetry, int BH_num, double **Porgbr, double **Porg0,
+   void Regrid(int Symmetry, int BH_num, double **Porgbr, double **Porg0,
-               MyList<var> *OldList, MyList<var> *StateList,
+               MyList<var> *OldList, MyList<var> *StateList,
-               MyList<var> *FutureList, MyList<var> *tmList, bool BB,
+               MyList<var> *FutureList, MyList<var> *tmList, bool BB,
-               monitor *ErrorMonitor);
+               monitor *ErrorMonitor);
-   void Regrid_fake(int Symmetry, int BH_num, double **Porgbr, double **Porg0,
+   void Regrid_fake(int Symmetry, int BH_num, double **Porgbr, double **Porg0,
-                    MyList<var> *OldList, MyList<var> *StateList,
+                    MyList<var> *OldList, MyList<var> *StateList,
-                    MyList<var> *FutureList, MyList<var> *tmList, bool BB,
+                    MyList<var> *FutureList, MyList<var> *tmList, bool BB,
-                    monitor *ErrorMonitor);
+                    monitor *ErrorMonitor);
-   void recompose_cgh(int nprocs, bool *lev_flag,
+   void recompose_cgh(int nprocs, bool *lev_flag,
-                      MyList<var> *OldList, MyList<var> *StateList,
+                      MyList<var> *OldList, MyList<var> *StateList,
-                      MyList<var> *FutureList, MyList<var> *tmList,
+                      MyList<var> *FutureList, MyList<var> *tmList,
-                      int Symmetry, bool BB);
+                      int Symmetry, bool BB);
-   void recompose_cgh_fake(int nprocs, bool *lev_flag,
+   void recompose_cgh_fake(int nprocs, bool *lev_flag,
-                           MyList<var> *OldList, MyList<var> *StateList,
+                           MyList<var> *OldList, MyList<var> *StateList,
-                           MyList<var> *FutureList, MyList<var> *tmList,
+                           MyList<var> *FutureList, MyList<var> *tmList,
-                           int Symmetry, bool BB);
+                           int Symmetry, bool BB);
-   void read_bbox(int Symmetry, char *filename);
+   void read_bbox(int Symmetry, char *filename);
-   MyList<Patch> *construct_patchlist(int lev, int Symmetry);
+   MyList<Patch> *construct_patchlist(int lev, int Symmetry);
-   bool Interp_One_Point(MyList<var> *VarList,
+   bool Interp_One_Point(MyList<var> *VarList,
-                         double *XX, /*input global Cartesian coordinate*/
+                         double *XX, /*input global Cartesian coordinate*/
-                         double *Shellf, int Symmetry);
+                         double *Shellf, int Symmetry);
-   void recompose_cgh_Onelevel(int nprocs, int lev,
+   void recompose_cgh_Onelevel(int nprocs, int lev,
-                               MyList<var> *OldList, MyList<var> *StateList,
+                               MyList<var> *OldList, MyList<var> *StateList,
-                               MyList<var> *FutureList, MyList<var> *tmList,
+                               MyList<var> *FutureList, MyList<var> *tmList,
-                               int Symmetry, bool BB);
+                               int Symmetry, bool BB);
-   void Regrid_Onelevel(int lev, int Symmetry, int BH_num, double **Porgbr, double **Porg0,
+   void Regrid_Onelevel(int lev, int Symmetry, int BH_num, double **Porgbr, double **Porg0,
-                        MyList<var> *OldList, MyList<var> *StateList,
+                        MyList<var> *OldList, MyList<var> *StateList,
-                        MyList<var> *FutureList, MyList<var> *tmList, bool BB,
+                        MyList<var> *FutureList, MyList<var> *tmList, bool BB,
-                        monitor *ErrorMonitor);
+                        monitor *ErrorMonitor);
-   void Regrid_Onelevel_aux(int lev, int Symmetry, int BH_num, double **Porgbr, double **Porg0,
+   void Regrid_Onelevel_aux(int lev, int Symmetry, int BH_num, double **Porgbr, double **Porg0,
-                            MyList<var> *OldList, MyList<var> *StateList,
+                            MyList<var> *OldList, MyList<var> *StateList,
-                            MyList<var> *FutureList, MyList<var> *tmList, bool BB,
+                            MyList<var> *FutureList, MyList<var> *tmList, bool BB,
-                            monitor *ErrorMonitor);
+                            monitor *ErrorMonitor);
-   void settrfls(const int lev);
+   void settrfls(const int lev);
-
+
-#if (PSTR == 1 || PSTR == 2 || PSTR == 3)
+#if (PSTR == 1 || PSTR == 2 || PSTR == 3)
-   void construct_mylev(int nprocs);
+   void construct_mylev(int nprocs);
-#endif
+#endif
-};
+
-
+   // Load balancing support
-#endif /* CGH_H */
+   bool enable_load_balance;         // Enable load balancing
+   int load_balance_check_interval;  // Check interval (in time steps)
+   int current_time_step;            // Current time step counter
+   double *rank_interp_times;        // Store interpolation times for each rank
+   int *heavy_ranks;                 // Store heavy rank numbers
+   int num_heavy_ranks;              // Number of heavy ranks
+
+   void init_load_balance(int nprocs);
+   void update_interp_time(int rank, double time);
+   bool check_and_rebalance(int nprocs, int lev,
+                           MyList<var> *OldList, MyList<var> *StateList,
+                           MyList<var> *FutureList, MyList<var> *tmList,
+                           int Symmetry, bool BB);
+};
+
+#endif /* CGH_H */

AMSS_NCKU_source/diff_new.f90

@@ -69,10 +69,12 @@
   fy = ZEO
   fz = ZEO
 
+!DIR$ SIMD VECTORLENGTHFOR(KNOWN_INTEGER=8)
+!DIR$ UNROLL PARTIAL(4)
   do k=1,ex(3)-1
   do j=1,ex(2)-1
   do i=1,ex(1)-1
-! x direction   
+! x direction
         if(i+1 <= imax .and. i-1 >= imin)then
 !
 !              - f(i-1) + f(i+1)
@@ -371,6 +373,8 @@
   fxz = ZEO
   fyz = ZEO
 
+!DIR$ SIMD VECTORLENGTHFOR(KNOWN_INTEGER=8)
+!DIR$ UNROLL PARTIAL(4)
   do k=1,ex(3)-1
   do j=1,ex(2)-1
   do i=1,ex(1)-1
@@ -1000,7 +1004,86 @@
   do k=1,ex(3)-1
   do j=1,ex(2)-1
   do i=1,ex(1)-1
+#if 0  
+! x direction   
+        if(i+2 <= imax .and. i-2 >= imin)then
+!
+!              f(i-2) - 8 f(i-1) + 8 f(i+1) - f(i+2)
+!  fx(i) = ---------------------------------------------
+!                             12 dx
+      fx(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 .and. i-1 >= imin)then
+!
+!              - f(i-1) + f(i+1)
+!  fx(i) = --------------------------------
+!                     2 dx
+      fx(i,j,k)=d2dx*(-fh(i-1,j,k)+fh(i+1,j,k))
+
+! set imax and imin 0
+    endif
+! y direction   
+        if(j+2 <= jmax .and. j-2 >= jmin)then
+
+      fy(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 .and. j-1 >= jmin)then
+
+     fy(i,j,k)=d2dy*(-fh(i,j-1,k)+fh(i,j+1,k))
+
+! set jmax and jmin 0
+    endif
+! z direction   
+        if(k+2 <= kmax .and. k-2 >= kmin)then
+
+      fz(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 .and. k-1 >= kmin)then
+
+      fz(i,j,k)=d2dz*(-fh(i,j,k-1)+fh(i,j,k+1))
+
+! set kmax and kmin 0
+    endif
+#elif 0
+! x direction   
+        if(i+2 <= imax .and. i-2 >= imin)then
+!
+!              f(i-2) - 8 f(i-1) + 8 f(i+1) - f(i+2)
+!  fx(i) = ---------------------------------------------
+!                             12 dx
+      fx(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+3 <= imax .and. i-1 >= imin)then
+      fx(i,j,k)=d12dx*(-3.d0*fh(i-1,j,k)-1.d1*fh(i,j,k)+1.8d1*fh(i+1,j,k)-6.d0*fh(i+2,j,k)+fh(i+3,j,k))
+    elseif(i+1 <= imax .and. i-3 >= imin)then
+      fx(i,j,k)=d12dx*( 3.d0*fh(i+1,j,k)+1.d1*fh(i,j,k)-1.8d1*fh(i-1,j,k)+6.d0*fh(i-2,j,k)-fh(i-3,j,k))
+! set imax and imin 0
+    endif
+! y direction   
+        if(j+2 <= jmax .and. j-2 >= jmin)then
+
+      fy(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+3 <= jmax .and. j-1 >= jmin)then
+      fy(i,j,k)=d12dy*(-3.d0*fh(i,j-1,k)-1.d1*fh(i,j,k)+1.8d1*fh(i,j+1,k)-6.d0*fh(i,j+2,k)+fh(i,j+3,k))
+    elseif(j+1 <= jmax .and. j-3 >= jmin)then
+      fy(i,j,k)=d12dy*( 3.d0*fh(i,j+1,k)+1.d1*fh(i,j,k)-1.8d1*fh(i,j-1,k)+6.d0*fh(i,j-2,k)-fh(i,j-3,k))
+
+! set jmax and jmin 0
+    endif
+! z direction   
+        if(k+2 <= kmax .and. k-2 >= kmin)then
+
+      fz(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+3 <= kmax .and. k-1 >= kmin)then
+      fz(i,j,k)=d12dz*(-3.d0*fh(i,j,k-1)-1.d1*fh(i,j,k)+1.8d1*fh(i,j,k+1)-6.d0*fh(i,j,k+2)+fh(i,j,k+3))
+    elseif(k+1 <= kmax .and. k-3 >= kmin)then
+      fz(i,j,k)=d12dz*( 3.d0*fh(i,j,k+1)+1.d1*fh(i,j,k)-1.8d1*fh(i,j,k-1)+6.d0*fh(i,j,k-2)-fh(i,j,k-3))
+
+! set kmax and kmin 0
+    endif
+#else
 ! for bam comparison
    if(i+2 <= imax .and. i-2 >= imin .and. &
       j+2 <= jmax .and. j-2 >= jmin .and. &
@@ -1015,7 +1098,7 @@
       fy(i,j,k)=d2dy*(-fh(i,j-1,k)+fh(i,j+1,k))
       fz(i,j,k)=d2dz*(-fh(i,j,k-1)+fh(i,j,k+1))
    endif
-
+#endif
   enddo
   enddo
   enddo
@@ -1325,7 +1408,85 @@
   do k=1,ex(3)-1
   do j=1,ex(2)-1
   do i=1,ex(1)-1
+#if 0  
+!~~~~~~ fxx
+        if(i+2 <= imax .and. i-2 >= imin)then
+!
+!               - f(i-2) + 16 f(i-1) - 30 f(i) + 16 f(i+1) - f(i+2)
+!  fxx(i) = ----------------------------------------------------------
+!                                  12 dx^2 
+   fxx(i,j,k) = Fdxdx*(-fh(i-2,j,k)+F16*fh(i-1,j,k)-F30*fh(i,j,k) &
+                       -fh(i+2,j,k)+F16*fh(i+1,j,k)              )
+   elseif(i+1 <= imax .and. i-1 >= imin)then
+!
+!               f(i-1) - 2 f(i) + f(i+1)
+!  fxx(i) = --------------------------------
+!                         dx^2 
+   fxx(i,j,k) = Sdxdx*(fh(i-1,j,k)-TWO*fh(i,j,k) &
+                      +fh(i+1,j,k)              )
+   endif
 
+
+!~~~~~~ fyy
+        if(j+2 <= jmax .and. j-2 >= jmin)then
+
+   fyy(i,j,k) = Fdydy*(-fh(i,j-2,k)+F16*fh(i,j-1,k)-F30*fh(i,j,k) &
+                       -fh(i,j+2,k)+F16*fh(i,j+1,k)              )
+   elseif(j+1 <= jmax .and. j-1 >= jmin)then
+
+   fyy(i,j,k) = Sdydy*(fh(i,j-1,k)-TWO*fh(i,j,k) &
+                      +fh(i,j+1,k)              )
+   endif
+
+!~~~~~~ fzz
+        if(k+2 <= kmax .and. k-2 >= kmin)then
+
+   fzz(i,j,k) = Fdzdz*(-fh(i,j,k-2)+F16*fh(i,j,k-1)-F30*fh(i,j,k) &
+                       -fh(i,j,k+2)+F16*fh(i,j,k+1)              )
+   elseif(k+1 <= kmax .and. k-1 >= kmin)then
+
+   fzz(i,j,k) = Sdzdz*(fh(i,j,k-1)-TWO*fh(i,j,k) &
+                      +fh(i,j,k+1)              )
+   endif
+!~~~~~~ fxy
+       if(i+2 <= imax .and. i-2 >= imin .and. j+2 <= jmax .and. j-2 >= jmin)then
+!
+!                 ( f(i-2,j-2) - 8 f(i-1,j-2) + 8 f(i+1,j-2) - f(i+2,j-2) )
+!             - 8 ( f(i-2,j-1) - 8 f(i-1,j-1) + 8 f(i+1,j-1) - f(i+2,j-1) )
+!             + 8 ( f(i-2,j+1) - 8 f(i-1,j+1) + 8 f(i+1,j+1) - f(i+2,j+1) )
+!             -   ( f(i-2,j+2) - 8 f(i-1,j+2) + 8 f(i+1,j+2) - f(i+2,j+2) )
+!  fxy(i,j) = ----------------------------------------------------------------
+!                                  144 dx dy
+   fxy(i,j,k) = Fdxdy*(     (fh(i-2,j-2,k)-F8*fh(i-1,j-2,k)+F8*fh(i+1,j-2,k)-fh(i+2,j-2,k))  &
+                       -F8 *(fh(i-2,j-1,k)-F8*fh(i-1,j-1,k)+F8*fh(i+1,j-1,k)-fh(i+2,j-1,k))  &
+                       +F8 *(fh(i-2,j+1,k)-F8*fh(i-1,j+1,k)+F8*fh(i+1,j+1,k)-fh(i+2,j+1,k))  &
+                       -    (fh(i-2,j+2,k)-F8*fh(i-1,j+2,k)+F8*fh(i+1,j+2,k)-fh(i+2,j+2,k)))
+
+   elseif(i+1 <= imax .and. i-1 >= imin .and. j+1 <= jmax .and. j-1 >= jmin)then
+!                 f(i-1,j-1) - f(i+1,j-1) - f(i-1,j+1) + f(i+1,j+1) 
+!  fxy(i,j) = -----------------------------------------------------------
+!                                      4 dx dy
+   fxy(i,j,k) = Sdxdy*(fh(i-1,j-1,k)-fh(i+1,j-1,k)-fh(i-1,j+1,k)+fh(i+1,j+1,k))
+   endif
+!~~~~~~ fxz
+       if(i+2 <= imax .and. i-2 >= imin .and. k+2 <= kmax .and. k-2 >= kmin)then
+   fxz(i,j,k) = Fdxdz*(     (fh(i-2,j,k-2)-F8*fh(i-1,j,k-2)+F8*fh(i+1,j,k-2)-fh(i+2,j,k-2))  &
+                       -F8 *(fh(i-2,j,k-1)-F8*fh(i-1,j,k-1)+F8*fh(i+1,j,k-1)-fh(i+2,j,k-1))  &
+                       +F8 *(fh(i-2,j,k+1)-F8*fh(i-1,j,k+1)+F8*fh(i+1,j,k+1)-fh(i+2,j,k+1))  &
+                       -    (fh(i-2,j,k+2)-F8*fh(i-1,j,k+2)+F8*fh(i+1,j,k+2)-fh(i+2,j,k+2)))
+   elseif(i+1 <= imax .and. i-1 >= imin .and. k+1 <= kmax .and. k-1 >= kmin)then
+   fxz(i,j,k) = Sdxdz*(fh(i-1,j,k-1)-fh(i+1,j,k-1)-fh(i-1,j,k+1)+fh(i+1,j,k+1))
+   endif
+!~~~~~~ fyz
+       if(j+2 <= jmax .and. j-2 >= jmin .and. k+2 <= kmax .and. k-2 >= kmin)then
+   fyz(i,j,k) = Fdydz*(     (fh(i,j-2,k-2)-F8*fh(i,j-1,k-2)+F8*fh(i,j+1,k-2)-fh(i,j+2,k-2))  &
+                       -F8 *(fh(i,j-2,k-1)-F8*fh(i,j-1,k-1)+F8*fh(i,j+1,k-1)-fh(i,j+2,k-1))  &
+                       +F8 *(fh(i,j-2,k+1)-F8*fh(i,j-1,k+1)+F8*fh(i,j+1,k+1)-fh(i,j+2,k+1))  &
+                       -    (fh(i,j-2,k+2)-F8*fh(i,j-1,k+2)+F8*fh(i,j+1,k+2)-fh(i,j+2,k+2)))
+   elseif(j+1 <= jmax .and. j-1 >= jmin .and. k+1 <= kmax .and. k-1 >= kmin)then
+   fyz(i,j,k) = Sdydz*(fh(i,j-1,k-1)-fh(i,j+1,k-1)-fh(i,j-1,k+1)+fh(i,j+1,k+1))
+   endif 
+#else
 ! for bam comparison
    if(i+2 <= imax .and. i-2 >= imin .and. &
       j+2 <= jmax .and. j-2 >= jmin .and. &
@@ -1361,7 +1522,7 @@
    fxz(i,j,k) = Sdxdz*(fh(i-1,j,k-1)-fh(i+1,j,k-1)-fh(i-1,j,k+1)+fh(i+1,j,k+1))
    fyz(i,j,k) = Sdydz*(fh(i,j-1,k-1)-fh(i,j+1,k-1)-fh(i,j-1,k+1)+fh(i,j+1,k+1))
    endif
-
+#endif
    enddo
    enddo
    enddo

AMSS_NCKU_source/fmisc.f90

@@ -326,8 +326,7 @@ subroutine symmetry_bd(ord,extc,func,funcc,SoA)
 
   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)
-    funcc(-i,1:extc(2),1:extc(3)) = funcc(i+2,1:extc(2),1:extc(3))*SoA(1)
    enddo
    do i=0,ord-1
       funcc(:,-i,1:extc(3)) = funcc(:,i+2,1:extc(3))*SoA(2)
@@ -884,13 +883,17 @@ subroutine symmetry_bd(ord,extc,func,funcc,SoA)
 
   integer::i
 
+!DIR$ SIMD VECTORLENGTHFOR(KNOWN_INTEGER=8)
   funcc(1:extc(1),1:extc(2),1:extc(3)) = func
+!DIR$ SIMD VECTORLENGTHFOR(KNOWN_INTEGER=8)
    do i=0,ord-1
       funcc(-i,1:extc(2),1:extc(3)) = funcc(i+1,1:extc(2),1:extc(3))*SoA(1)
    enddo
+!DIR$ SIMD VECTORLENGTHFOR(KNOWN_INTEGER=8)
    do i=0,ord-1
       funcc(:,-i,1:extc(3)) = funcc(:,i+1,1:extc(3))*SoA(2)
    enddo
+!DIR$ SIMD VECTORLENGTHFOR(KNOWN_INTEGER=8)
    do i=0,ord-1
       funcc(:,:,-i) = funcc(:,:,i+1)*SoA(3)
    enddo
@@ -1113,6 +1116,7 @@ end subroutine d2dump
 ! Lagrangian polynomial interpolation
 !------------------------------------------------------------------------------
 
+!DIR$ ATTRIBUTES FORCEINLINE :: polint
   subroutine polint(xa, ya, x, y, dy, ordn)
   implicit none
 

AMSS_NCKU_source/kodiss.f90

@@ -6,6 +6,103 @@
 ! Vertex or Cell is distinguished in routine symmetry_bd which locates in
 ! file "fmisc.f90"
 
+#if (ghost_width == 2)
+! second order code
+
+!------------------------------------------------------------------------------------------------------------------------------
+!usual type Kreiss-Oliger type numerical dissipation
+!We support cell center only
+!  (D_+D_-)^2 =
+!   f(i-2) - 4 f(i-1) + 6 f(i) - 4 f(i+1) + f(i+2)
+! ------------------------------------------------------
+!                       dx^4
+!------------------------------------------------------------------------------------------------------------------------------
+! do not add dissipation near boundary
+subroutine kodis(ex,X,Y,Z,f,f_rhs,SoA,Symmetry,eps)
+
+implicit none
+! argument variables
+integer,intent(in) :: Symmetry
+integer,dimension(3),intent(in)::ex
+real*8, dimension(1:3), intent(in) :: SoA
+double precision,intent(in),dimension(ex(1))::X
+double precision,intent(in),dimension(ex(2))::Y
+double precision,intent(in),dimension(ex(3))::Z
+double precision,intent(in),dimension(ex(1),ex(2),ex(3))::f
+double precision,intent(inout),dimension(ex(1),ex(2),ex(3))::f_rhs
+real*8,intent(in) :: eps
+
+!~~~~~~ other variables
+
+  real*8 :: dX,dY,dZ
+  real*8,dimension(-1:ex(1),-1:ex(2),-1:ex(3))   :: fh
+  integer :: imin,jmin,kmin,imax,jmax,kmax
+  integer, parameter :: NO_SYMM = 0, EQ_SYMM = 1, OCTANT = 2
+  real*8,parameter   :: cof = 1.6d1 ! 2^4
+  real*8,  parameter :: F4=4.d0,F6=6.d0
+  integer::i,j,k
+
+  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
+
+  call symmetry_bd(2,ex,f,fh,SoA)
+
+!   f(i-2) - 4 f(i-1) + 6 f(i) - 4 f(i+1) + f(i+2)
+! ------------------------------------------------------
+!                       dx^4
+
+!  note the sign (-1)^r-1, now r=2
+!DIR$ SIMD VECTORLENGTHFOR(KNOWN_INTEGER=8)
+!DIR$ UNROLL PARTIAL(4)
+  do k=1,ex(3)
+  do j=1,ex(2)
+  do i=1,ex(1)
+
+  if(i-2 >= imin .and. i+2 <= imax .and. &
+     j-2 >= jmin .and. j+2 <= jmax .and. &
+     k-2 >= kmin .and. k+2 <= kmax) then
+! x direction
+   f_rhs(i,j,k)       = f_rhs(i,j,k) - eps/dX/cof * (     &
+                                (fh(i-2,j,k)+fh(i+2,j,k)) &
+                         - F4 * (fh(i-1,j,k)+fh(i+1,j,k)) &
+                         + F6 *  fh(i,j,k) )
+! y direction
+
+   f_rhs(i,j,k)       = f_rhs(i,j,k) - eps/dY/cof * (     &
+                                (fh(i,j-2,k)+fh(i,j+2,k)) &
+                         - F4 * (fh(i,j-1,k)+fh(i,j+1,k)) &
+                         + F6 *  fh(i,j,k) )
+! z direction
+
+   f_rhs(i,j,k)       = f_rhs(i,j,k) - eps/dZ/cof * (     &
+                                (fh(i,j,k-2)+fh(i,j,k+2)) &
+                         - F4 * (fh(i,j,k-1)+fh(i,j,k+1)) &
+                         + F6 *  fh(i,j,k) )
+
+  endif
+
+  enddo
+  enddo
+  enddo
+
+  return
+
+end subroutine kodis
+
+#elif (ghost_width == 3)
 ! fourth order code
 
 !---------------------------------------------------------------------------------------------
@@ -61,7 +158,7 @@ integer, parameter :: NO_SYMM=0, OCTANT=2
   if(Symmetry > NO_SYMM .and. dabs(Z(1)) < dZ) kmin = -2
   if(Symmetry == OCTANT .and. dabs(X(1)) < dX) imin = -2
   if(Symmetry == OCTANT .and. dabs(Y(1)) < dY) jmin = -2
-  !print*,'imin,jmin,kmin=',imin,jmin,kmin
+
   call symmetry_bd(3,ex,f,fh,SoA)
 
   do k=1,ex(3)
@@ -71,7 +168,28 @@ integer, parameter :: NO_SYMM=0, OCTANT=2
   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
+#if 0     
+! x direction
+   f_rhs(i,j,k)       = f_rhs(i,j,k) + eps/dX/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)            )
+! y direction
 
+   f_rhs(i,j,k)       = f_rhs(i,j,k) + eps/dY/cof * (     &
+                              (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)            )
+! z direction
+
+   f_rhs(i,j,k)       = f_rhs(i,j,k) + eps/dZ/cof * (     &
+                              (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)            )
+#else
 ! calculation order if important ?
    f_rhs(i,j,k)       = f_rhs(i,j,k) + eps/cof *( (     &
                               (fh(i-3,j,k)+fh(i+3,j,k)) - &
@@ -88,7 +206,7 @@ integer, parameter :: NO_SYMM=0, OCTANT=2
                           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
   endif
 
   enddo
@@ -99,6 +217,218 @@ integer, parameter :: NO_SYMM=0, OCTANT=2
 
   end subroutine kodis
 
+#elif (ghost_width == 4)
+! sixth order code
+!------------------------------------------------------------------------------------------------------------------------------
+!usual type Kreiss-Oliger type numerical dissipation
+!We support cell center only
+!  (D_+D_-)^4 =
+!   f(i-4) - 8 f(i-3) + 28 f(i-2) - 56 f(i-1) + 70 f(i) - 56 f(i+1) + 28 f(i+2) - 8 f(i+3) + f(i+4)
+! ----------------------------------------------------------------------------------------------------------
+!                                              dx^8
+!------------------------------------------------------------------------------------------------------------------------------
+! do not add dissipation near boundary
+subroutine kodis(ex,X,Y,Z,f,f_rhs,SoA,Symmetry,eps)
 
+implicit none
+! argument variables
+integer,intent(in) :: Symmetry
+integer,dimension(3),intent(in)::ex
+real*8, dimension(1:3), intent(in) :: SoA
+double precision,intent(in),dimension(ex(1))::X
+double precision,intent(in),dimension(ex(2))::Y
+double precision,intent(in),dimension(ex(3))::Z
+double precision,intent(in),dimension(ex(1),ex(2),ex(3))::f
+double precision,intent(inout),dimension(ex(1),ex(2),ex(3))::f_rhs
+real*8,intent(in) :: eps
 
+!~~~~~~ other variables
 
+  real*8 :: dX,dY,dZ
+  real*8,dimension(-3:ex(1),-3:ex(2),-3:ex(3))   :: fh
+  integer :: imin,jmin,kmin,imax,jmax,kmax
+  integer, parameter :: NO_SYMM = 0, EQ_SYMM = 1, OCTANT = 2
+  real*8,parameter   :: cof = 2.56d2 ! 2^8
+  real*8,  parameter :: F8=8.d0,F28=2.8d1,F56=5.6d1,F70=7.d1
+  integer::i,j,k
+
+  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 = -3
+  if(Symmetry > EQ_SYMM .and. dabs(X(1)) < dX) imin = -3
+  if(Symmetry > EQ_SYMM .and. dabs(Y(1)) < dY) jmin = -3
+
+  call symmetry_bd(4,ex,f,fh,SoA)
+
+!   f(i-4) - 8 f(i-3) + 28 f(i-2) - 56 f(i-1) + 70 f(i) - 56 f(i+1) + 28 f(i+2) - 8 f(i+3) + f(i+4)
+! ----------------------------------------------------------------------------------------------------------
+!                                              dx^8
+
+!  note the sign (-1)^r-1, now r=4
+  do k=1,ex(3)
+  do j=1,ex(2)
+  do i=1,ex(1)
+
+  if(i>imin+3 .and. i < imax-3 .and. &
+     j>jmin+3 .and. j < jmax-3 .and. &
+     k>kmin+3 .and. k < kmax-3) then
+! x direction
+   f_rhs(i,j,k)       = f_rhs(i,j,k) - eps/dX/cof * (     &
+                                (fh(i-4,j,k)+fh(i+4,j,k)) &
+                         - F8 * (fh(i-3,j,k)+fh(i+3,j,k)) &
+                         +F28 * (fh(i-2,j,k)+fh(i+2,j,k)) &
+                         -F56 * (fh(i-1,j,k)+fh(i+1,j,k)) &
+                         +F70 *  fh(i,j,k) )
+! y direction
+
+   f_rhs(i,j,k)       = f_rhs(i,j,k) - eps/dY/cof * (     &
+                                (fh(i,j-4,k)+fh(i,j+4,k)) &
+                         - F8 * (fh(i,j-3,k)+fh(i,j+3,k)) &
+                         +F28 * (fh(i,j-2,k)+fh(i,j+2,k)) &
+                         -F56 * (fh(i,j-1,k)+fh(i,j+1,k)) &
+                         +F70 *  fh(i,j,k) )
+! z direction
+
+   f_rhs(i,j,k)       = f_rhs(i,j,k) - eps/dZ/cof * (     &
+                                (fh(i,j,k-4)+fh(i,j,k+4)) &
+                         - F8 * (fh(i,j,k-3)+fh(i,j,k+3)) &
+                         +F28 * (fh(i,j,k-2)+fh(i,j,k+2)) &
+                         -F56 * (fh(i,j,k-1)+fh(i,j,k+1)) &
+                         +F70 *  fh(i,j,k) )
+
+  endif
+
+  enddo
+  enddo
+  enddo
+
+  return
+
+end subroutine kodis
+
+#elif (ghost_width == 5)
+! eighth order code
+!------------------------------------------------------------------------------------------------------------------------------
+!usual type Kreiss-Oliger type numerical dissipation
+!We support cell center only
+! Note the notation D_+ and D_- [P240 of B. Gustafsson, H.-O. Kreiss, and J. Oliger, Time
+! Dependent Problems and Difference Methods (Wiley, New York, 1995).]
+! D_+ = (f(i+1) - f(i))/h
+! D_- = (f(i) - f(i-1))/h
+! then we have D_+D_- = D_-D_+ = (f(i+1) - 2f(i) + f(i-1))/h^2
+! for nth order accurate finite difference code, we need r =n/2+1
+!              D_+^rD_-^r = (D_+D_-)^r 
+! following the tradiation of PRD 77, 024027 (BB's calibration paper, Eq.(64),
+!  correct some typo according to above book) :
+! + eps*(-1)^(r-1)*h^(2r-1)/2^(2r)*(D_+D_-)^r 
+!
+!
+! this is for 8th order accurate finite difference scheme
+!  (D_+D_-)^5 =
+!  f(i-5) - 10 f(i-4) + 45 f(i-3) - 120 f(i-2) + 210 f(i-1) - 252 f(i) + 210 f(i+1) - 120 f(i+2) + 45 f(i+3) - 10 f(i+4) + f(i+5)
+! -------------------------------------------------------------------------------------------------------------------------------
+!                                                              dx^10
+!---------------------------------------------------------------------------------------------------------------------------------
+! do not add dissipation near boundary
+subroutine kodis(ex,X,Y,Z,f,f_rhs,SoA,Symmetry,eps)
+
+implicit none
+! argument variables
+integer,intent(in) :: Symmetry
+integer,dimension(3),intent(in)::ex
+real*8, dimension(1:3), intent(in) :: SoA
+double precision,intent(in),dimension(ex(1))::X
+double precision,intent(in),dimension(ex(2))::Y
+double precision,intent(in),dimension(ex(3))::Z
+double precision,intent(in),dimension(ex(1),ex(2),ex(3))::f
+double precision,intent(inout),dimension(ex(1),ex(2),ex(3))::f_rhs
+real*8,intent(in) :: eps
+
+!~~~~~~ other variables
+
+  real*8 :: dX,dY,dZ
+  real*8,dimension(-4:ex(1),-4:ex(2),-4:ex(3))   :: fh
+  integer :: imin,jmin,kmin,imax,jmax,kmax
+  integer, parameter :: NO_SYMM = 0, EQ_SYMM = 1, OCTANT = 2
+  real*8,parameter   :: cof = 1.024d3 ! 2^2r = 2^10
+  real*8,  parameter :: F10=1.d1,F45=4.5d1,F120=1.2d2,F210=2.1d2,F252=2.52d2
+  integer::i,j,k
+
+  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 = -4
+  if(Symmetry > EQ_SYMM .and. dabs(X(1)) < dX) imin = -4
+  if(Symmetry > EQ_SYMM .and. dabs(Y(1)) < dY) jmin = -4
+
+  call symmetry_bd(5,ex,f,fh,SoA)
+
+!  f(i-5) - 10 f(i-4) + 45 f(i-3) - 120 f(i-2) + 210 f(i-1) - 252 f(i) + 210 f(i+1) - 120 f(i+2) + 45 f(i+3) - 10 f(i+4) + f(i+5)
+! -------------------------------------------------------------------------------------------------------------------------------
+!                                                              dx^10
+
+!  note the sign (-1)^r-1, now r=5
+  do k=1,ex(3)
+  do j=1,ex(2)
+  do i=1,ex(1)
+
+  if(i>imin+4 .and. i < imax-4 .and. &
+     j>jmin+4 .and. j < jmax-4 .and. &
+     k>kmin+4 .and. k < kmax-4) then
+! x direction
+   f_rhs(i,j,k)       = f_rhs(i,j,k) + eps/dX/cof * (      &
+                                 (fh(i-5,j,k)+fh(i+5,j,k)) &
+                         - F10 * (fh(i-4,j,k)+fh(i+4,j,k)) &
+                         + F45 * (fh(i-3,j,k)+fh(i+3,j,k)) &
+                         - F120* (fh(i-2,j,k)+fh(i+2,j,k)) &
+                         + F210* (fh(i-1,j,k)+fh(i+1,j,k)) &
+                         - F252 * fh(i,j,k) )
+! y direction
+
+   f_rhs(i,j,k)       = f_rhs(i,j,k) + eps/dY/cof * (      &
+                                 (fh(i,j-5,k)+fh(i,j+5,k)) &
+                         - F10 * (fh(i,j-4,k)+fh(i,j+4,k)) &
+                         + F45 * (fh(i,j-3,k)+fh(i,j+3,k)) &
+                         - F120* (fh(i,j-2,k)+fh(i,j+2,k)) &
+                         + F210* (fh(i,j-1,k)+fh(i,j+1,k)) &
+                         - F252 * fh(i,j,k) )
+! z direction
+
+   f_rhs(i,j,k)       = f_rhs(i,j,k) + eps/dZ/cof * (      &
+                                 (fh(i,j,k-5)+fh(i,j,k+5)) &
+                         - F10 * (fh(i,j,k-4)+fh(i,j,k+4)) &
+                         + F45 * (fh(i,j,k-3)+fh(i,j,k+3)) &
+                         - F120* (fh(i,j,k-2)+fh(i,j,k+2)) &
+                         + F210* (fh(i,j,k-1)+fh(i,j,k+1)) &
+                         - F252 * fh(i,j,k) )
+
+  endif
+
+  enddo
+  enddo
+  enddo
+
+  return
+
+end subroutine kodis
+
+#endif  

AMSS_NCKU_source/lopsidediff.f90

@@ -7,7 +7,163 @@
 ! Vertex or Cell is distinguished in routine symmetry_bd which locates in
 ! file "fmisc.f90"
 
+#if (ghost_width == 2)
+! second order code
 
+!-----------------------------------------------------------------------------
+!         v
+! D f = ------[ - 3 f  + 4 f   - f     ]
+!  i     2dx         i      i+v   i+2v
+!
+! where
+!
+!        i
+!      |B |
+! v = -----
+!        i
+!       B
+!
+!-----------------------------------------------------------------------------
+subroutine lopsided(ex,X,Y,Z,f,f_rhs,Sfx,Sfy,Sfz,Symmetry,SoA)
+  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
+
+!~~~~~~> local variables:
+! note index -1,0, so we have 2 extra points
+  real*8,dimension(-1:ex(1),-1:ex(2),-1:ex(3))   :: fh
+  integer :: imin,jmin,kmin,imax,jmax,kmax,i,j,k
+  real*8 :: dX,dY,dZ
+  real*8 :: d2dx,d2dy,d2dz
+  real*8,  parameter :: ZEO=0.d0,ONE=1.d0,TWO=2.d0,THR=3.d0,FOUR=4.d0
+  integer, parameter :: NO_SYMM = 0, EQ_SYMM = 1, OCTANT = 2
+
+  dX = X(2)-X(1)
+  dY = Y(2)-Y(1)
+  dZ = Z(2)-Z(1)
+
+  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 = -1
+  if(Symmetry > EQ_SYMM .and. dabs(X(1)) < dX) imin = -1
+  if(Symmetry > EQ_SYMM .and. dabs(Y(1)) < dY) jmin = -1
+
+  call symmetry_bd(2,ex,f,fh,SoA)
+
+! 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+2 <= imax .and. i >= imin)then
+!         v
+! D f = ------[ - 3 f  + 4 f   - f     ]
+!  i     2dx         i      i+v   i+2v
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                           &
+                  Sfx(i,j,k)*d2dx*(-THR*fh(i,j,k)+FOUR*fh(i+1,j,k)-fh(i+2,j,k))
+       elseif(i+1 <= imax .and. i >= imin)then
+!         v
+! D f = ------[ - f  + f   ]
+!  i      dx       i    i+v
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                           &
+                  Sfx(i,j,k)*d2dx*(-fh(i,j,k)+fh(i+1,j,k))
+
+       endif
+
+    elseif(Sfx(i,j,k) <= ZEO)then
+      if( i-2 >= imin .and. i <= imax)then
+     f_rhs(i,j,k)=f_rhs(i,j,k)-                           &
+                  Sfx(i,j,k)*d2dx*(-THR*fh(i,j,k)+FOUR*fh(i-1,j,k)-fh(i-2,j,k))
+      elseif(i-1 >= imin .and. i <= imax)then
+     f_rhs(i,j,k)=f_rhs(i,j,k)-                           &
+                  Sfx(i,j,k)*d2dx*(-fh(i,j,k)+fh(i-1,j,k))
+      endif
+
+! set imax and imin 0
+    endif
+
+! y direction   
+    if(Sfy(i,j,k) >= ZEO)then
+       if( j+2 <= jmax .and. j >= jmin)then
+!         v
+! D f = ------[ - 3 f  + 4 f   - f     ]
+!  i     2dx         i      i+v   i+2v
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                           &
+                  Sfy(i,j,k)*d2dy*(-THR*fh(i,j,k)+FOUR*fh(i,j+1,k)-fh(i,j+2,k))
+       elseif(j+1 <= jmax .and. j >= jmin)then
+!         v
+! D f = ------[ - f  + f   ]
+!  i      dx       i    i+v
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                           &
+                  Sfy(i,j,k)*d2dy*(-fh(i,j,k)+fh(i,j+1,k))
+       endif
+
+    elseif(Sfy(i,j,k) <= ZEO)then
+      if( j-2 >= jmin .and. j <= jmax)then
+     f_rhs(i,j,k)=f_rhs(i,j,k)-                           &
+                  Sfy(i,j,k)*d2dy*(-THR*fh(i,j,k)+FOUR*fh(i,j-1,k)-fh(i,j-2,k))
+      elseif(j-1 >= jmin .and. j <= jmax)then
+     f_rhs(i,j,k)=f_rhs(i,j,k)-                           &
+                  Sfy(i,j,k)*d2dy*(-fh(i,j,k)+fh(i,j-1,k))
+      endif
+
+! set jmin and jmax 0
+     endif
+!! z direction   
+    if(Sfz(i,j,k) >= ZEO)then
+      if( k+2 <= kmax .and. k >= kmin)then
+!         v
+! D f = ------[ - 3 f  + 4 f   - f     ]
+!  i     2dx         i      i+v   i+2v
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                           &
+                  Sfz(i,j,k)*d2dz*(-THR*fh(i,j,k)+FOUR*fh(i,j,k+1)-fh(i,j,k+2))
+       elseif(k+1 <= kmax .and. k >= kmin)then
+!         v
+! D f = ------[ - f  + f   ]
+!  i      dx       i    i+v
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                           &
+                  Sfz(i,j,k)*d2dz*(-fh(i,j,k)+fh(i,j,k+1))
+       endif
+
+    elseif(Sfz(i,j,k) <= ZEO)then
+      if( k-2 >= kmin .and. k <= kmax)then
+     f_rhs(i,j,k)=f_rhs(i,j,k)-                           &
+                  Sfz(i,j,k)*d2dz*(-THR*fh(i,j,k)+FOUR*fh(i,j,k-1)-fh(i,j,k-2))
+      elseif(k-1 >= kmin .and. k <= kmax)then
+     f_rhs(i,j,k)=f_rhs(i,j,k)-                           &
+                  Sfz(i,j,k)*d2dz*(-fh(i,j,k)+fh(i,j,k-1))
+      endif
+
+! set kmin and kmax 0
+     endif
+
+  enddo
+  enddo
+  enddo
+
+  return
+
+  end subroutine lopsided
+
+#elif (ghost_width == 3)
 ! fourth order code
 
 !-----------------------------------------------------------------------------
@@ -80,7 +236,89 @@ subroutine lopsided(ex,X,Y,Z,f,f_rhs,Sfx,Sfy,Sfz,Symmetry,SoA)
   do k=1,ex(3)-1
   do j=1,ex(2)-1
   do i=1,ex(1)-1
+#if 0  
+!! old code
+! x direction   
+    if(Sfx(i,j,k) >= ZEO .and. i+3 <= imax .and. i-1 >= imin)then
+!         v
+! D f = ------[ - 3f    - 10f  + 18f    - 6f     + f     ]
+!  i     12dx       i-v      i      i+v     i+2v    i+3v
+     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(Sfx(i,j,k) <= ZEO .and. i-3 >= imin .and. 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))
+
+     elseif(i+2 <= imax .and. i-2 >= imin)then
+!
+!              f(i-2) - 8 f(i-1) + 8 f(i+1) - f(i+2)
+!  fx(i) = ---------------------------------------------
+!                             12 dx
+     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 .and. i-1 >= imin)then
+!
+!              - f(i-1) + f(i+1)
+!  fx(i) = --------------------------------
+!                     2 dx
+     f_rhs(i,j,k)=f_rhs(i,j,k) + Sfx(i,j,k)*d2dx*(-fh(i-1,j,k)+fh(i+1,j,k))
+
+! set imax and imin 0
+    endif
+
+! y direction   
+    if(Sfy(i,j,k) >= ZEO .and. j+3 <= jmax .and. j-1 >= jmin)then
+!         v
+! D f = ------[ - 3f    - 10f  + 18f    - 6f     + f     ]
+!  i     12dx       i-v      i      i+v     i+2v    i+3v
+     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(Sfy(i,j,k) <= ZEO .and. j-3 >= jmin .and. 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))
+
+     elseif(j+2 <= jmax .and. 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 <= jmax .and. j-1 >= jmin)then
+
+     f_rhs(i,j,k)=f_rhs(i,j,k) + Sfy(i,j,k)*d2dy*(-fh(i,j-1,k)+fh(i,j+1,k))
+! set jmin and jmax 0
+     endif
+!! z direction   
+    if(Sfz(i,j,k) >= ZEO .and. k+3 <= kmax .and. k-1 >= kmin)then
+!         v
+! D f = ------[ - 3f    - 10f  + 18f    - 6f     + f     ]
+!  i     12dx       i-v      i      i+v     i+2v    i+3v
+     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(Sfz(i,j,k) <= ZEO .and. k-3 >= kmin .and. 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))
+
+     elseif(k+2 <= kmax .and. 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 <= kmax .and. k-1 >= kmin)then
+
+     f_rhs(i,j,k)=f_rhs(i,j,k)+Sfz(i,j,k)*d2dz*(-fh(i,j,k-1)+fh(i,j,k+1))
+! set kmin and kmax 0
+     endif
+#else
 !! new code, 2012dec27, based on bam
 ! x direction   
     if(Sfx(i,j,k) > ZEO)then
@@ -240,6 +478,7 @@ subroutine lopsided(ex,X,Y,Z,f,f_rhs,Sfx,Sfy,Sfz,Symmetry,SoA)
 ! set kmax and kmin 0
      endif
    endif
+#endif
   enddo
   enddo
   enddo
@@ -247,3 +486,612 @@ subroutine lopsided(ex,X,Y,Z,f,f_rhs,Sfx,Sfy,Sfz,Symmetry,SoA)
   return
 
   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
+!         v
+! D f = ------[ 2f     - 24f    - 35f  + 80f    - 30f     + 8f     - f    ]
+!  i     60dx     i-2v      i-v      i      i+v      i+2v     i+3v    i+4v
+!
+! where
+!
+!        i
+!      |B |
+! v = -----
+!        i
+!       B
+!
+!-----------------------------------------------------------------------------
+subroutine lopsided(ex,X,Y,Z,f,f_rhs,Sfx,Sfy,Sfz,Symmetry,SoA)
+  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
+
+!~~~~~~> local variables:
+
+  real*8,dimension(-3:ex(1),-3:ex(2),-3:ex(3))   :: fh
+  integer :: imin,jmin,kmin,imax,jmax,kmax,i,j,k
+  real*8 :: dX,dY,dZ
+  real*8 :: d60dx,d60dy,d60dz,d12dx,d12dy,d12dz,d2dx,d2dy,d2dz
+  real*8,  parameter :: ZEO=0.d0,ONE=1.d0, F60=6.d1
+  real*8,  parameter :: TWO=2.d0,F24=2.4d1,F35=3.5d1,F80=8.d1,F30=3.d1,EIT=8.d0
+  real*8,  parameter ::  F9=9.d0,F45=4.5d1,F12=1.2d1
+  real*8,  parameter ::  F10=1.d1,F77=7.7d1,F150=1.5d2,F100=1.d2,F50=5.d1,F15=1.5d1
+  integer, parameter :: NO_SYMM = 0, EQ_SYMM = 1, OCTANT = 2
+
+  dX = X(2)-X(1)
+  dY = Y(2)-Y(1)
+  dZ = Z(2)-Z(1)
+
+  d60dx = ONE/F60/dX
+  d60dy = ONE/F60/dY
+  d60dz = ONE/F60/dZ
+
+  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 = -3
+  if(Symmetry > EQ_SYMM .and. dabs(X(1)) < dX) imin = -3
+  if(Symmetry > EQ_SYMM .and. dabs(Y(1)) < dY) jmin = -3
+
+  call symmetry_bd(4,ex,f,fh,SoA)
+
+! 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 .and. i+4 <= imax .and. i-2 >= imin)then
+!         v
+! D f = ------[ 2f     - 24f    - 35f  + 80f    - 30f     + 8f     - f    ]
+!  i     60dx     i-2v      i-v      i      i+v      i+2v     i+3v    i+4v
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                                                             &
+                  Sfx(i,j,k)*d60dx*(TWO*fh(i-2,j,k)-F24*fh(i-1,j,k)-F35*fh(i,j,k)+F80*fh(i+1,j,k) &
+                                   -F30*fh(i+2,j,k)+EIT*fh(i+3,j,k)-    fh(i+4,j,k))
+    elseif(Sfx(i,j,k) >= ZEO .and. i+5 <= imax .and. i-1 >= imin)then
+!         v
+! D f = ------[-10f    - 77f  + 150f    - 100f     + 50f     -15f     + 2f    ]
+!  i     60dx      i-v      i       i+v       i+2v      i+3v     i+4v    i+5v
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                                                                        &
+                  Sfx(i,j,k)*d60dx*(-F10*fh(i-1,j,k)-F77*fh(i  ,j,k)+F150*fh(i+1,j,k)-F100*fh(i+2,j,k) &
+                                    +F50*fh(i+3,j,k)-F15*fh(i+4,j,k)+ TWO*fh(i+5,j,k))
+
+    elseif(Sfx(i,j,k) <= ZEO .and. i-4 >= imin .and. i+2 <= imax)then
+     f_rhs(i,j,k)=f_rhs(i,j,k)-                                                                   &
+                  Sfx(i,j,k)*d60dx*(TWO*fh(i+2,j,k)-F24*fh(i+1,j,k)-F35*fh(i,j,k)+F80*fh(i-1,j,k) &
+                                   -F30*fh(i-2,j,k)+EIT*fh(i-3,j,k)-    fh(i-4,j,k))
+    elseif(Sfx(i,j,k) <= ZEO .and. i-5 >= imin .and. i+1 <= imax)then
+     f_rhs(i,j,k)=f_rhs(i,j,k)-                                                                        &
+                  Sfx(i,j,k)*d60dx*(-F10*fh(i+1,j,k)-F77*fh(i  ,j,k)+F150*fh(i-1,j,k)-F100*fh(i-2,j,k) &
+                                    +F50*fh(i-3,j,k)-F15*fh(i-4,j,k)+ TWO*fh(i-5,j,k))
+
+     elseif(i+3 <= imax .and. i-3 >= imin)then
+!           - f(i-3) + 9 f(i-2) - 45 f(i-1) + 45 f(i+1) - 9 f(i+2) + f(i+3)
+!  fx(i) = -----------------------------------------------------------------
+!                                        60 dx
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                                                                              &
+                  Sfx(i,j,k)*d60dx*(-fh(i-3,j,k)+F9*fh(i-2,j,k)-F45*fh(i-1,j,k)+F45*fh(i+1,j,k)-F9*fh(i+2,j,k)+fh(i+3,j,k))
+
+     elseif(i+2 <= imax .and. i-2 >= imin)then
+!
+!              f(i-2) - 8 f(i-1) + 8 f(i+1) - f(i+2)
+!  fx(i) = ---------------------------------------------
+!                             12 dx
+     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 .and. i-1 >= imin)then
+!
+!              - f(i-1) + f(i+1)
+!  fx(i) = --------------------------------
+!                     2 dx
+     f_rhs(i,j,k)=f_rhs(i,j,k) + Sfx(i,j,k)*d2dx*(-fh(i-1,j,k)+fh(i+1,j,k))
+
+! set imax and imin 0
+    endif
+
+! y direction   
+     if(Sfy(i,j,k) >= ZEO .and. j+4 <= jmax .and. j-2 >= jmin)then
+
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                                                                   &
+                  Sfy(i,j,k)*d60dy*(TWO*fh(i,j-2,k)-F24*fh(i,j-1,k)-F35*fh(i,j,k)+F80*fh(i,j+1,k) &
+                                   -F30*fh(i,j+2,k)+EIT*fh(i,j+3,k)-    fh(i,j+4,k))
+     elseif(Sfy(i,j,k) >= ZEO .and. j+5 <= jmax .and. j-1 >= jmin)then
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                                                                        &
+                  Sfy(i,j,k)*d60dy*(-F10*fh(i,j-1,k)-F77*fh(i,j  ,k)+F150*fh(i,j+1,k)-F100*fh(i,j+2,k) &
+                                    +F50*fh(i,j+3,k)-F15*fh(i,j+4,k)+ TWO*fh(i,j+5,k))
+
+     elseif(Sfy(i,j,k) <= ZEO .and. j-4 >= jmin .and. j+2 <= jmax)then
+
+     f_rhs(i,j,k)=f_rhs(i,j,k)-                                                                   &
+                  Sfy(i,j,k)*d60dy*(TWO*fh(i,j+2,k)-F24*fh(i,j+1,k)-F35*fh(i,j,k)+F80*fh(i,j-1,k) &
+                                   -F30*fh(i,j-2,k)+EIT*fh(i,j-3,k)-    fh(i,j-4,k))
+
+     elseif(Sfy(i,j,k) <= ZEO .and. j-5 >= jmin .and. j+1 <= jmax)then
+
+     f_rhs(i,j,k)=f_rhs(i,j,k)-                                                                        &
+                  Sfy(i,j,k)*d60dy*(-F10*fh(i,j+1,k)-F77*fh(i,j  ,k)+F150*fh(i,j-1,k)-F100*fh(i,j-2,k) &
+                                    +F50*fh(i,j-3,k)-F15*fh(i,j-4,k)+ TWO*fh(i,j-5,k))
+
+     elseif(j+3 <= jmax .and. j-3 >= jmin)then
+          
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                                                                                         &
+                  Sfy(i,j,k)*d60dy*(-fh(i,j-3,k)+F9*fh(i,j-2,k)-F45*fh(i,j-1,k)+F45*fh(i,j+1,k)-F9*fh(i,j+2,k)+fh(i,j+3,k))
+
+     elseif(j+2 <= jmax .and. 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 <= jmax .and. j-1 >= jmin)then
+
+     f_rhs(i,j,k)=f_rhs(i,j,k) + Sfy(i,j,k)*d2dy*(-fh(i,j-1,k)+fh(i,j+1,k))
+! set jmin and jmax 0
+     endif
+!! z direction   
+     if(Sfz(i,j,k) >= ZEO .and. k+4 <= kmax .and. k-2 >= kmin)then
+
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                                                                   &
+                  Sfz(i,j,k)*d60dz*(TWO*fh(i,j,k-2)-F24*fh(i,j,k-1)-F35*fh(i,j,k)+F80*fh(i,j,k+1) &
+                                   -F30*fh(i,j,k+2)+EIT*fh(i,j,k+3)-    fh(i,j,k+4))
+     elseif(Sfz(i,j,k) >= ZEO .and. k+5 <= kmax .and. k-1 >= kmin)then
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                                                                        &
+                  Sfz(i,j,k)*d60dz*(-F10*fh(i,j,k-1)-F77*fh(i,j,k  )+F150*fh(i,j,k+1)-F100*fh(i,j,k+2) &
+                                    +F50*fh(i,j,k+3)-F15*fh(i,j,k+4)+ TWO*fh(i,j,k+5))
+
+     elseif(Sfz(i,j,k) <= ZEO .and. k-4 >= kmin .and. k+2 <= kmax)then
+
+     f_rhs(i,j,k)=f_rhs(i,j,k)-                                                                   &
+                  Sfz(i,j,k)*d60dz*(TWO*fh(i,j,k+2)-F24*fh(i,j,k+1)-F35*fh(i,j,k)+F80*fh(i,j,k-1) &
+                                   -F30*fh(i,j,k-2)+EIT*fh(i,j,k-3)-    fh(i,j,k-4))
+
+     elseif(Sfz(i,j,k) <= ZEO .and. k-5 >= kmin .and. k+1 <= kmax)then
+
+     f_rhs(i,j,k)=f_rhs(i,j,k)-                                                                        &
+                  Sfz(i,j,k)*d60dz*(-F10*fh(i,j,k+1)-F77*fh(i,j,k  )+F150*fh(i,j,k-1)-F100*fh(i,j,k-2) &
+                                    +F50*fh(i,j,k-3)-F15*fh(i,j,k-4)+ TWO*fh(i,j,k-5))
+     
+     elseif(k+3 <= kmax .and. k-3 >= kmin)then
+
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                                                                                         &
+                  Sfz(i,j,k)*d60dz*(-fh(i,j,k-3)+F9*fh(i,j,k-2)-F45*fh(i,j,k-1)+F45*fh(i,j,k+1)-F9*fh(i,j,k+2)+fh(i,j,k+3))
+
+     elseif(k+2 <= kmax .and. 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 <= kmax .and. k-1 >= kmin)then
+
+     f_rhs(i,j,k)=f_rhs(i,j,k)+Sfz(i,j,k)*d2dz*(-fh(i,j,k-1)+fh(i,j,k+1))
+! set kmin and kmax 0
+     endif
+
+  enddo
+  enddo
+  enddo
+
+  return
+
+  end subroutine lopsided
+
+#elif (ghost_width == 5)
+! eighth order code
+!-----------------------------------------------------------------------------
+! PRD 77, 024034 (2008)
+! Compute advection terms in right hand sides of field equations
+!        v [ - 5 f(i-3v) + 60 f(i-2v) - 420 f(i-v) - 378 f(i) + 1050 f(i+v) - 420 f(i+2v) + 140 f(i+3v) - 30 f(i+4v) + 3 f(i+5v)]
+! D f = --------------------------------------------------------------------------------------------------------------------------
+!  i                                                             840 dx           
+!
+! where
+!
+!        i
+!      |B |
+! v = -----
+!        i
+!       B
+!
+!-----------------------------------------------------------------------------
+subroutine lopsided(ex,X,Y,Z,f,f_rhs,Sfx,Sfy,Sfz,Symmetry,SoA)
+  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
+
+!~~~~~~> local variables:
+
+  real*8,dimension(-4:ex(1),-4:ex(2),-4:ex(3))   :: fh
+  integer :: imin,jmin,kmin,imax,jmax,kmax,i,j,k
+  real*8 :: dX,dY,dZ
+  real*8 :: d840dx,d840dy,d840dz,d60dx,d60dy,d60dz,d12dx,d12dy,d12dz,d2dx,d2dy,d2dz
+  real*8,  parameter :: ZEO=0.d0,ONE=1.d0, F60=6.d1
+  real*8,  parameter :: TWO=2.d0,F30=3.d1,EIT=8.d0
+  real*8,  parameter ::  F9=9.d0,F45=4.5d1,F12=1.2d1,F140=1.4d2,THR=3.d0
+  real*8,  parameter :: F840=8.4d2,F5=5.d0,F420=4.2d2,F378=3.78d2,F1050=1.05d3
+  real*8,  parameter :: F32=3.2d1,F168=1.68d2,F672=6.72d2
+  integer, parameter :: NO_SYMM = 0, EQ_SYMM = 1, OCTANT = 2
+
+  dX = X(2)-X(1)
+  dY = Y(2)-Y(1)
+  dZ = Z(2)-Z(1)
+
+  d840dx = ONE/F840/dX
+  d840dy = ONE/F840/dY
+  d840dz = ONE/F840/dZ
+
+  d60dx = ONE/F60/dX
+  d60dy = ONE/F60/dY
+  d60dz = ONE/F60/dZ
+
+  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 = -4
+  if(Symmetry > EQ_SYMM .and. dabs(X(1)) < dX) imin = -4
+  if(Symmetry > EQ_SYMM .and. dabs(Y(1)) < dY) jmin = -4
+
+  call symmetry_bd(5,ex,f,fh,SoA)
+
+! 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 .and. i+5 <= imax .and. i-3 >= imin)then
+!        v [ - 5 f(i-3v) + 60 f(i-2v) - 420 f(i-v) - 378 f(i) + 1050 f(i+v) - 420 f(i+2v) + 140 f(i+3v) - 30 f(i+4v) + 3 f(i+5v)]
+! D f = --------------------------------------------------------------------------------------------------------------------------
+!  i                                                             840 dx    
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                                                                         &
+                  Sfx(i,j,k)*d840dx*(-F5*fh(i-3,j,k)+F60 *fh(i-2,j,k)-F420*fh(i-1,j,k)-F378*fh(i  ,j,k) &
+                                  +F1050*fh(i+1,j,k)-F420*fh(i+2,j,k)+F140*fh(i+3,j,k)-F30 *fh(i+4,j,k)+THR*fh(i+5,j,k))
+
+    elseif(Sfx(i,j,k) <= ZEO .and. i-5 >= imin .and. i+3 <= imax)then
+     f_rhs(i,j,k)=f_rhs(i,j,k)-                                                                          &
+                  Sfx(i,j,k)*d840dx*(-F5*fh(i+3,j,k)+F60 *fh(i+2,j,k)-F420*fh(i+1,j,k)-F378*fh(i   ,j,k) &
+                                  +F1050*fh(i-1,j,k)-F420*fh(i-2,j,k)+F140*fh(i-3,j,k)- F30*fh(i-4,j,k)+THR*fh(i-5,j,k))
+
+    elseif(i+4 <= imax .and. i-4 >= imin)then
+!           3 f(i-4) - 32 f(i-3) + 168 f(i-2) - 672 f(i-1) + 672 f(i+1) - 168 f(i+2) + 32 f(i+3) - 3 f(i+4)
+!  fx(i) = -------------------------------------------------------------------------------------------------
+!                                                        840 dx
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                                                                              &
+                  Sfx(i,j,k)*d840dx*( THR*fh(i-4,j,k)-F32 *fh(i-3,j,k)+F168*fh(i-2,j,k)-F672*fh(i-1,j,k)+    &
+                                     F672*fh(i+1,j,k)-F168*fh(i+2,j,k)+F32 *fh(i+3,j,k)-THR *fh(i+4,j,k))
+
+     elseif(i+3 <= imax .and. i-3 >= imin)then
+!           - f(i-3) + 9 f(i-2) - 45 f(i-1) + 45 f(i+1) - 9 f(i+2) + f(i+3)
+!  fx(i) = -----------------------------------------------------------------
+!                                        60 dx
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                                                                              &
+                  Sfx(i,j,k)*d60dx*(-fh(i-3,j,k)+F9*fh(i-2,j,k)-F45*fh(i-1,j,k)+F45*fh(i+1,j,k)-F9*fh(i+2,j,k)+fh(i+3,j,k))
+
+     elseif(i+2 <= imax .and. i-2 >= imin)then
+!
+!              f(i-2) - 8 f(i-1) + 8 f(i+1) - f(i+2)
+!  fx(i) = ---------------------------------------------
+!                             12 dx
+     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 .and. i-1 >= imin)then
+!
+!              - f(i-1) + f(i+1)
+!  fx(i) = --------------------------------
+!                     2 dx
+     f_rhs(i,j,k)=f_rhs(i,j,k) + Sfx(i,j,k)*d2dx*(-fh(i-1,j,k)+fh(i+1,j,k))
+
+! set imax and imin 0
+    endif
+
+! y direction   
+    if(Sfy(i,j,k) >= ZEO .and. j+5 <= jmax .and. j-3 >= jmin)then
+
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                                                                         &
+                  Sfy(i,j,k)*d840dy*(-F5*fh(i,j-3,k)+F60 *fh(i,j-2,k)-F420*fh(i,j-1,k)-F378*fh(i,j  ,k) &
+                                  +F1050*fh(i,j+1,k)-F420*fh(i,j+2,k)+F140*fh(i,j+3,k)-F30 *fh(i,j+4,k)+THR*fh(i,j+5,k))
+
+    elseif(Sfy(i,j,k) <= ZEO .and. j-5 >= jmin .and. j+3 <= jmax)then
+     f_rhs(i,j,k)=f_rhs(i,j,k)-                                                                         &
+                  Sfy(i,j,k)*d840dy*(-F5*fh(i,j+3,k)+F60 *fh(i,j+2,k)-F420*fh(i,j+1,k)-F378*fh(i,j  ,k) &
+                                  +F1050*fh(i,j-1,k)-F420*fh(i,j-2,k)+F140*fh(i,j-3,k)- F30*fh(i,j-4,k)+THR*fh(i,j-5,k))
+
+    elseif(j+4 <= jmax .and. j-4 >= jmin)then
+
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                                                                              &
+                  Sfy(i,j,k)*d840dy*( THR*fh(i,j-4,k)-F32 *fh(i,j-3,k)+F168*fh(i,j-2,k)-F672*fh(i,j-1,k)+    &
+                                     F672*fh(i,j+1,k)-F168*fh(i,j+2,k)+F32 *fh(i,j+3,k)-THR *fh(i,j+4,k))
+
+     elseif(j+3 <= jmax .and. j-3 >= jmin)then
+          
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                                                                                         &
+                  Sfy(i,j,k)*d60dy*(-fh(i,j-3,k)+F9*fh(i,j-2,k)-F45*fh(i,j-1,k)+F45*fh(i,j+1,k)-F9*fh(i,j+2,k)+fh(i,j+3,k))
+
+     elseif(j+2 <= jmax .and. 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 <= jmax .and. j-1 >= jmin)then
+
+     f_rhs(i,j,k)=f_rhs(i,j,k) + Sfy(i,j,k)*d2dy*(-fh(i,j-1,k)+fh(i,j+1,k))
+! set jmin and jmax 0
+     endif
+!! z direction   
+    if(Sfz(i,j,k) >= ZEO .and. k+5 <= kmax .and. k-3 >= kmin)then
+
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                                                                         &
+                  Sfz(i,j,k)*d840dz*(-F5*fh(i,j,k-3)+F60 *fh(i,j,k-2)-F420*fh(i,j,k-1)-F378*fh(i,j,k  ) &
+                                  +F1050*fh(i,j,k+1)-F420*fh(i,j,k+2)+F140*fh(i,j,k+3)-F30 *fh(i,j,k+4)+THR*fh(i,j,k+5))
+
+    elseif(Sfz(i,j,k) <= ZEO .and. k-5 >= kmin .and. k+3 <= kmax)then
+     f_rhs(i,j,k)=f_rhs(i,j,k)-                                                                         &
+                  Sfz(i,j,k)*d840dz*(-F5*fh(i,j,k+3)+F60 *fh(i,j,k+2)-F420*fh(i,j,k+1)-F378*fh(i,j,k  ) &
+                                  +F1050*fh(i,j,k-1)-F420*fh(i,j,k-2)+F140*fh(i,j,k-3)- F30*fh(i,j,k-4)+THR*fh(i,j,k-5))
+
+    elseif(k+4 <= kmax .and. k-4 >= kmin)then
+
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                                                                              &
+                  Sfz(i,j,k)*d840dz*( THR*fh(i,j,k-4)-F32 *fh(i,j,k-3)+F168*fh(i,j,k-2)-F672*fh(i,j,k-1)+    &
+                                     F672*fh(i,j,k+1)-F168*fh(i,j,k+2)+F32 *fh(i,j,k+3)-THR *fh(i,j,k+4))
+     
+     elseif(k+3 <= kmax .and. k-3 >= kmin)then
+
+     f_rhs(i,j,k)=f_rhs(i,j,k)+                                                                                         &
+                  Sfz(i,j,k)*d60dz*(-fh(i,j,k-3)+F9*fh(i,j,k-2)-F45*fh(i,j,k-1)+F45*fh(i,j,k+1)-F9*fh(i,j,k+2)+fh(i,j,k+3))
+
+     elseif(k+2 <= kmax .and. 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 <= kmax .and. k-1 >= kmin)then
+
+     f_rhs(i,j,k)=f_rhs(i,j,k)+Sfz(i,j,k)*d2dz*(-fh(i,j,k-1)+fh(i,j,k+1))
+! set kmin and kmax 0
+     endif
+
+  enddo
+  enddo
+  enddo
+
+  return
+
+  end subroutine lopsided
+
+#endif  

AMSS_NCKU_source/makefile.inc

@@ -10,14 +10,15 @@ filein  = -I/usr/include/ -I${MKLROOT}/include
 ## Added -lifcore for Intel Fortran runtime and -limf for Intel math library
 LDLIBS  = -L${MKLROOT}/lib -lmkl_intel_lp64 -lmkl_sequential -lmkl_core -lifcore -limf -lpthread -lm -ldl
 
-## Aggressive optimization flags:
+## Aggressive optimization flags + PGO Phase 2 (profile-guided optimization)
-## -O3: Maximum optimization
+## -fprofile-instr-use: use collected profile data to guide optimization decisions
-## -xHost: Optimize for the host CPU architecture (Intel/AMD compatible)
+##   (branch prediction, basic block layout, inlining, loop unrolling)
-## -fp-model fast=2: Aggressive floating-point optimizations
+PROFDATA     = ../../pgo_profile/default.profdata
-## -fma: Enable fused multiply-add instructions
 CXXAPPFLAGS  = -O3 -xHost -fp-model fast=2 -fma -ipo \
+               -fprofile-instr-use=$(PROFDATA) \
                -Dfortran3 -Dnewc -I${MKLROOT}/include
 f90appflags  = -O3 -xHost -fp-model fast=2 -fma -ipo \
+               -fprofile-instr-use=$(PROFDATA) \
                -align array64byte -fpp -I${MKLROOT}/include
 f90          = ifx
 f77          = ifx

makefile_and_run.py

@@ -15,12 +15,13 @@ import time
 ## 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 0-111"
+NUMACTL_CPU_BIND = "taskset -c 16-47,64-95"
 
 ## Build parallelism configuration
 ## Use nohz_full cores (4-55, 60-111) for compilation: 52 + 52 = 104 cores
 ## Set make -j to utilize available cores for faster builds
-BUILD_JOBS = 104
+BUILD_JOBS = 96
 
 
 ##################################################################
@@ -117,6 +118,7 @@ def run_ABE():
     
     if (input_data.GPU_Calculation == "no"):
         mpi_command         = NUMACTL_CPU_BIND + " mpirun -np " + str(input_data.MPI_processes) + " ./ABE"
+        #mpi_command         = " mpirun -np " + str(input_data.MPI_processes) + " ./ABE"
         mpi_command_outfile = "ABE_out.log"
     elif (input_data.GPU_Calculation == "yes"):
         mpi_command         = NUMACTL_CPU_BIND + " mpirun -np " + str(input_data.MPI_processes) + " ./ABEGPU"
@@ -158,7 +160,8 @@ def run_TwoPunctureABE():
     print(                                                          )
     
     ## Define the command to run
-    TwoPuncture_command         = NUMACTL_CPU_BIND + " ./TwoPunctureABE"
+    #TwoPuncture_command         = NUMACTL_CPU_BIND + " ./TwoPunctureABE"
+    TwoPuncture_command         = " ./TwoPunctureABE"
     TwoPuncture_command_outfile = "TwoPunctureABE_out.log"
 
     ## Execute the command with subprocess.Popen and stream output

parallel_plot_helper.py

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

pgo_profile/PGO_Profile_Analysis.md

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

plot_GW_strain_amplitude_xiaoqu.py

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

plot_binary_data.py

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

plot_xiaoqu.py

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