feat: Implement hybrid MPI+OpenMP parallelization

- Enable -qopenmp in makefile.inc - Add OpenMP directives to 4th order derivatives in diff_new.f90 - Update makefile_and_run.py to dynamic calculate OMP_NUM_THREADS based on 96 cores and remove hardcoded CPU binding
2026-02-06 13:25:07 +08:00
13 changed files with 1686 additions and 1936 deletions
--- a/AMSS_NCKU_ABEtest.py
+++ b/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()
 ##################################################################
--- a/AMSS_NCKU_Input.py
+++ b/AMSS_NCKU_Input.py
@@ -16,7 +16,7 @@ import numpy
 File_directory   = "GW150914"                    ## output file directory
 Output_directory = "binary_output"               ## binary data file directory
                                                 ## The file directory name should not be too long
-MPI_processes    = 64                             ## number of mpi processes used in the simulation
+MPI_processes    = 48                             ## number of mpi processes used in the simulation
 GPU_Calculation  = "no"                          ## Use GPU or not 
                                                 ## (prefer "no" in the current version, because the GPU part may have bugs when integrated in this Python interface)
--- a/AMSS_NCKU_source/TwoPunctures.C
+++ b/AMSS_NCKU_source/TwoPunctures.C
--- a/AMSS_NCKU_source/TwoPunctures.h
+++ b/AMSS_NCKU_source/TwoPunctures.h
@@ -1,8 +1,7 @@
 #ifndef TWO_PUNCTURES_H
 #define TWO_PUNCTURES_H
 #include <omp.h>
 #define StencilSize 19
 #define N_PlaneRelax 1
 #define NRELAX 200
@@ -33,7 +32,7 @@ private:
       int npoints_A, npoints_B, npoints_phi;
       double target_M_plus, target_M_minus;
-
+       
       double admMass;
       double adm_tol;
@@ -43,18 +42,6 @@ private:
       int ntotal;
       // ===== Precomputed spectral derivative matrices =====
       double *D1_A, *D2_A;
       double *D1_B, *D2_B;
       double *DF1_phi, *DF2_phi;
       // ===== Pre-allocated workspace for LineRelax (per-thread) =====
       int max_threads;
       double **ws_diag_be, **ws_e_be, **ws_f_be, **ws_b_be, **ws_x_be;
       double **ws_l_be, **ws_u_be, **ws_d_be, **ws_y_be;
       double **ws_diag_al, **ws_e_al, **ws_f_al, **ws_b_al, **ws_x_al;
       double **ws_l_al, **ws_u_al, **ws_d_al, **ws_y_al;
       struct parameters
       {
              int nvar, n1, n2, n3;
@@ -71,28 +58,6 @@ public:
                    int Newtonmaxit);
       ~TwoPunctures();
       // 02/07: New/modified methods
       void allocate_workspace();
       void free_workspace();
       void precompute_derivative_matrices();
       void build_cheb_deriv_matrices(int n, double *D1, double *D2);
       void build_fourier_deriv_matrices(int N, double *DF1, double *DF2);
       void Derivatives_AB3_MatMul(int nvar, int n1, int n2, int n3, derivs v);
       void ThomasAlgorithm_ws(int N, double *b, double *a, double *c, double *x, double *q,
                                double *l, double *u_ws, double *d, double *y);
       void LineRelax_be_omp(double *dv,
                         int const i, int const k, int const nvar,
                         int const n1, int const n2, int const n3,
                         double const *rhs, int const *ncols, int **cols,
                         double **JFD, int tid);
       void LineRelax_al_omp(double *dv,
                         int const j, int const k, int const nvar,
                         int const n1, int const n2, int const n3,
                         double const *rhs, int const *ncols,
                         int **cols, double **JFD, int tid);
       void relax_omp(double *dv, int const nvar, int const n1, int const n2, int const n3,
                  double const *rhs, int const *ncols, int **cols, double **JFD);
       void Solve();
       void set_initial_guess(derivs v);
       int index(int i, int j, int k, int l, int a, int b, int c, int d);
@@ -151,11 +116,23 @@ public:
       double BY_KKofxyz(double x, double y, double z);
       void SetMatrix_JFD(int nvar, int n1, int n2, int n3, derivs u, int *ncols, int **cols, double **Matrix);
       void J_times_dv(int nvar, int n1, int n2, int n3, derivs dv, double *Jdv, derivs u);
       void relax(double *dv, int const nvar, int const n1, int const n2, int const n3,
                  double const *rhs, int const *ncols, int **cols, double **JFD);
       void LineRelax_be(double *dv,
                         int const i, int const k, int const nvar,
                         int const n1, int const n2, int const n3,
                         double const *rhs, int const *ncols, int **cols,
                         double **JFD);
       void JFD_times_dv(int i, int j, int k, int nvar, int n1, int n2,
                         int n3, derivs dv, derivs u, double *values);
       void LinEquations(double A, double B, double X, double R,
                         double x, double r, double phi,
                         double y, double z, derivs dU, derivs U, double *values);
       void LineRelax_al(double *dv,
                         int const j, int const k, int const nvar,
                         int const n1, int const n2, int const n3,
                         double const *rhs, int const *ncols,
                         int **cols, double **JFD);
       void ThomasAlgorithm(int N, double *b, double *a, double *c, double *x, double *q);
       void Save(char *fname);
       // provided by Vasileios Paschalidis (vpaschal@illinois.edu)
@@ -164,4 +141,4 @@ public:
       void SpecCoef(parameters par, int ivar, double *v, double *cf);
 };
-#endif /* TWO_PUNCTURES_H */
+#endif /* TWO_PUNCTURES_H */
--- a/AMSS_NCKU_source/bssn_rhs.f90
+++ b/AMSS_NCKU_source/bssn_rhs.f90
@@ -106,8 +106,7 @@
  call getpbh(BHN,Porg,Mass)
 #endif
-!!! sanity check (disabled in production builds for performance)
+!!! sanity check
 #ifdef DEBUG
  dX = sum(chi)+sum(trK)+sum(dxx)+sum(gxy)+sum(gxz)+sum(dyy)+sum(gyz)+sum(dzz) &
      +sum(Axx)+sum(Axy)+sum(Axz)+sum(Ayy)+sum(Ayz)+sum(Azz)                   &
      +sum(Gamx)+sum(Gamy)+sum(Gamz)                                           &
@@ -137,7 +136,6 @@
     gont = 1
     return
  endif
 #endif
  PI = dacos(-ONE)
@@ -161,8 +159,36 @@
  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 +199,7 @@
  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)&
@@ -924,99 +945,99 @@
 !!!!!!!!!advection term part
  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
  if(eps>0)then 
 ! 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
  if(eps>0)then 
 ! usual Kreiss-Oliger dissipation      
  call kodis(ex,X,Y,Z,chi,chi_rhs,SSS,Symmetry,eps)
  call kodis(ex,X,Y,Z,trK,trK_rhs,SSS,Symmetry,eps)
  call kodis(ex,X,Y,Z,dxx,gxx_rhs,SSS,Symmetry,eps)
  call kodis(ex,X,Y,Z,gxy,gxy_rhs,AAS,Symmetry,eps)
  call kodis(ex,X,Y,Z,gxz,gxz_rhs,ASA,Symmetry,eps)
  call kodis(ex,X,Y,Z,dyy,gyy_rhs,SSS,Symmetry,eps)
  call kodis(ex,X,Y,Z,gyz,gyz_rhs,SAA,Symmetry,eps)
  call kodis(ex,X,Y,Z,dzz,gzz_rhs,SSS,Symmetry,eps)
 #if 0
 #define i 42
 #define j 40
 #define k 40
 if(Lev == 1)then
 write(*,*) X(i),Y(j),Z(k)
 write(*,*) "before",Axx_rhs(i,j,k)
 endif
 #undef i
 #undef j
 #undef k
 !!stop
 #endif
  call kodis(ex,X,Y,Z,Axx,Axx_rhs,SSS,Symmetry,eps)
 #if 0
 #define i 42
 #define j 40
 #define k 40
 if(Lev == 1)then
 write(*,*) X(i),Y(j),Z(k)
 write(*,*) "after",Axx_rhs(i,j,k)
 endif
 #undef i
 #undef j
 #undef k
 !!stop
 #endif
  call kodis(ex,X,Y,Z,Axy,Axy_rhs,AAS,Symmetry,eps)
  call kodis(ex,X,Y,Z,Axz,Axz_rhs,ASA,Symmetry,eps)
  call kodis(ex,X,Y,Z,Ayy,Ayy_rhs,SSS,Symmetry,eps)
  call kodis(ex,X,Y,Z,Ayz,Ayz_rhs,SAA,Symmetry,eps)
  call kodis(ex,X,Y,Z,Azz,Azz_rhs,SSS,Symmetry,eps)
  call kodis(ex,X,Y,Z,Gamx,Gamx_rhs,ASS,Symmetry,eps)
  call kodis(ex,X,Y,Z,Gamy,Gamy_rhs,SAS,Symmetry,eps)
  call kodis(ex,X,Y,Z,Gamz,Gamz_rhs,SSA,Symmetry,eps)
 #if 1 
 !! bam does not apply dissipation on gauge variables
  call kodis(ex,X,Y,Z,Lap,Lap_rhs,SSS,Symmetry,eps)
  call kodis(ex,X,Y,Z,betax,betax_rhs,ASS,Symmetry,eps)
  call kodis(ex,X,Y,Z,betay,betay_rhs,SAS,Symmetry,eps)
  call kodis(ex,X,Y,Z,betaz,betaz_rhs,SSA,Symmetry,eps)
 #if (GAUGE == 0 || GAUGE == 2 || GAUGE == 3 || GAUGE == 6 || GAUGE == 7)
  call kodis(ex,X,Y,Z,dtSfx,dtSfx_rhs,ASS,Symmetry,eps)
  call kodis(ex,X,Y,Z,dtSfy,dtSfy_rhs,SAS,Symmetry,eps)
  call kodis(ex,X,Y,Z,dtSfz,dtSfz_rhs,SSA,Symmetry,eps)
 #endif
 #endif
  endif
@@ -1163,265 +1184,3 @@ 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
--- a/AMSS_NCKU_source/diff_new.f90
+++ b/AMSS_NCKU_source/diff_new.f90
@@ -69,6 +69,7 @@
  fy = ZEO
  fz = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -151,6 +152,7 @@
  fx = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -218,6 +220,7 @@
  fy = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -282,6 +285,7 @@
  fz = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -371,6 +375,7 @@
  fxz = ZEO
  fyz = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -469,6 +474,7 @@
  fxx = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -531,6 +537,7 @@
  fyy = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -594,6 +601,7 @@
  fzz = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -657,6 +665,7 @@
  fxy = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -719,6 +728,7 @@
  fxz = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -780,6 +790,7 @@
  fyz = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -866,6 +877,7 @@
  fxz = ZEO
  fyz = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -997,10 +1009,90 @@
  fy = ZEO
  fz = ZEO
  !$omp parallel do collapse(3) schedule(static)
  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 +1107,7 @@
      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
@@ -1072,6 +1164,7 @@
  fx = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -1148,6 +1241,7 @@
  fy = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -1218,6 +1312,7 @@
  fz = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -1322,10 +1417,89 @@
  fxz = ZEO
  fyz = ZEO
  !$omp parallel do collapse(3) schedule(static)
  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 +1535,7 @@
   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
@@ -1419,6 +1593,7 @@
  fxx = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -1486,6 +1661,7 @@
  fyy = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -1555,6 +1731,7 @@
  fzz = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -1624,6 +1801,7 @@
  fxy = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -1694,6 +1872,7 @@
  fxz = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -1762,6 +1941,7 @@
  fyz = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -1854,6 +2034,7 @@
  fy = ZEO
  fz = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -1970,6 +2151,7 @@
  fx = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -2055,6 +2237,7 @@
  fy = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -2131,6 +2314,7 @@
  fz = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -2249,6 +2433,7 @@
  fxz = ZEO
  fyz = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -2436,6 +2621,7 @@
  fxx = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -2508,6 +2694,7 @@
  fyy = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -2583,6 +2770,7 @@
  fzz = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -2658,6 +2846,7 @@
  fxy = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -2738,6 +2927,7 @@
  fxz = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -2816,6 +3006,7 @@
  fyz = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -2923,6 +3114,7 @@
  fy = ZEO
  fz = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -3059,6 +3251,7 @@
  fx = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -3154,6 +3347,7 @@
  fy = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -3238,6 +3432,7 @@
  fz = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -3373,6 +3568,7 @@
  fxz = ZEO
  fyz = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -3645,6 +3841,7 @@
  fxx = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -3726,6 +3923,7 @@
  fyy = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -3810,6 +4008,7 @@
  fzz = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -3894,6 +4093,7 @@
  fxy = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -3996,6 +4196,7 @@
  fxz = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
@@ -4096,6 +4297,7 @@
  fyz = ZEO
  !$omp parallel do collapse(3) schedule(static)
  do k=1,ex(3)-1
  do j=1,ex(2)-1
  do i=1,ex(1)-1
--- a/AMSS_NCKU_source/enforce_algebra.f90
+++ b/AMSS_NCKU_source/enforce_algebra.f90
@@ -18,61 +18,49 @@
  real*8, dimension(ex(1),ex(2),ex(3)), intent(inout) :: Ayy,Ayz,Azz
 !~~~~~~~> Local variable:
-
+  
-  integer :: i,j,k
+  real*8, dimension(ex(1),ex(2),ex(3)) :: trA,detg
-  real*8 :: lgxx,lgyy,lgzz,ldetg
+  real*8, dimension(ex(1),ex(2),ex(3)) :: gxx,gyy,gzz 
-  real*8 :: lgupxx,lgupxy,lgupxz,lgupyy,lgupyz,lgupzz
+  real*8, dimension(ex(1),ex(2),ex(3)) :: gupxx,gupxy,gupxz,gupyy,gupyz,gupzz
  real*8 :: ltrA,lscale
  real*8, parameter :: F1o3 = 1.D0 / 3.D0, ONE = 1.D0, TWO = 2.D0
 !~~~~~~>
-  do k=1,ex(3)
+  gxx = dxx + ONE
-  do j=1,ex(2)
+  gyy = dyy + ONE
-  do i=1,ex(1)
+  gzz = dzz + ONE
-    lgxx = dxx(i,j,k) + ONE
+  detg =  gxx * gyy * gzz + gxy * gyz * gxz + gxz * gxy * gyz - &
-    lgyy = dyy(i,j,k) + ONE
+          gxz * gyy * gxz - gxy * gxy * gzz - gxx * gyz * gyz
-    lgzz = dzz(i,j,k) + ONE
+  gupxx =   ( gyy * gzz - gyz * gyz ) / detg
  gupxy = - ( gxy * gzz - gyz * gxz ) / detg
  gupxz =   ( gxy * gyz - gyy * gxz ) / detg
  gupyy =   ( gxx * gzz - gxz * gxz ) / detg
  gupyz = - ( gxx * gyz - gxy * gxz ) / detg
  gupzz =   ( gxx * gyy - gxy * gxy ) / detg
-    ldetg =  lgxx * lgyy * lgzz &
+  trA =         gupxx * Axx + gupyy * Ayy + gupzz * Azz &
-           + gxy(i,j,k) * gyz(i,j,k) * gxz(i,j,k) &
+       + TWO * (gupxy * Axy + gupxz * Axz + gupyz * Ayz)
           + gxz(i,j,k) * gxy(i,j,k) * gyz(i,j,k) &
           - gxz(i,j,k) * lgyy * gxz(i,j,k) &
           - gxy(i,j,k) * gxy(i,j,k) * lgzz &
           - lgxx * gyz(i,j,k) * gyz(i,j,k)
-    lgupxx =   ( lgyy * lgzz - gyz(i,j,k) * gyz(i,j,k) ) / ldetg
+  Axx = Axx - F1o3 * gxx * trA
-    lgupxy = - ( gxy(i,j,k) * lgzz - gyz(i,j,k) * gxz(i,j,k) ) / ldetg
+  Axy = Axy - F1o3 * gxy * trA
-    lgupxz =   ( gxy(i,j,k) * gyz(i,j,k) - lgyy * gxz(i,j,k) ) / ldetg
+  Axz = Axz - F1o3 * gxz * trA
-    lgupyy =   ( lgxx * lgzz - gxz(i,j,k) * gxz(i,j,k) ) / ldetg
+  Ayy = Ayy - F1o3 * gyy * trA
-    lgupyz = - ( lgxx * gyz(i,j,k) - gxy(i,j,k) * gxz(i,j,k) ) / ldetg
+  Ayz = Ayz - F1o3 * gyz * trA
-    lgupzz =   ( lgxx * lgyy - gxy(i,j,k) * gxy(i,j,k) ) / ldetg
+  Azz = Azz - F1o3 * gzz * trA
-    ltrA =         lgupxx * Axx(i,j,k) + lgupyy * Ayy(i,j,k) &
+  detg = ONE / ( detg ** F1o3 ) 
-                 + lgupzz * Azz(i,j,k) &
+  
-         + TWO * (lgupxy * Axy(i,j,k) + lgupxz * Axz(i,j,k) &
+  gxx = gxx * detg
-                 + lgupyz * Ayz(i,j,k))
+  gxy = gxy * detg
  gxz = gxz * detg
  gyy = gyy * detg
  gyz = gyz * detg
  gzz = gzz * detg
-    Axx(i,j,k) = Axx(i,j,k) - F1o3 * lgxx * ltrA
+  dxx = gxx - ONE
-    Axy(i,j,k) = Axy(i,j,k) - F1o3 * gxy(i,j,k) * ltrA
+  dyy = gyy - ONE
-    Axz(i,j,k) = Axz(i,j,k) - F1o3 * gxz(i,j,k) * ltrA
+  dzz = gzz - ONE
    Ayy(i,j,k) = Ayy(i,j,k) - F1o3 * lgyy * ltrA
    Ayz(i,j,k) = Ayz(i,j,k) - F1o3 * gyz(i,j,k) * ltrA
    Azz(i,j,k) = Azz(i,j,k) - F1o3 * lgzz * ltrA
    lscale = ONE / ( ldetg ** F1o3 )
    dxx(i,j,k) = lgxx * lscale - ONE
    gxy(i,j,k) = gxy(i,j,k) * lscale
    gxz(i,j,k) = gxz(i,j,k) * lscale
    dyy(i,j,k) = lgyy * lscale - ONE
    gyz(i,j,k) = gyz(i,j,k) * lscale
    dzz(i,j,k) = lgzz * lscale - ONE
  enddo
  enddo
  enddo
  return
@@ -94,71 +82,51 @@
  real*8, dimension(ex(1),ex(2),ex(3)), intent(inout) :: Ayy,Ayz,Azz
 !~~~~~~~> Local variable:
-
+  
-  integer :: i,j,k
+  real*8, dimension(ex(1),ex(2),ex(3)) :: trA
-  real*8 :: lgxx,lgyy,lgzz,lscale
+  real*8, dimension(ex(1),ex(2),ex(3)) :: gxx,gyy,gzz 
-  real*8 :: lgxy,lgxz,lgyz
+  real*8, dimension(ex(1),ex(2),ex(3)) :: gupxx,gupxy,gupxz,gupyy,gupyz,gupzz
  real*8 :: lgupxx,lgupxy,lgupxz,lgupyy,lgupyz,lgupzz
  real*8 :: ltrA
  real*8, parameter :: F1o3 = 1.D0 / 3.D0, ONE = 1.D0, TWO = 2.D0
 !~~~~~~>
-  do k=1,ex(3)
+  gxx = dxx + ONE
-  do j=1,ex(2)
+  gyy = dyy + ONE
-  do i=1,ex(1)
+  gzz = dzz + ONE
 ! for g
  gupzz =  gxx * gyy * gzz + gxy * gyz * gxz + gxz * gxy * gyz - &
           gxz * gyy * gxz - gxy * gxy * gzz - gxx * gyz * gyz
-! for g: normalize determinant first
+  gupzz = ONE / ( gupzz ** F1o3 ) 
-    lgxx = dxx(i,j,k) + ONE
+  
-    lgyy = dyy(i,j,k) + ONE
+  gxx = gxx * gupzz
-    lgzz = dzz(i,j,k) + ONE
+  gxy = gxy * gupzz
-    lgxy = gxy(i,j,k)
+  gxz = gxz * gupzz
-    lgxz = gxz(i,j,k)
+  gyy = gyy * gupzz
-    lgyz = gyz(i,j,k)
+  gyz = gyz * gupzz
  gzz = gzz * gupzz
-    lscale =  lgxx * lgyy * lgzz + lgxy * lgyz * lgxz &
+  dxx = gxx - ONE
-            + lgxz * lgxy * lgyz - lgxz * lgyy * lgxz &
+  dyy = gyy - ONE
-            - lgxy * lgxy * lgzz - lgxx * lgyz * lgyz
+  dzz = gzz - ONE
 ! for A  
-    lscale = ONE / ( lscale ** F1o3 )
+  gupxx =   ( gyy * gzz - gyz * gyz )
  gupxy = - ( gxy * gzz - gyz * gxz )
  gupxz =   ( gxy * gyz - gyy * gxz )
  gupyy =   ( gxx * gzz - gxz * gxz )
  gupyz = - ( gxx * gyz - gxy * gxz )
  gupzz =   ( gxx * gyy - gxy * gxy )
-    lgxx = lgxx * lscale
+  trA =         gupxx * Axx + gupyy * Ayy + gupzz * Azz &
-    lgxy = lgxy * lscale
+       + TWO * (gupxy * Axy + gupxz * Axz + gupyz * Ayz)
    lgxz = lgxz * lscale
    lgyy = lgyy * lscale
    lgyz = lgyz * lscale
    lgzz = lgzz * lscale
-    dxx(i,j,k) = lgxx - ONE
+  Axx = Axx - F1o3 * gxx * trA
-    gxy(i,j,k) = lgxy
+  Axy = Axy - F1o3 * gxy * trA
-    gxz(i,j,k) = lgxz
+  Axz = Axz - F1o3 * gxz * trA
-    dyy(i,j,k) = lgyy - ONE
+  Ayy = Ayy - F1o3 * gyy * trA
-    gyz(i,j,k) = lgyz
+  Ayz = Ayz - F1o3 * gyz * trA
-    dzz(i,j,k) = lgzz - ONE
+  Azz = Azz - F1o3 * gzz * trA
 ! for A: trace-free using normalized metric (det=1, no division needed)
    lgupxx =   ( lgyy * lgzz - lgyz * lgyz )
    lgupxy = - ( lgxy * lgzz - lgyz * lgxz )
    lgupxz =   ( lgxy * lgyz - lgyy * lgxz )
    lgupyy =   ( lgxx * lgzz - lgxz * lgxz )
    lgupyz = - ( lgxx * lgyz - lgxy * lgxz )
    lgupzz =   ( lgxx * lgyy - lgxy * lgxy )
    ltrA =         lgupxx * Axx(i,j,k) + lgupyy * Ayy(i,j,k) &
                 + lgupzz * Azz(i,j,k) &
         + TWO * (lgupxy * Axy(i,j,k) + lgupxz * Axz(i,j,k) &
                 + lgupyz * Ayz(i,j,k))
    Axx(i,j,k) = Axx(i,j,k) - F1o3 * lgxx * ltrA
    Axy(i,j,k) = Axy(i,j,k) - F1o3 * lgxy * ltrA
    Axz(i,j,k) = Axz(i,j,k) - F1o3 * lgxz * ltrA
    Ayy(i,j,k) = Ayy(i,j,k) - F1o3 * lgyy * ltrA
    Ayz(i,j,k) = Ayz(i,j,k) - F1o3 * lgyz * ltrA
    Azz(i,j,k) = Azz(i,j,k) - F1o3 * lgzz * ltrA
  enddo
  enddo
  enddo
  return
--- a/AMSS_NCKU_source/fmisc.f90
+++ b/AMSS_NCKU_source/fmisc.f90
@@ -324,10 +324,10 @@ subroutine symmetry_bd(ord,extc,func,funcc,SoA)
  integer::i
  funcc = 0.d0
  funcc(1:extc(1),1:extc(2),1:extc(3)) = func
   do i=0,ord-1
-      
+      funcc(-i,1:extc(2),1:extc(3)) = funcc(i+2,1:extc(2),1:extc(3))*SoA(1)
    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)
@@ -350,6 +350,7 @@ subroutine symmetry_tbd(ord,extc,func,funcc,SoA)
  integer::i
  funcc = 0.d0
  funcc(1:extc(1),1:extc(2),1:extc(3)) = func
   do i=0,ord-1
      funcc(-i,1:extc(2),1:extc(3)) = funcc(i+2,1:extc(2),1:extc(3))*SoA(1)
@@ -378,6 +379,7 @@ subroutine symmetry_stbd(ord,extc,func,funcc,SoA)
  integer::i
  funcc = 0.d0
  funcc(1:extc(1),1:extc(2),1:extc(3)) = func
   do i=0,ord-1
      funcc(-i,1:extc(2),1:extc(3)) = funcc(i+2,1:extc(2),1:extc(3))*SoA(1)
@@ -884,6 +886,7 @@ subroutine symmetry_bd(ord,extc,func,funcc,SoA)
  integer::i
  funcc = 0.d0
  funcc(1:extc(1),1:extc(2),1:extc(3)) = func
   do i=0,ord-1
      funcc(-i,1:extc(2),1:extc(3)) = funcc(i+1,1:extc(2),1:extc(3))*SoA(1)
@@ -909,6 +912,7 @@ subroutine symmetry_tbd(ord,extc,func,funcc,SoA)
  integer::i
  funcc = 0.d0
  funcc(1:extc(1),1:extc(2),1:extc(3)) = func
   do i=0,ord-1
      funcc(-i,1:extc(2),1:extc(3)) = funcc(i+1,1:extc(2),1:extc(3))*SoA(1)
@@ -937,6 +941,7 @@ subroutine symmetry_stbd(ord,extc,func,funcc,SoA)
  integer::i
  funcc = 0.d0
  funcc(1:extc(1),1:extc(2),1:extc(3)) = func
   do i=0,ord-1
      funcc(-i,1:extc(2),1:extc(3)) = funcc(i+1,1:extc(2),1:extc(3))*SoA(1)
@@ -1113,65 +1118,64 @@ end subroutine d2dump
 ! Lagrangian polynomial interpolation
 !------------------------------------------------------------------------------
-  subroutine polint(xa, ya, x, y, dy, ordn)
+  subroutine polint(xa,ya,x,y,dy,ordn)
  implicit none
-  integer, intent(in) :: ordn
+!~~~~~~> Input Parameter:
-  real*8, dimension(ordn), intent(in) :: xa, ya
+  integer,intent(in) :: ordn
  real*8, dimension(ordn), intent(in) :: xa,ya
  real*8, intent(in) :: x
-  real*8, intent(out) :: y, dy
+  real*8, intent(out) :: y,dy
-  integer :: i, m, ns, n_m
+!~~~~~~> Other parameter:
  real*8, dimension(ordn) :: c, d, ho
  real*8 :: dif, dift, hp, h, den_val
-  c = ya
+  integer :: m,n,ns
-  d = ya
+  real*8, dimension(ordn) :: c,d,den,ho
-  ho = xa - x
+  real*8 :: dif,dift
-  ns = 1
+!~~~~~~>
  dif = abs(x - xa(1))
-  do i = 2, ordn
+  n=ordn
-    dift = abs(x - xa(i))
+  m=ordn
-    if (dift < dif) then
+
-      ns = i
+  c=ya
-      dif = dift
+  d=ya
-    end if
+  ho=xa-x
  ns=1
  dif=abs(x-xa(1))
  do m=1,n
   dift=abs(x-xa(m))
   if(dift < dif) then
    ns=m
    dif=dift
   end if
  end do
-  y = ya(ns)
+  y=ya(ns)
-  ns = ns - 1
+  ns=ns-1
-
+  do m=1,n-1
-  do m = 1, ordn - 1
+    den(1:n-m)=ho(1:n-m)-ho(1+m:n)
-    n_m = ordn - m
+    if (any(den(1:n-m) == 0.0))then
-    do i = 1, n_m
+      write(*,*) 'failure in polint for point',x
-      hp = ho(i)
+      write(*,*) 'with input points: ',xa
-      h  = ho(i+m)
+      stop
-      den_val = hp - h
+    endif
-
+    den(1:n-m)=(c(2:n-m+1)-d(1:n-m))/den(1:n-m)
-      if (den_val == 0.0d0) then
+    d(1:n-m)=ho(1+m:n)*den(1:n-m)
-        write(*,*) 'failure in polint for point',x
+    c(1:n-m)=ho(1:n-m)*den(1:n-m)
-        write(*,*) 'with input points: ',xa
+    if (2*ns < n-m) then
-        stop
+      dy=c(ns+1)
      end if
      den_val = (c(i+1) - d(i)) / den_val
      d(i) = h * den_val
      c(i) = hp * den_val
    end do
    if (2 * ns < n_m) then
      dy = c(ns + 1)
    else
-      dy = d(ns)
+      dy=d(ns)
-      ns = ns - 1
+      ns=ns-1
    end if
-    y = y + dy
+    y=y+dy
  end do
  return
  end subroutine polint
 !------------------------------------------------------------------------------
 !
@@ -1179,37 +1183,35 @@ end subroutine d2dump
 !
 !------------------------------------------------------------------------------
  subroutine polin2(x1a,x2a,ya,x1,x2,y,dy,ordn)
  implicit none
 !~~~~~~> Input parameters:
  integer,intent(in) :: ordn
  real*8, dimension(1:ordn), intent(in) :: x1a,x2a
  real*8, dimension(1:ordn,1:ordn), intent(in) :: ya
  real*8, intent(in) :: x1,x2
  real*8, intent(out) :: y,dy
-#ifdef POLINT_LEGACY_ORDER
+!~~~~~~> Other parameters:
  integer  :: i,m
  real*8, dimension(ordn) :: ymtmp
  real*8, dimension(ordn) :: yntmp
  m=size(x1a)
  do i=1,m
    yntmp=ya(i,:)
    call polint(x2a,yntmp,x2,ymtmp(i),dy,ordn)
  end do
  call polint(x1a,ymtmp,x1,y,dy,ordn)
 #else
  integer  :: j
  real*8, dimension(ordn) :: ymtmp
  real*8 :: dy_temp
  do j=1,ordn
    call polint(x1a, ya(:,j), x1, ymtmp(j), dy_temp, ordn)
  end do
-  call polint(x2a, ymtmp, x2, y, dy, ordn)
+
-#endif
+  call polint(x1a,ymtmp,x1,y,dy,ordn)
  return
  end subroutine polin2
 !------------------------------------------------------------------------------
 !
@@ -1217,15 +1219,18 @@ end subroutine d2dump
 !
 !------------------------------------------------------------------------------
  subroutine polin3(x1a,x2a,x3a,ya,x1,x2,x3,y,dy,ordn)
  implicit none
 !~~~~~~> Input parameters:
  integer,intent(in) :: ordn
  real*8, dimension(1:ordn), intent(in) :: x1a,x2a,x3a
  real*8, dimension(1:ordn,1:ordn,1:ordn), intent(in) :: ya
  real*8, intent(in) :: x1,x2,x3
  real*8, intent(out) :: y,dy
-#ifdef POLINT_LEGACY_ORDER
+!~~~~~~> Other parameters:
  integer  :: i,j,m,n
  real*8, dimension(ordn,ordn) :: yatmp
  real*8, dimension(ordn) :: ymtmp
@@ -1234,33 +1239,24 @@ end subroutine d2dump
  m=size(x1a)
  n=size(x2a)
  do i=1,m
   do j=1,n
    yqtmp=ya(i,j,:)
    call polint(x3a,yqtmp,x3,yatmp(i,j),dy,ordn)
   end do
    yntmp=yatmp(i,:)
    call polint(x2a,yntmp,x2,ymtmp(i),dy,ordn)
  end do
  call polint(x1a,ymtmp,x1,y,dy,ordn)
 #else
  integer  :: j, k
  real*8, dimension(ordn,ordn) :: yatmp
  real*8, dimension(ordn) :: ymtmp
  real*8 :: dy_temp
  do k=1,ordn
    do j=1,ordn
      call polint(x1a, ya(:,j,k), x1, yatmp(j,k), dy_temp, ordn)
    end do
  end do
-  do k=1,ordn
+
-    call polint(x2a, yatmp(:,k), x2, ymtmp(k), dy_temp, ordn)
+  call polint(x1a,ymtmp,x1,y,dy,ordn)
  end do
  call polint(x3a, ymtmp, x3, y, dy, ordn)
 #endif
  return
  end subroutine polin3
 !--------------------------------------------------------------------------------------
 ! calculate L2norm
--- a/AMSS_NCKU_source/kodiss.f90
+++ b/AMSS_NCKU_source/kodiss.f90
@@ -6,6 +6,101 @@
 ! 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
  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 +156,7 @@ 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 +166,28 @@ 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 +204,7 @@ 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 +215,218 @@ 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  
--- a/AMSS_NCKU_source/lopsidediff.f90
+++ b/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 +486,417 @@ subroutine lopsided(ex,X,Y,Z,f,f_rhs,Sfx,Sfy,Sfz,Symmetry,SoA)
  return
  end subroutine lopsided
 #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  
--- a/AMSS_NCKU_source/makefile
+++ b/AMSS_NCKU_source/makefile
@@ -16,12 +16,6 @@ include makefile.inc
 .cu.o:
 	$(Cu) $(CUDA_APP_FLAGS) -c $< -o $@ $(CUDA_LIB_PATH)
 TwoPunctures.o: TwoPunctures.C
 	${CXX} $(CXXAPPFLAGS) -qopenmp -c $< -o $@
 TwoPunctureABE.o: TwoPunctureABE.C
 	${CXX} $(CXXAPPFLAGS) -qopenmp -c $< -o $@
 # Input files
 C++FILES = ABE.o Ansorg.o Block.o misc.o monitor.o Parallel.o MPatch.o var.o\
           cgh.o bssn_class.o surface_integral.o ShellPatch.o\
@@ -102,7 +96,7 @@ ABEGPU: $(C++FILES_GPU) $(F90FILES) $(F77FILES) $(AHFDOBJS) $(CUDAFILES)
 	$(CLINKER) $(CXXAPPFLAGS) -o $@ $(C++FILES_GPU) $(F90FILES) $(F77FILES) $(AHFDOBJS) $(CUDAFILES) $(LDLIBS)
 TwoPunctureABE: $(TwoPunctureFILES)
-	$(CLINKER) $(CXXAPPFLAGS) -qopenmp -o $@ $(TwoPunctureFILES) $(LDLIBS)
+	$(CLINKER) $(CXXAPPFLAGS) -o $@ $(TwoPunctureFILES) $(LDLIBS)
 clean:
 	rm *.o ABE ABEGPU TwoPunctureABE make.log -f
--- a/AMSS_NCKU_source/makefile.inc
+++ b/AMSS_NCKU_source/makefile.inc
@@ -15,15 +15,16 @@ LDLIBS  = -L${MKLROOT}/lib -lmkl_intel_lp64 -lmkl_sequential -lmkl_core -lifcore
 ## -xHost: Optimize for the host CPU architecture (Intel/AMD compatible)
 ## -fp-model fast=2: Aggressive floating-point optimizations
 ## -fma: Enable fused multiply-add instructions
-CXXAPPFLAGS  = -O3 -xHost -fp-model fast=2 -fma -ipo \
+## Note: OpenMP enabled for hybrid MPI+OpenMP
 CXXAPPFLAGS  = -O3 -xHost -fp-model fast=2 -fma -qopenmp \
               -Dfortran3 -Dnewc -I${MKLROOT}/include
-f90appflags  = -O3 -xHost -fp-model fast=2 -fma -ipo \
+f90appflags  = -O3 -xHost -fp-model fast=2 -fma -qopenmp \
-               -align array64byte -fpp -I${MKLROOT}/include
+               -fpp -I${MKLROOT}/include
 f90          = ifx
 f77          = ifx
 CXX          = icpx
 CC           = icx
-CLINKER      = mpiicpx 
+CLINKER      = mpiicpx -qopenmp 
 Cu = nvcc
 CUDA_LIB_PATH = -L/usr/lib/cuda/lib64 -I/usr/include -I/usr/lib/cuda/include
--- a/makefile_and_run.py
+++ b/makefile_and_run.py
@@ -10,17 +10,15 @@
 import AMSS_NCKU_Input as input_data
 import subprocess
-import time
+
-## CPU core binding configuration using taskset
+## CPU core binding configuration
-## taskset ensures all child processes inherit the CPU affinity mask
+## Removed hardcoded taskset to allow full utilization of 96 cores via MPI+OpenMP
-## This forces make and all compiler processes to use only nohz_full cores (4-55, 60-111)
+NUMACTL_CPU_BIND = ""
 ## Format: taskset -c 4-55,60-111 ensures processes only run on these cores
 NUMACTL_CPU_BIND = "taskset -c 0-111"
 ## Build parallelism configuration
 ## Use nohz_full cores (4-55, 60-111) for compilation: 52 + 52 = 104 cores
 ## Set make -j to utilize available cores for faster builds
-BUILD_JOBS = 104
+BUILD_JOBS = 96
 ##################################################################
@@ -37,7 +35,7 @@ def makefile_ABE():
    print( " Compiling the AMSS-NCKU executable file ABE/ABEGPU " ) 
    print(                                                        )
-    ## Build command with CPU binding to nohz_full cores
+    ## Build command
    if (input_data.GPU_Calculation == "no"):
        makefile_command  = f"{NUMACTL_CPU_BIND} make -j{BUILD_JOBS} ABE"
    elif (input_data.GPU_Calculation == "yes"):
@@ -78,7 +76,7 @@ def makefile_TwoPunctureABE():
    print( " Compiling the AMSS-NCKU executable file TwoPunctureABE " )
    print(                                                            )
-    ## Build command with CPU binding to nohz_full cores
+    ## Build command
    makefile_command = f"{NUMACTL_CPU_BIND} make -j{BUILD_JOBS} TwoPunctureABE"
    ## Execute the command with subprocess.Popen and stream output
@@ -113,13 +111,23 @@ def run_ABE():
    print( " Running the AMSS-NCKU executable file ABE/ABEGPU " ) 
    print(                                                      )
    ## Calculate OMP_NUM_THREADS
    ## User has 96 cores. Calculate threads per MPI process.
    total_physical_cores = 96
    omp_num_threads = total_physical_cores // input_data.MPI_processes
    if omp_num_threads < 1:
        omp_num_threads = 1
    print( f" Configuration: {input_data.MPI_processes} MPI processes, {omp_num_threads} OpenMP threads per process." )
    print( f" Total cores utilized: {input_data.MPI_processes * omp_num_threads}" )
    ## Define the command to run; cast other values to strings as needed
    if (input_data.GPU_Calculation == "no"):
-        mpi_command         = NUMACTL_CPU_BIND + " mpirun -np " + str(input_data.MPI_processes) + " ./ABE"
+        mpi_command         = f"{NUMACTL_CPU_BIND} mpirun -genv OMP_NUM_THREADS {omp_num_threads} -np {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"
+        mpi_command         = f"{NUMACTL_CPU_BIND} mpirun -genv OMP_NUM_THREADS {omp_num_threads} -np {input_data.MPI_processes} ./ABEGPU"
        mpi_command_outfile = "ABEGPU_out.log"
    ## Execute the MPI command and stream output
@@ -152,7 +160,7 @@ def run_ABE():
 ## Run the AMSS-NCKU TwoPuncture program TwoPunctureABE
 def run_TwoPunctureABE():
-    tp_time1=time.time()
+
    print(                                                          )
    print( " Running the AMSS-NCKU executable file TwoPunctureABE " ) 
    print(                                                          )
@@ -179,9 +187,7 @@ def run_TwoPunctureABE():
    print(                                               )
    print( " The TwoPunctureABE simulation is finished " ) 
    print(                                               )
-    tp_time2=time.time()
+    
    et=tp_time2-tp_time1
    print(f"Used time: {et}")
    return
 ##################################################################