Switch legacy build to GCC and OpenMPI

AMSS_NCKU_source/FFT.f90

@@ -37,51 +37,56 @@ close(77)
 end program checkFFT
 #endif
 
-!-------------
-! Optimized FFT using Intel oneMKL DFTI
-! Mathematical equivalence: Standard DFT definition
-!   Forward (isign=1):  X[k] = sum_{n=0}^{N-1} x[n] * exp(-2*pi*i*k*n/N)
-!   Backward (isign=-1): X[k] = sum_{n=0}^{N-1} x[n] * exp(+2*pi*i*k*n/N)
-! Input/Output: dataa is interleaved complex array [Re(0),Im(0),Re(1),Im(1),...]
-!-------------
 SUBROUTINE four1(dataa,nn,isign)
-use MKL_DFTI
 implicit none
-INTEGER, intent(in) :: isign, nn
-DOUBLE PRECISION, dimension(2*nn), intent(inout) :: dataa
-
-type(DFTI_DESCRIPTOR), pointer :: desc
-integer :: status
-
-! Create DFTI descriptor for 1D complex-to-complex transform
-status = DftiCreateDescriptor(desc, DFTI_DOUBLE, DFTI_COMPLEX, 1, nn)
-if (status /= 0) return
-
-! Set input/output storage as interleaved complex (default)
-status = DftiSetValue(desc, DFTI_PLACEMENT, DFTI_INPLACE)
-if (status /= 0) then
-   status = DftiFreeDescriptor(desc)
-   return
+INTEGER::isign,nn
+double precision,dimension(2*nn)::dataa
+INTEGER::i,istep,j,m,mmax,n
+double precision::tempi,tempr
+DOUBLE PRECISION::theta,wi,wpi,wpr,wr,wtemp
+n=2*nn
+j=1
+do i=1,n,2
+  if(j.gt.i)then
+     tempr=dataa(j)
+     tempi=dataa(j+1)
+     dataa(j)=dataa(i)
+     dataa(j+1)=dataa(i+1)
+     dataa(i)=tempr
+     dataa(i+1)=tempi
+  endif
+  m=nn
+1 if ((m.ge.2).and.(j.gt.m)) then
+  j=j-m
+  m=m/2
+goto 1
+  endif
+j=j+m
+enddo
+mmax=2
+2  if (n.gt.mmax) then
+     istep=2*mmax
+     theta=6.28318530717959d0/(isign*mmax)
+     wpr=-2.d0*sin(0.5d0*theta)**2
+     wpi=sin(theta)
+     wr=1.d0
+     wi=0.d0
+     do m=1,mmax,2
+       do i=m,n,istep
+         j=i+mmax
+         tempr=sngl(wr)*dataa(j)-sngl(wi)*dataa(j+1)
+         tempi=sngl(wr)*dataa(j+1)+sngl(wi)*dataa(j)
+         dataa(j)=dataa(i)-tempr
+         dataa(j+1)=dataa(i+1)-tempi
+         dataa(i)=dataa(i)+tempr
+         dataa(i+1)=dataa(i+1)+tempi
+       enddo
+          wtemp=wr
+          wr=wr*wpr-wi*wpi+wr
+          wi=wi*wpr+wtemp*wpi+wi
+     enddo
+mmax=istep
+goto 2
 endif
-
-! Commit the descriptor
-status = DftiCommitDescriptor(desc)
-if (status /= 0) then
-   status = DftiFreeDescriptor(desc)
-   return
-endif
-
-! Execute FFT based on direction
-if (isign == 1) then
-   ! Forward FFT: exp(-2*pi*i*k*n/N)
-   status = DftiComputeForward(desc, dataa)
-else
-   ! Backward FFT: exp(+2*pi*i*k*n/N)
-   status = DftiComputeBackward(desc, dataa)
-endif
-
-! Free descriptor
-status = DftiFreeDescriptor(desc)
-
 return
 END SUBROUTINE four1

AMSS_NCKU_source/Parallel.C

@@ -5284,6 +5284,41 @@ double Parallel::L2Norm(Patch *Pat, var *vf)
 
   return tvf;
 }
+void Parallel::L2Norm7(Patch *Pat, var **vf, double *norms)
+{
+  int myrank;
+  MPI_Comm_rank(MPI_COMM_WORLD, &myrank);
+
+  double tvf[7], dtvf[7];
+  int BDW = ghost_width;
+  for (int i = 0; i < 7; i++)
+    dtvf[i] = 0;
+
+  MyList<Block> *BP = Pat->blb;
+  while (BP)
+  {
+    Block *cg = BP->data;
+    if (myrank == cg->rank)
+    {
+      f_l2normhelper7(cg->shape, cg->X[0], cg->X[1], cg->X[2],
+                      Pat->bbox[0], Pat->bbox[1], Pat->bbox[2],
+                      Pat->bbox[3], Pat->bbox[4], Pat->bbox[5],
+                      cg->fgfs[vf[0]->sgfn], cg->fgfs[vf[1]->sgfn], cg->fgfs[vf[2]->sgfn],
+                      cg->fgfs[vf[3]->sgfn], cg->fgfs[vf[4]->sgfn], cg->fgfs[vf[5]->sgfn],
+                      cg->fgfs[vf[6]->sgfn], tvf, BDW);
+      for (int i = 0; i < 7; i++)
+        dtvf[i] += tvf[i];
+    }
+    if (BP == Pat->ble)
+      break;
+    BP = BP->next;
+  }
+
+  MPI_Allreduce(dtvf, tvf, 7, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
+
+  for (int i = 0; i < 7; i++)
+    norms[i] = sqrt(tvf[i]);
+}
 double Parallel::L2Norm(Patch *Pat, var *vf, MPI_Comm Comm_here)
 {
   int myrank;
@@ -5315,6 +5350,41 @@ double Parallel::L2Norm(Patch *Pat, var *vf, MPI_Comm Comm_here)
 
   return tvf;
 }
+void Parallel::L2Norm7(Patch *Pat, var **vf, double *norms, MPI_Comm Comm_here)
+{
+  int myrank;
+  MPI_Comm_rank(MPI_COMM_WORLD, &myrank);
+
+  double tvf[7], dtvf[7];
+  int BDW = ghost_width;
+  for (int i = 0; i < 7; i++)
+    dtvf[i] = 0;
+
+  MyList<Block> *BP = Pat->blb;
+  while (BP)
+  {
+    Block *cg = BP->data;
+    if (myrank == cg->rank)
+    {
+      f_l2normhelper7(cg->shape, cg->X[0], cg->X[1], cg->X[2],
+                      Pat->bbox[0], Pat->bbox[1], Pat->bbox[2],
+                      Pat->bbox[3], Pat->bbox[4], Pat->bbox[5],
+                      cg->fgfs[vf[0]->sgfn], cg->fgfs[vf[1]->sgfn], cg->fgfs[vf[2]->sgfn],
+                      cg->fgfs[vf[3]->sgfn], cg->fgfs[vf[4]->sgfn], cg->fgfs[vf[5]->sgfn],
+                      cg->fgfs[vf[6]->sgfn], tvf, BDW);
+      for (int i = 0; i < 7; i++)
+        dtvf[i] += tvf[i];
+    }
+    if (BP == Pat->ble)
+      break;
+    BP = BP->next;
+  }
+
+  MPI_Allreduce(dtvf, tvf, 7, MPI_DOUBLE, MPI_SUM, Comm_here);
+
+  for (int i = 0; i < 7; i++)
+    norms[i] = sqrt(tvf[i]);
+}
 void Parallel::checkgsl(MyList<Parallel::gridseg> *pp, bool first_only)
 {
   int myrank = 0;

AMSS_NCKU_source/Parallel.h

@@ -183,6 +183,7 @@ namespace Parallel
                          MyList<Parallel::gridseg> **out_src, MyList<Parallel::gridseg> **out_dst);
   void PeriodicBD(Patch *Pat, MyList<var> *VarList, int Symmetry);
   double L2Norm(Patch *Pat, var *vf);
+  void L2Norm7(Patch *Pat, var **vf, double *norms);
   void checkgsl(MyList<Parallel::gridseg> *pp, bool first_only);
   void checkvarl(MyList<var> *pp, bool first_only);
   MyList<Parallel::gridseg> *divide_gsl(MyList<Parallel::gridseg> *p, Patch *Pat);
@@ -218,6 +219,7 @@ namespace Parallel
   void checkpatchlist(MyList<Patch> *PatL, bool buflog);
 
   double L2Norm(Patch *Pat, var *vf, MPI_Comm Comm_here);
+  void L2Norm7(Patch *Pat, var **vf, double *norms, MPI_Comm Comm_here);
   bool PatList_Interp_Points(MyList<Patch> *PatL, MyList<var> *VarList,
                              int NN, double **XX,
                              double *Shellf, int Symmetry, MPI_Comm Comm_here);

AMSS_NCKU_source/ShellPatch.C

@@ -3472,6 +3472,43 @@ double ShellPatch::L2Norm(var *vf)
 
   return tvf;
 }
+void ShellPatch::L2Norm7(var **vf, double *norms)
+{
+  double tvf[7], dtvf[7];
+  int BDW = overghost;
+  for (int i = 0; i < 7; i++)
+    dtvf[i] = 0;
+
+  MyList<ss_patch> *sPp = PatL;
+  while (sPp)
+  {
+    MyList<Block> *Bp = sPp->data->blb;
+    while (Bp)
+    {
+      Block *cg = Bp->data;
+      if (myrank == cg->rank)
+      {
+        f_l2normhelper7(cg->shape, cg->X[0], cg->X[1], cg->X[2],
+                        sPp->data->bbox[0], sPp->data->bbox[1], sPp->data->bbox[2],
+                        sPp->data->bbox[3], sPp->data->bbox[4], sPp->data->bbox[5],
+                        cg->fgfs[vf[0]->sgfn], cg->fgfs[vf[1]->sgfn], cg->fgfs[vf[2]->sgfn],
+                        cg->fgfs[vf[3]->sgfn], cg->fgfs[vf[4]->sgfn], cg->fgfs[vf[5]->sgfn],
+                        cg->fgfs[vf[6]->sgfn], tvf, BDW);
+        for (int i = 0; i < 7; i++)
+          dtvf[i] += tvf[i];
+      }
+      if (Bp == sPp->data->ble)
+        break;
+      Bp = Bp->next;
+    }
+    sPp = sPp->next;
+  }
+
+  MPI_Allreduce(dtvf, tvf, 7, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
+
+  for (int i = 0; i < 7; i++)
+    norms[i] = sqrt(tvf[i]);
+}
 
 // find maximum of abstract value, XX store position for maximum, Shellf store maximum themselvs
 void ShellPatch::Find_Maximum(MyList<var> *VarList, double *XX,

AMSS_NCKU_source/ShellPatch.h

@@ -198,6 +198,7 @@ public:
    void write_Pablo_file_ss(int *ext, double xmin, double xmax, double ymin, double ymax, double zmin, double zmax,
                             char *filename, int sst);
    double L2Norm(var *vf);
+   void L2Norm7(var **vf, double *norms);
    void Find_Maximum(MyList<var> *VarList, double *XX, double *Shellf);
 };
 

AMSS_NCKU_source/TwoPunctures.C

@@ -27,7 +27,21 @@ using namespace std;
 #endif
 
 #include "TwoPunctures.h"
-#include <mkl_cblas.h>
+
+extern "C" {
+double cblas_ddot(const int, const double *, const int, const double *, const int);
+double cblas_dnrm2(const int, const double *, const int);
+void cblas_dgemm(const int, const int, const int,
+                 const int, const int, const int,
+                 const double, const double *, const int,
+                 const double *, const int, const double,
+                 double *, const int);
+}
+
+enum {
+  CblasRowMajor = 101,
+  CblasNoTrans = 111
+};
 
 TwoPunctures::TwoPunctures(double mp, double mm, double b,
                            double P_plusx, double P_plusy, double P_plusz,

AMSS_NCKU_source/bssn_class.C

@@ -41,6 +41,239 @@ using namespace std;
 #include "derivatives.h"
 #include "ricci_gamma.h"
 
+// Compile-time switch for per-timestep memory usage collection/printing.
+// Default is OFF to reduce overhead in production runs.
+#ifndef BSSN_ENABLE_MEM_USAGE_LOG
+#define BSSN_ENABLE_MEM_USAGE_LOG 0
+#endif
+
+#ifndef BSSN_FINE_TIMING
+#define BSSN_FINE_TIMING 0
+#endif
+
+#ifndef BSSN_FINE_TIMING_EVERY
+#define BSSN_FINE_TIMING_EVERY 1
+#endif
+
+#ifndef BSSN_FINE_TIMING_TOPN
+#define BSSN_FINE_TIMING_TOPN 8
+#endif
+
+#ifndef BSSN_KERNEL_FINE_TIMING
+#define BSSN_KERNEL_FINE_TIMING 0
+#endif
+
+#ifndef BSSN_ENABLE_STDIN_ABORT_POLL
+#define BSSN_ENABLE_STDIN_ABORT_POLL 0
+#endif
+
+#if BSSN_FINE_TIMING
+namespace step_timing
+{
+  enum Bucket
+  {
+    TB_ANALYSIS_PSI4 = 0,
+    TB_ANALYSIS_SURFACE,
+    TB_ANALYSIS_IO,
+    TB_BH_PREDICTOR,
+    TB_PREDICTOR_RHS,
+    TB_PREDICTOR_SYNC,
+    TB_BH_CORRECTOR,
+    TB_CORRECTOR_RHS,
+    TB_CORRECTOR_SYNC,
+    TB_STATE_SWAP,
+    TB_RESTRICT_PROLONG,
+    TB_CONSTRAINT_OUT,
+    TB_DUMP_3D,
+    TB_DUMP_2D,
+    TB_CHECKPOINT,
+    TB_REGRID,
+    TB_COUNT
+  };
+
+  static double local_bucket_seconds[TB_COUNT];
+
+  static const char *bucket_labels[TB_COUNT] =
+  {
+    "analysis_psi4",
+    "analysis_surface",
+    "analysis_io",
+    "bh_predictor",
+    "predictor_rhs",
+    "predictor_sync",
+    "bh_corrector",
+    "corrector_rhs",
+    "corrector_sync",
+    "state_swap",
+    "restrict_prolong",
+    "constraint_out",
+    "dump_3d",
+    "dump_2d",
+    "checkpoint",
+    "regrid"
+  };
+
+  void reset()
+  {
+    for (int i = 0; i < TB_COUNT; i++)
+      local_bucket_seconds[i] = 0.0;
+  }
+
+  void add(Bucket bucket, double seconds)
+  {
+    local_bucket_seconds[int(bucket)] += seconds;
+  }
+
+  void report(int myrank, int nprocs, monitor *TimingMonitor,
+              int step_index, double phys_time, double step_wall_seconds)
+  {
+    double max_bucket_seconds[TB_COUNT];
+    double avg_bucket_seconds[TB_COUNT];
+
+    MPI_Reduce(local_bucket_seconds, max_bucket_seconds, TB_COUNT, MPI_DOUBLE, MPI_MAX, 0, MPI_COMM_WORLD);
+    MPI_Reduce(local_bucket_seconds, avg_bucket_seconds, TB_COUNT, MPI_DOUBLE, MPI_SUM, 0, MPI_COMM_WORLD);
+
+    if (myrank != 0)
+      return;
+
+    for (int i = 0; i < TB_COUNT; i++)
+      avg_bucket_seconds[i] /= Mymax(1, nprocs);
+
+    if (TimingMonitor)
+    {
+      double row[2 + 2 * TB_COUNT];
+      row[0] = double(step_index);
+      row[1] = step_wall_seconds;
+      for (int i = 0; i < TB_COUNT; i++)
+      {
+        row[2 + i] = max_bucket_seconds[i];
+        row[2 + TB_COUNT + i] = avg_bucket_seconds[i];
+      }
+      TimingMonitor->writefile(phys_time, 2 + 2 * TB_COUNT, row);
+    }
+
+    double residual = step_wall_seconds;
+    for (int i = 0; i < TB_COUNT; i++)
+      residual -= max_bucket_seconds[i];
+    if (residual < 0.0)
+      residual = 0.0;
+
+    int order[TB_COUNT];
+    for (int i = 0; i < TB_COUNT; i++)
+      order[i] = i;
+
+    for (int i = 0; i < TB_COUNT - 1; i++)
+      for (int j = i + 1; j < TB_COUNT; j++)
+        if (max_bucket_seconds[order[j]] > max_bucket_seconds[order[i]])
+        {
+          int tmp = order[i];
+          order[i] = order[j];
+          order[j] = tmp;
+        }
+
+    ios::fmtflags old_flags = cout.flags();
+    streamsize old_precision = cout.precision();
+
+    cout << " Fine timing hot spots (max rank wall estimate):" << endl;
+    const int topn = Mymin(BSSN_FINE_TIMING_TOPN, TB_COUNT);
+    for (int i = 0; i < topn; i++)
+    {
+      const int ib = order[i];
+      const double frac = (step_wall_seconds > 0.0) ? (100.0 * max_bucket_seconds[ib] / step_wall_seconds) : 0.0;
+      cout << "  "
+           << setw(20) << left << bucket_labels[ib]
+           << " = " << setw(10) << right << setprecision(6) << max_bucket_seconds[ib]
+           << " s  (" << setw(6) << setprecision(4) << frac << "%)" << endl;
+    }
+    if (residual > 1.0e-6)
+    {
+      const double frac = (step_wall_seconds > 0.0) ? (100.0 * residual / step_wall_seconds) : 0.0;
+      cout << "  "
+           << setw(20) << left << "unprofiled_residual"
+           << " = " << setw(10) << right << setprecision(6) << residual
+           << " s  (" << setw(6) << setprecision(4) << frac << "%)" << endl;
+    }
+    cout << endl;
+    cout.flags(old_flags);
+    cout.precision(old_precision);
+  }
+}
+
+#define STEP_TIMER_DECL(var_name) const double var_name = MPI_Wtime()
+#define STEP_TIMER_ADD(bucket_name, var_name) step_timing::add(step_timing::bucket_name, MPI_Wtime() - (var_name))
+#else
+#define STEP_TIMER_DECL(var_name)
+#define STEP_TIMER_ADD(bucket_name, var_name)
+#endif
+
+#if BSSN_KERNEL_FINE_TIMING
+namespace rhs_kernel_timing_report
+{
+  void report(int myrank, int nprocs, int step_index, double step_wall_seconds)
+  {
+    const int bucket_count = f_bssn_rhs_kernel_timing_bucket_count();
+    const double *local_bucket_seconds = f_bssn_rhs_kernel_timing_local_seconds();
+
+    if (bucket_count <= 0 || !local_bucket_seconds)
+      return;
+
+    double *max_bucket_seconds = new double[bucket_count];
+    double *avg_bucket_seconds = new double[bucket_count];
+    int *order = new int[bucket_count];
+
+    MPI_Reduce((void *)local_bucket_seconds, max_bucket_seconds, bucket_count, MPI_DOUBLE, MPI_MAX, 0, MPI_COMM_WORLD);
+    MPI_Reduce((void *)local_bucket_seconds, avg_bucket_seconds, bucket_count, MPI_DOUBLE, MPI_SUM, 0, MPI_COMM_WORLD);
+
+    if (myrank == 0)
+    {
+      double kernel_total = 0.0;
+      for (int i = 0; i < bucket_count; ++i)
+      {
+        avg_bucket_seconds[i] /= Mymax(1, nprocs);
+        order[i] = i;
+        kernel_total += max_bucket_seconds[i];
+      }
+
+      for (int i = 0; i < bucket_count - 1; ++i)
+        for (int j = i + 1; j < bucket_count; ++j)
+          if (max_bucket_seconds[order[j]] > max_bucket_seconds[order[i]])
+          {
+            int tmp = order[i];
+            order[i] = order[j];
+            order[j] = tmp;
+          }
+
+      ios::fmtflags old_flags = cout.flags();
+      streamsize old_precision = cout.precision();
+
+      const double kernel_frac = (step_wall_seconds > 0.0) ? (100.0 * kernel_total / step_wall_seconds) : 0.0;
+      cout << " RHS kernel split (max-rank accumulated over step " << step_index << "): total "
+           << setprecision(6) << kernel_total << " s  (" << setprecision(4)
+           << kernel_frac << "% of coarse step)" << endl;
+
+      const int topn = Mymin(BSSN_FINE_TIMING_TOPN, bucket_count);
+      for (int i = 0; i < topn; ++i)
+      {
+        const int ib = order[i];
+        const double frac = (kernel_total > 0.0) ? (100.0 * max_bucket_seconds[ib] / kernel_total) : 0.0;
+        cout << "  "
+             << setw(20) << left << f_bssn_rhs_kernel_timing_label(ib)
+             << " = " << setw(10) << right << setprecision(6) << max_bucket_seconds[ib]
+             << " s  (" << setw(6) << setprecision(4) << frac << "% of kernel)" << endl;
+      }
+      cout << endl;
+
+      cout.flags(old_flags);
+      cout.precision(old_precision);
+    }
+
+    delete[] max_bucket_seconds;
+    delete[] avg_bucket_seconds;
+    delete[] order;
+  }
+}
+#endif
+
 //================================================================================================
 
 // define bssn_class
@@ -59,6 +292,7 @@ bssn_class::bssn_class(double Couranti, double StartTimei, double TotalTimei,
                          xc(0), yc(0), zc(0), xr(0), yr(0), zr(0), trigger(0), dTT(0), dumpid(0),
 #endif
                          a_lev(a_levi), maxl(maxli), decn(decni), maxrex(maxrexi), drex(drexi),
+                         ConstraintRefreshLevels(0),
                          CheckPoint(0)
                          // CheckPoint(0)
 {
@@ -101,6 +335,24 @@ bssn_class::bssn_class(double Couranti, double StartTimei, double TotalTimei,
     a_stream.str("");
     a_stream << setw(15) << "# time Ham Px Py Pz Gx Gy Gz";
     ConVMonitor = new monitor("bssn_constraint.dat", myrank, a_stream.str());
+
+#if BSSN_FINE_TIMING
+    a_stream.clear();
+    a_stream.str("");
+    a_stream << setw(8) << "# step";
+    a_stream << setw(14) << "wall";
+    for (int ib = 0; ib < step_timing::TB_COUNT; ib++)
+      a_stream << setw(18) << step_timing::bucket_labels[ib];
+    for (int ib = 0; ib < step_timing::TB_COUNT; ib++)
+    {
+      char str_avg[64];
+      sprintf(str_avg, "avg_%s", step_timing::bucket_labels[ib]);
+      a_stream << setw(18) << str_avg;
+    }
+    TimingMonitor = new monitor("bssn_step_timing.dat", myrank, a_stream.str());
+#else
+    TimingMonitor = 0;
+#endif
   }
   // setup sphere integration engine
   Waveshell = new surface_integral(Symmetry);
@@ -696,6 +948,9 @@ void bssn_class::Initialize()
     }
   }
   GH = new cgh(0, ngfs, Symmetry, pname, checkrun, ErrorMonitor);
+  ConstraintRefreshLevels = new int[GH->levels];
+  for (int il = 0; il < GH->levels; il++)
+    ConstraintRefreshLevels[il] = 0;
   if (checkrun)
     CheckPoint->readcheck_cgh(PhysTime, GH, myrank, nprocs, Symmetry);
   else
@@ -785,6 +1040,8 @@ bssn_class::~bssn_class()
   DumpList->clearList();
   ConstraintList->clearList();
 
+  delete[] ConstraintRefreshLevels;
+
   delete phio;
   delete trKo;
   delete gxxo;
@@ -1044,6 +1301,7 @@ bssn_class::~bssn_class()
   delete BHMonitor;
   delete MAPMonitor;
   delete ConVMonitor;
+  delete TimingMonitor;
   delete Waveshell;
 
   delete CheckPoint;
@@ -2129,8 +2387,10 @@ void bssn_class::Evolve(int Steps)
   #endif
   */
   
+#if BSSN_ENABLE_MEM_USAGE_LOG
   perf bssn_perf;
   size_t current_min, current_avg, current_max, peak_min, peak_avg, peak_max;
+#endif
 
   for (int lev = 0; lev < GH->levels; lev++)
     GH->Lt[lev] = PhysTime;
@@ -2139,6 +2399,15 @@ void bssn_class::Evolve(int Steps)
 
   for (int ncount = 1; ncount < Steps + 1; ncount++)
   {
+#if BSSN_FINE_TIMING
+    step_timing::reset();
+#endif
+#if BSSN_KERNEL_FINE_TIMING
+    f_bssn_rhs_kernel_timing_reset();
+#endif
+#if (BSSN_FINE_TIMING || BSSN_KERNEL_FINE_TIMING)
+    const double step_wall_start = MPI_Wtime();
+#endif
     // special for large mass ratio consideration
     //     if(fabs(Porg0[0][0]-Porg0[1][0])+fabs(Porg0[0][1]-Porg0[1][1])+fabs(Porg0[0][2]-Porg0[1][2])<1e-6) 
     //     { GH->levels=GH->movls; }
@@ -2165,6 +2434,7 @@ void bssn_class::Evolve(int Steps)
     // When LastDump >= DumpTime, output corresponding binary data
     if (LastDump >= DumpTime)
     {
+      STEP_TIMER_DECL(timer_dump3d);
       //       misc::tillherecheck("before Dump_Data");
 
       for (int lev = 0; lev < GH->levels; lev++)
@@ -2172,6 +2442,7 @@ void bssn_class::Evolve(int Steps)
 #ifdef WithShell
       SH->Dump_Data(DumpList, 0, PhysTime, dT_mon);
 #endif
+      STEP_TIMER_ADD(TB_DUMP_3D, timer_dump3d);
 
       LastDump = 0;
 
@@ -2184,10 +2455,12 @@ void bssn_class::Evolve(int Steps)
     // When Last2dDump >= d2DumpTime, output corresponding 2D data
     if (Last2dDump >= d2DumpTime)
     {
+      STEP_TIMER_DECL(timer_dump2d);
       //       misc::tillherecheck("before 2dDump_Data");
 
       for (int lev = 0; lev < GH->levels; lev++)
         Parallel::d2Dump_Data(GH->PatL[lev], DumpList, 0, PhysTime, dT_mon);
+      STEP_TIMER_ADD(TB_DUMP_2D, timer_dump2d);
 
       Last2dDump = 0;
 
@@ -2212,10 +2485,12 @@ void bssn_class::Evolve(int Steps)
       break;
 
 #if (REGLEV == 1)
+    STEP_TIMER_DECL(timer_regrid);
     GH->Regrid(Symmetry, BH_num, Porgbr, Porg0,
                SynchList_cor, OldStateList, StateList, SynchList_pre,
                fgt(PhysTime - dT_mon, StartTime, dT_mon / 2), ErrorMonitor);
     for (int il = 0; il < GH->levels; il++) { sync_cache_pre[il].invalidate(); sync_cache_cor[il].invalidate(); sync_cache_rp_coarse[il].invalidate(); sync_cache_rp_fine[il].invalidate(); sync_cache_restrict[il].invalidate(); sync_cache_outbd[il].invalidate(); }
+    STEP_TIMER_ADD(TB_REGRID, timer_regrid);
 #endif
 
 #if (REGLEV == 0 && (PSTR == 1 || PSTR == 2))
@@ -2224,6 +2499,7 @@ void bssn_class::Evolve(int Steps)
 //		fgt(PhysTime-dT_mon,StartTime,dT_mon/2),ErrorMonitor);
 #endif
 
+#if BSSN_ENABLE_MEM_USAGE_LOG
     // Retrieve memory usage information used during computation; master process prints it
     bssn_perf.MemoryUsage(&current_min, &current_avg, &current_max,
                           &peak_min, &peak_avg, &peak_max, nprocs);
@@ -2239,6 +2515,7 @@ void bssn_class::Evolve(int Steps)
              (double)peak_max / (1024.0 * 1024.0));
       cout << endl;
     }
+#endif
     
     // Output puncture positions at each step
     if (myrank == 0)
@@ -2253,10 +2530,13 @@ void bssn_class::Evolve(int Steps)
              << endl;
       }
       cout << endl;
+#if BSSN_ENABLE_STDIN_ABORT_POLL
       cout << " If you think the physical evolution time is enough for this simulation, please input 'stop' in the terminal to stop the MPI processes in the next evolution step ! " << endl;
+#endif
       // cout << endl;
     }
     
+#if BSSN_ENABLE_STDIN_ABORT_POLL
     ////////////////////////////////////////////////////////////
     // If an "abort" command is detected on stdin, terminate MPI processes
     ////////////////////////////////////////////////////////////
@@ -2284,10 +2564,12 @@ void bssn_class::Evolve(int Steps)
     }
         
     ////////////////////////////////////////////////////////////
+#endif
 
     // When LastCheck >= CheckTime, perform runtime checks and output status data
     if (LastCheck >= CheckTime)
     {
+      STEP_TIMER_DECL(timer_checkpoint);
       LastCheck = 0;
 
       CheckPoint->write_Black_Hole_position(BH_num_input, BH_num, Porg0, Porgbr, Mass);
@@ -2296,7 +2578,20 @@ void bssn_class::Evolve(int Steps)
       CheckPoint->writecheck_sh(PhysTime, SH);
 #endif
       CheckPoint->write_bssn(LastDump, Last2dDump, LastAnas);
+      STEP_TIMER_ADD(TB_CHECKPOINT, timer_checkpoint);
     }
+
+#if (BSSN_FINE_TIMING || BSSN_KERNEL_FINE_TIMING)
+    const double step_wall_seconds = MPI_Wtime() - step_wall_start;
+#endif
+#if BSSN_FINE_TIMING
+    if (ncount % BSSN_FINE_TIMING_EVERY == 0)
+      step_timing::report(myrank, nprocs, TimingMonitor, ncount, PhysTime, step_wall_seconds);
+#endif
+#if BSSN_KERNEL_FINE_TIMING
+    if (ncount % BSSN_FINE_TIMING_EVERY == 0)
+      rhs_kernel_timing_report::report(myrank, nprocs, ncount, step_wall_seconds);
+#endif
   }
   /*
   #ifdef With_AHF
@@ -2428,10 +2723,16 @@ void bssn_class::RecursiveStep(int lev)
 #endif
 
 #if (REGLEV == 0)
+  STEP_TIMER_DECL(timer_regrid_onelevel);
   if (GH->Regrid_Onelevel(lev, Symmetry, BH_num, Porgbr, Porg0,
                       SynchList_cor, OldStateList, StateList, SynchList_pre,
                       fgt(PhysTime - dT_lev, StartTime, dT_lev / 2), ErrorMonitor))
+  {
+  if (ConstraintRefreshLevels)
+    ConstraintRefreshLevels[lev] = 1;
   for (int il = 0; il < GH->levels; il++) { sync_cache_pre[il].invalidate(); sync_cache_cor[il].invalidate(); sync_cache_rp_coarse[il].invalidate(); sync_cache_rp_fine[il].invalidate(); sync_cache_restrict[il].invalidate(); sync_cache_outbd[il].invalidate(); }
+  }
+  STEP_TIMER_ADD(TB_REGRID, timer_regrid_onelevel);
 #endif
 }
 
@@ -3024,6 +3325,7 @@ void bssn_class::Step(int lev, int YN)
 
 // new code 2013-2-15, zjcao
 #if (MAPBH == 1)
+  STEP_TIMER_DECL(timer_bh_predictor);
   // for black hole position
   if (BH_num > 0 && lev == GH->levels - 1)
   {
@@ -3054,6 +3356,7 @@ void bssn_class::Step(int lev, int YN)
       }
     }
   }
+  STEP_TIMER_ADD(TB_BH_PREDICTOR, timer_bh_predictor);
 
   // data analysis part
   // Warning NOTE: the variables1 are used as temp storege room
@@ -3076,6 +3379,7 @@ void bssn_class::Step(int lev, int YN)
   int ERROR = 0;
 
   MyList<ss_patch> *sPp;
+  STEP_TIMER_DECL(timer_predictor_rhs);
   // Predictor
   MyList<Patch> *Pp = GH->PatL[lev];
   while (Pp)
@@ -3351,6 +3655,9 @@ void bssn_class::Step(int lev, int YN)
   }
 #endif
 
+  STEP_TIMER_ADD(TB_PREDICTOR_RHS, timer_predictor_rhs);
+
+  STEP_TIMER_DECL(timer_predictor_sync);
   Parallel::AsyncSyncState async_pre;
   Parallel::Sync_start(GH->PatL[lev], SynchList_pre, Symmetry, sync_cache_pre[lev], async_pre);
 
@@ -3388,6 +3695,7 @@ void bssn_class::Step(int lev, int YN)
     }
   }
 #endif
+  STEP_TIMER_ADD(TB_PREDICTOR_SYNC, timer_predictor_sync);
 
 #if (MAPBH == 0)
   // for black hole position
@@ -3432,6 +3740,7 @@ void bssn_class::Step(int lev, int YN)
   // corrector
   for (iter_count = 1; iter_count < 4; iter_count++)
   {
+    STEP_TIMER_DECL(timer_corrector_rhs);
     // for RK4: t0, t0+dt/2, t0+dt/2, t0+dt;
     if (iter_count == 1 || iter_count == 3)
       TRK4 += dT_lev / 2;
@@ -3711,6 +4020,9 @@ void bssn_class::Step(int lev, int YN)
     }
 #endif
 
+    STEP_TIMER_ADD(TB_CORRECTOR_RHS, timer_corrector_rhs);
+
+    STEP_TIMER_DECL(timer_corrector_sync);
     Parallel::AsyncSyncState async_cor;
     Parallel::Sync_start(GH->PatL[lev], SynchList_cor, Symmetry, sync_cache_cor[lev], async_cor);
 
@@ -3750,8 +4062,10 @@ void bssn_class::Step(int lev, int YN)
       }
     }
 #endif
+    STEP_TIMER_ADD(TB_CORRECTOR_SYNC, timer_corrector_sync);
 
 #if (MAPBH == 0)
+    STEP_TIMER_DECL(timer_bh_corrector);
     // for black hole position
     if (BH_num > 0 && lev == GH->levels - 1)
     {
@@ -3784,11 +4098,13 @@ void bssn_class::Step(int lev, int YN)
         }
       }
     }
+    STEP_TIMER_ADD(TB_BH_CORRECTOR, timer_bh_corrector);
 #endif
 
     // swap time level
     if (iter_count < 3)
     {
+      STEP_TIMER_DECL(timer_state_swap);
       Pp = GH->PatL[lev];
       while (Pp)
       {
@@ -3835,9 +4151,11 @@ void bssn_class::Step(int lev, int YN)
         }
       }
 #endif
+      STEP_TIMER_ADD(TB_STATE_SWAP, timer_state_swap);
     }
   }
 #if (RPS == 0)
+  STEP_TIMER_DECL(timer_restrict_prolong);
   // mesh refinement boundary part
   RestrictProlong(lev, YN, BB);
 
@@ -3858,6 +4176,7 @@ void bssn_class::Step(int lev, int YN)
   }
 #endif
 
+  STEP_TIMER_ADD(TB_RESTRICT_PROLONG, timer_restrict_prolong);
 #endif
   // note the data structure before update
   // SynchList_cor 1   -----------
@@ -3866,6 +4185,7 @@ void bssn_class::Step(int lev, int YN)
   //
   // OldStateList  old -----------
   // update
+  STEP_TIMER_DECL(timer_state_commit);
   Pp = GH->PatL[lev];
   while (Pp)
   {
@@ -3922,6 +4242,7 @@ void bssn_class::Step(int lev, int YN)
       Porg0[ithBH][2] = Porg1[ithBH][2];
     }
   }
+  STEP_TIMER_ADD(TB_STATE_SWAP, timer_state_commit);
 }
 
 //================================================================================================
@@ -4248,7 +4569,9 @@ void bssn_class::Step(int lev, int YN)
     }
   }
 #endif
+  STEP_TIMER_ADD(TB_PREDICTOR_SYNC, timer_predictor_sync);
 
+  STEP_TIMER_DECL(timer_bh_predictor);
   // for black hole position
   if (BH_num > 0 && lev == GH->levels - 1)
   {
@@ -4287,6 +4610,7 @@ void bssn_class::Step(int lev, int YN)
   {
     AnalysisStuff(lev, dT_lev);
   }
+  STEP_TIMER_ADD(TB_BH_PREDICTOR, timer_bh_predictor);
   // corrector
   for (iter_count = 1; iter_count < 3; iter_count++)
   {
@@ -5757,6 +6081,7 @@ void bssn_class::RestrictProlong(int lev, int YN, bool BB,
 //
 // SynchList_cor  old -----------
 {
+  STEP_TIMER_DECL(timer_restrict_prolong);
 #if (PSTR == 1 || PSTR == 2)
 //  stringstream a_stream;
 //  a_stream.setf(ios::left);
@@ -5899,6 +6224,7 @@ void bssn_class::RestrictProlong(int lev, int YN, bool BB,
 //    misc::tillherecheck(GH->Commlev[GH->mylev],GH->start_rank[GH->mylev],a_stream.str());
 #endif
   }
+  STEP_TIMER_ADD(TB_RESTRICT_PROLONG, timer_restrict_prolong);
 }
 
 //================================================================================================
@@ -5918,6 +6244,7 @@ void bssn_class::RestrictProlong_aux(int lev, int YN, bool BB,
 //
 // SynchList_cor  old -----------
 {
+  STEP_TIMER_DECL(timer_restrict_prolong);
   //  misc::tillherecheck(GH->Commlev[lev],GH->start_rank[lev],"starting RestrictProlong_aux");
 
   if (lev >= GH->levels - 1)
@@ -5986,6 +6313,7 @@ void bssn_class::RestrictProlong_aux(int lev, int YN, bool BB,
 
     Parallel::Sync_cached(GH->PatL[lev], SL, Symmetry, sync_cache_rp_fine[lev]);
   }
+  STEP_TIMER_ADD(TB_RESTRICT_PROLONG, timer_restrict_prolong);
 }
 
 //================================================================================================
@@ -5996,6 +6324,7 @@ void bssn_class::RestrictProlong_aux(int lev, int YN, bool BB,
 
 void bssn_class::RestrictProlong(int lev, int YN, bool BB)
 {
+  STEP_TIMER_DECL(timer_restrict_prolong);
   double dT_lev = dT * pow(0.5, Mymax(lev, trfls));
   // we assume  for fine
   // SynchList_cor 1   -----------
@@ -6075,6 +6404,7 @@ void bssn_class::RestrictProlong(int lev, int YN, bool BB)
 
     Parallel::Sync_cached(GH->PatL[lev], SynchList_cor, Symmetry, sync_cache_rp_fine[lev]);
   }
+  STEP_TIMER_ADD(TB_RESTRICT_PROLONG, timer_restrict_prolong);
 }
 
 //================================================================================================
@@ -6825,18 +7155,15 @@ void bssn_class::compute_Porg_rhs(double **BH_PS,double **BH_RHS,var *forx,var *
 
 void bssn_class::compute_Porg_rhs(double **BH_PS, double **BH_RHS, var *forx, var *fory, var *forz, int ilev)
 {
-  const int InList = 3;
+  MyList<var> DG_List_x(forx);
+  MyList<var> DG_List_y(fory);
+  MyList<var> DG_List_z(forz);
+  DG_List_x.next = &DG_List_y;
+  DG_List_y.next = &DG_List_z;
 
-  MyList<var> *DG_List = new MyList<var>(forx);
-  DG_List->insert(fory);
-  DG_List->insert(forz);
-
-  double *x1, *y1, *z1;
-  double *shellf;
-  shellf = new double[3];
-  double *pox[3];
-  for (int i = 0; i < 3; i++)
-    pox[i] = new double[1];
+  double shellf[3];
+  double pox_buf[3][1];
+  double *pox[3] = {pox_buf[0], pox_buf[1], pox_buf[2]};
 
   for (int n = 0; n < BH_num; n++)
   {
@@ -6847,9 +7174,9 @@ void bssn_class::compute_Porg_rhs(double **BH_PS, double **BH_RHS, var *forx, va
     int lev = ilev;
 
 #if (PSTR == 0)
-    while (!Parallel::PatList_Interp_Points(GH->PatL[lev], DG_List, 1, pox, shellf, Symmetry))
+    while (!Parallel::PatList_Interp_Points(GH->PatL[lev], &DG_List_x, 1, pox, shellf, Symmetry))
 #elif (PSTR == 1 || PSTR == 2 || PSTR == 3)
-    while (!Parallel::PatList_Interp_Points(GH->PatL[lev], DG_List, 1, pox, shellf, Symmetry, GH->Commlev[lev]))
+    while (!Parallel::PatList_Interp_Points(GH->PatL[lev], &DG_List_x, 1, pox, shellf, Symmetry, GH->Commlev[lev]))
 #endif
     {
       lev--;
@@ -6858,7 +7185,7 @@ void bssn_class::compute_Porg_rhs(double **BH_PS, double **BH_RHS, var *forx, va
         ErrorMonitor->outfile << "fail to find black holes at t = " << PhysTime << endl;
         for (n = 0; n < BH_num; n++)
           ErrorMonitor->outfile << "(x,y,z) = (" 
-                                << pox[0][n] << "," << pox[1][n] << "," << pox[2][n] 
+                                << BH_PS[n][0] << "," << BH_PS[n][1] << "," << BH_PS[n][2] 
                                 << ")" << endl;
         break;
       }
@@ -6871,11 +7198,6 @@ void bssn_class::compute_Porg_rhs(double **BH_PS, double **BH_RHS, var *forx, va
       BH_RHS[n][2] = -shellf[2];
     }
   }
-
-  DG_List->clearList();
-  delete[] shellf;
-  for (int i = 0; i < 3; i++)
-    delete[] pox[i];
 }
 #endif
 
@@ -7098,6 +7420,10 @@ void bssn_class::AnalysisStuff(int lev, double dT_lev)
     IP = new double[NN];
     RoutMAP = new double[7];
     double Rex = maxrex;
+    bool patch_mass_prepared = false;
+#ifdef WithShell
+    bool shell_mass_prepared = false;
+#endif
     for (int i = 0; i < decn; i++)
     {
 #ifdef Point_Psi4
@@ -7125,7 +7451,8 @@ void bssn_class::AnalysisStuff(int lev, double dT_lev)
                                  gxx0, gxy0, gxz0, gyy0, gyz0, gzz0,
                                  Axx0, Axy0, Axz0, Ayy0, Ayz0, Azz0,
                                  Gmx0, Gmy0, Gmz0, Sfx1, Sfy1, Sfz1, // here we can not touch rhs variables, but 1 variables
-                                 RoutMAP, ErrorMonitor);
+                                 RoutMAP, ErrorMonitor, !patch_mass_prepared);
+        patch_mass_prepared = true;
       }
       else
       {
@@ -7133,44 +7460,52 @@ void bssn_class::AnalysisStuff(int lev, double dT_lev)
                                  gxx0, gxy0, gxz0, gyy0, gyz0, gzz0,
                                  Axx0, Axy0, Axz0, Ayy0, Ayz0, Azz0,
                                  Gmx0, Gmy0, Gmz0, Sfx1, Sfy1, Sfz1, // here we can not touch rhs variables, but 1 variables
-                                 RoutMAP, ErrorMonitor);
+                                 RoutMAP, ErrorMonitor, !shell_mass_prepared);
+        shell_mass_prepared = true;
       }
 #else
       Waveshell->surf_MassPAng(Rex, lev, GH, phi0, trK0,
                                gxx0, gxy0, gxz0, gyy0, gyz0, gzz0,
                                Axx0, Axy0, Axz0, Ayy0, Ayz0, Azz0,
                                Gmx0, Gmy0, Gmz0, Sfx1, Sfy1, Sfz1, // here we can not touch rhs variables, but 1 variables
-                               RoutMAP, ErrorMonitor);
+                               RoutMAP, ErrorMonitor, !patch_mass_prepared);
+      patch_mass_prepared = true;
 #endif
 #else
 //        misc::tillherecheck(GH->Commlev[lev],GH->start_rank[lev],"before surface integral");
 #ifdef WithShell
       if (lev > 0 || Rex < GH->bbox[0][0][3])
       {
-        Waveshell->surf_Wave(Rex, lev, GH, Rpsi4, Ipsi4, 2, maxl, NN, RP, IP, ErrorMonitor);
-        Waveshell->surf_MassPAng(Rex, lev, GH, phi0, trK0,
-                                 gxx0, gxy0, gxz0, gyy0, gyz0, gzz0,
-                                 Axx0, Axy0, Axz0, Ayy0, Ayz0, Azz0,
-                                 Gmx0, Gmy0, Gmz0, Sfx1, Sfy1, Sfz1, // here we can not touch rhs variables, but 1 variables
-                                 RoutMAP, ErrorMonitor);
+        Waveshell->surf_WaveMassPAng(Rex, lev, GH,
+                                     Rpsi4, Ipsi4, 2, maxl, NN, RP, IP,
+                                     phi0, trK0,
+                                     gxx0, gxy0, gxz0, gyy0, gyz0, gzz0,
+                                     Axx0, Axy0, Axz0, Ayy0, Ayz0, Azz0,
+                                     Gmx0, Gmy0, Gmz0, Sfx1, Sfy1, Sfz1,
+                                     RoutMAP, ErrorMonitor, !patch_mass_prepared);
+        patch_mass_prepared = true;
       }
       else
       {
-        Waveshell->surf_Wave(Rex, lev, SH, Rpsi4, Ipsi4, 2, maxl, NN, RP, IP, ErrorMonitor);
-        Waveshell->surf_MassPAng(Rex, lev, SH, phi0, trK0,
-                                 gxx0, gxy0, gxz0, gyy0, gyz0, gzz0,
-                                 Axx0, Axy0, Axz0, Ayy0, Ayz0, Azz0,
-                                 Gmx0, Gmy0, Gmz0, Sfx1, Sfy1, Sfz1, // here we can not touch rhs variables, but 1 variables
-                                 RoutMAP, ErrorMonitor);
+        Waveshell->surf_WaveMassPAng(Rex, lev, SH,
+                                     Rpsi4, Ipsi4, 2, maxl, NN, RP, IP,
+                                     phi0, trK0,
+                                     gxx0, gxy0, gxz0, gyy0, gyz0, gzz0,
+                                     Axx0, Axy0, Axz0, Ayy0, Ayz0, Azz0,
+                                     Gmx0, Gmy0, Gmz0, Sfx1, Sfy1, Sfz1,
+                                     RoutMAP, ErrorMonitor, !shell_mass_prepared);
+        shell_mass_prepared = true;
       }
 #else
 #if (PSTR == 0)
-      Waveshell->surf_Wave(Rex, lev, GH, Rpsi4, Ipsi4, 2, maxl, NN, RP, IP, ErrorMonitor);
-      Waveshell->surf_MassPAng(Rex, lev, GH, phi0, trK0,
-                               gxx0, gxy0, gxz0, gyy0, gyz0, gzz0,
-                               Axx0, Axy0, Axz0, Ayy0, Ayz0, Azz0,
-                               Gmx0, Gmy0, Gmz0, Sfx1, Sfy1, Sfz1, // here we can not touch rhs variables, but 1 variables
-                               RoutMAP, ErrorMonitor);
+      Waveshell->surf_WaveMassPAng(Rex, lev, GH,
+                                   Rpsi4, Ipsi4, 2, maxl, NN, RP, IP,
+                                   phi0, trK0,
+                                   gxx0, gxy0, gxz0, gyy0, gyz0, gzz0,
+                                   Axx0, Axy0, Axz0, Ayy0, Ayz0, Azz0,
+                                   Gmx0, Gmy0, Gmz0, Sfx1, Sfy1, Sfz1,
+                                   RoutMAP, ErrorMonitor, !patch_mass_prepared);
+      patch_mass_prepared = true;
 #elif (PSTR == 1 || PSTR == 2)
       Waveshell->surf_Wave(Rex, lev, GH, Rpsi4, Ipsi4, 2, maxl, NN, RP, IP, ErrorMonitor, GH->Commlev[lev]);
       //        misc::tillherecheck(GH->Commlev[lev],GH->start_rank[lev],"after surf_Wave");
@@ -7178,7 +7513,8 @@ void bssn_class::AnalysisStuff(int lev, double dT_lev)
                                gxx0, gxy0, gxz0, gyy0, gyz0, gzz0,
                                Axx0, Axy0, Axz0, Ayy0, Ayz0, Azz0,
                                Gmx0, Gmy0, Gmz0, Sfx1, Sfy1, Sfz1, // here we can not touch rhs variables, but 1 variables
-                               RoutMAP, ErrorMonitor, GH->Commlev[lev]);
+                               RoutMAP, ErrorMonitor, GH->Commlev[lev], !patch_mass_prepared);
+      patch_mass_prepared = true;
 #endif
 #endif
 //        misc::tillherecheck(GH->Commlev[lev],GH->start_rank[lev],"end surface integral");
@@ -7251,7 +7587,7 @@ void bssn_class::Constraint_Out()
     for (int lev = 0; lev < GH->levels; lev++)
     {
       // make sure the data consistent for higher levels
-      if (lev > 0) // if the constrait quantities can be reused from the step rhs calculation
+      if (lev > 0 && ConstraintRefreshLevels && ConstraintRefreshLevels[lev]) // only refresh levels whose grid layout changed after evolution
       {
         double TRK4 = PhysTime;
         double ndeps = numepsb;
@@ -7405,35 +7741,18 @@ void bssn_class::Constraint_Out()
 #if (PSTR == 1 || PSTR == 2)
     double ConV_h[7];
 #endif
+    var *ConstraintVars[7] = {Cons_Ham, Cons_Px, Cons_Py, Cons_Pz, Cons_Gx, Cons_Gy, Cons_Gz};
 
 #ifdef WithShell
-    ConV[0] = SH->L2Norm(Cons_Ham);
-    ConV[1] = SH->L2Norm(Cons_Px);
-    ConV[2] = SH->L2Norm(Cons_Py);
-    ConV[3] = SH->L2Norm(Cons_Pz);
-    ConV[4] = SH->L2Norm(Cons_Gx);
-    ConV[5] = SH->L2Norm(Cons_Gy);
-    ConV[6] = SH->L2Norm(Cons_Gz);
+    SH->L2Norm7(ConstraintVars, ConV);
     ConVMonitor->writefile(PhysTime, 7, ConV);
 #endif
     for (int levi = 0; levi < GH->levels; levi++)
     {
 #if (PSTR == 0)
-      ConV[0] = Parallel::L2Norm(GH->PatL[levi]->data, Cons_Ham);
-      ConV[1] = Parallel::L2Norm(GH->PatL[levi]->data, Cons_Px);
-      ConV[2] = Parallel::L2Norm(GH->PatL[levi]->data, Cons_Py);
-      ConV[3] = Parallel::L2Norm(GH->PatL[levi]->data, Cons_Pz);
-      ConV[4] = Parallel::L2Norm(GH->PatL[levi]->data, Cons_Gx);
-      ConV[5] = Parallel::L2Norm(GH->PatL[levi]->data, Cons_Gy);
-      ConV[6] = Parallel::L2Norm(GH->PatL[levi]->data, Cons_Gz);
+      Parallel::L2Norm7(GH->PatL[levi]->data, ConstraintVars, ConV);
 #elif (PSTR == 1 || PSTR == 2)
-      ConV[0] = Parallel::L2Norm(GH->PatL[levi]->data, Cons_Ham, GH->Commlev[levi]);
-      ConV[1] = Parallel::L2Norm(GH->PatL[levi]->data, Cons_Px, GH->Commlev[levi]);
-      ConV[2] = Parallel::L2Norm(GH->PatL[levi]->data, Cons_Py, GH->Commlev[levi]);
-      ConV[3] = Parallel::L2Norm(GH->PatL[levi]->data, Cons_Pz, GH->Commlev[levi]);
-      ConV[4] = Parallel::L2Norm(GH->PatL[levi]->data, Cons_Gx, GH->Commlev[levi]);
-      ConV[5] = Parallel::L2Norm(GH->PatL[levi]->data, Cons_Gy, GH->Commlev[levi]);
-      ConV[6] = Parallel::L2Norm(GH->PatL[levi]->data, Cons_Gz, GH->Commlev[levi]);
+      Parallel::L2Norm7(GH->PatL[levi]->data, ConstraintVars, ConV, GH->Commlev[levi]);
       //        misc::tillherecheck("before collect data to cpu0");
       // MPI_ALLREDUCE( sendbuf, recvbuf, count, datatype, op, comm), sendbu and recvbuf must be different
       if (levi > 0)
@@ -7464,6 +7783,9 @@ void bssn_class::Constraint_Out()
     Interp_Constraint(false);
 
     LastConsOut = 0;
+    if (ConstraintRefreshLevels)
+      for (int lev = 0; lev < GH->levels; lev++)
+        ConstraintRefreshLevels[lev] = 0;
   }
 }
 

AMSS_NCKU_source/bssn_class.h

@@ -48,6 +48,7 @@ public:
        double StartTime, TotalTime;
        double AnasTime, DumpTime, d2DumpTime, CheckTime;
        double LastAnas, LastConsOut;
+       int *ConstraintRefreshLevels;
        double Courant;
        double numepss, numepsb, numepsh;
        int Symmetry;
@@ -134,7 +135,7 @@ public:
        Parallel::SyncCache *sync_cache_outbd;      // cached OutBdLow2Hi in RestrictProlong
 
        monitor *ErrorMonitor, *Psi4Monitor, *BHMonitor, *MAPMonitor;
-       monitor *ConVMonitor;
+       monitor *ConVMonitor, *TimingMonitor;
        surface_integral *Waveshell;
        checkpoint *CheckPoint;
 

AMSS_NCKU_source/bssn_rhs.f90

@@ -62,6 +62,7 @@
   real*8, dimension(ex(1),ex(2),ex(3)),intent(inout) :: Gmx_Res, Gmy_Res, Gmz_Res
 !  gont = 0: success; gont = 1: something wrong
   integer::gont
+  integer :: i,j,k
 
 !~~~~~~> Other variables:
 
@@ -85,6 +86,13 @@
 
   real*8,dimension(3) ::SSS,AAS,ASA,SAA,ASS,SAS,SSA
   real*8            :: dX, dY, dZ, PI
+  real*8            :: divb_loc,det_loc
+  real*8            :: gupxx_loc,gupxy_loc,gupxz_loc,gupyy_loc,gupyz_loc,gupzz_loc
+  real*8            :: Rxx_loc,Rxy_loc,Rxz_loc,Ryy_loc,Ryz_loc,Rzz_loc
+  real*8            :: fxx_loc,fxy_loc,fxz_loc
+  real*8            :: Gamxa_loc,Gamya_loc,Gamza_loc
+  real*8            :: f_loc,chin_loc
+  real*8            :: l_fxx,l_fxy,l_fxz,l_fyy,l_fyz,l_fzz,S_loc
   real*8, parameter :: ZEO = 0.d0,ONE = 1.D0, TWO = 2.D0, FOUR = 4.D0
   real*8, parameter :: EIGHT = 8.D0, HALF = 0.5D0, THR = 3.d0
   real*8, parameter :: SYM = 1.D0, ANTI= - 1.D0
@@ -97,7 +105,7 @@
 #endif
 
 #if (GAUGE == 6 || GAUGE == 7)
-  integer :: BHN,i,j,k
+  integer :: BHN
   real*8, dimension(9) :: Porg
   real*8, dimension(3) :: Mass
   real*8 :: r1,r2,M,A,w1,w2,C1,C2
@@ -145,22 +153,24 @@
   dY = Y(2) - Y(1)
   dZ = Z(2) - Z(1)
 
-  alpn1 = Lap + ONE
-  chin1 = chi + ONE
-  gxx = dxx + ONE
-  gyy = dyy + ONE
-  gzz = dzz + ONE
+  do k=1,ex(3)
+  do j=1,ex(2)
+  do i=1,ex(1)
+    alpn1(i,j,k) = Lap(i,j,k) + ONE
+    chin1(i,j,k) = chi(i,j,k) + ONE
+    gxx(i,j,k) = dxx(i,j,k) + ONE
+    gyy(i,j,k) = dyy(i,j,k) + ONE
+    gzz(i,j,k) = dzz(i,j,k) + ONE
+  enddo
+  enddo
+  enddo
 
   call fderivs(ex,betax,betaxx,betaxy,betaxz,X,Y,Z,ANTI, SYM, SYM,Symmetry,Lev)
   call fderivs(ex,betay,betayx,betayy,betayz,X,Y,Z, SYM,ANTI, SYM,Symmetry,Lev)
   call fderivs(ex,betaz,betazx,betazy,betazz,X,Y,Z, SYM, SYM,ANTI,Symmetry,Lev)
   
-  div_beta = betaxx + betayy + betazz
- 
   call fderivs(ex,chi,chix,chiy,chiz,X,Y,Z,SYM,SYM,SYM,symmetry,Lev)
 
-  chi_rhs = F2o3 *chin1*( alpn1 * trK - div_beta ) !rhs for chi
-
   call fderivs(ex,dxx,gxxx,gxxy,gxxz,X,Y,Z,SYM ,SYM ,SYM ,Symmetry,Lev)
   call fderivs(ex,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)
@@ -168,151 +178,179 @@
   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)
+  do k=1,ex(3)
+  do j=1,ex(2)
+  do i=1,ex(1)
+    divb_loc = betaxx(i,j,k) + betayy(i,j,k) + betazz(i,j,k)
+    div_beta(i,j,k) = divb_loc
 
-  gyy_rhs = - TWO * alpn1 * Ayy    -  F2o3 * gyy * div_beta          + &
-              TWO *(  gxy * betaxy +   gyy * betayy +   gyz * betazy)
+    chi_rhs(i,j,k) = F2o3 * chin1(i,j,k) * (alpn1(i,j,k) * trK(i,j,k) - divb_loc)
 
-  gzz_rhs = - TWO * alpn1 * Azz    -  F2o3 * gzz * div_beta          + &
-              TWO *(  gxz * betaxz +   gyz * betayz +   gzz * betazz)
+    gxx_rhs(i,j,k) = - TWO * alpn1(i,j,k) * Axx(i,j,k) - F2o3 * gxx(i,j,k) * divb_loc + &
+         TWO * ( gxx(i,j,k) * betaxx(i,j,k) + gxy(i,j,k) * betayx(i,j,k) + gxz(i,j,k) * betazx(i,j,k) )
 
-  gxy_rhs = - TWO * alpn1 * Axy    +  F1o3 * gxy    * div_beta       + &
-                      gxx * betaxy                  +   gxz * betazy + &
-                                       gyy * betayx +   gyz * betazx   &
-                                                    -   gxy * betazz
+    gyy_rhs(i,j,k) = - TWO * alpn1(i,j,k) * Ayy(i,j,k) - F2o3 * gyy(i,j,k) * divb_loc + &
+         TWO * ( gxy(i,j,k) * betaxy(i,j,k) + gyy(i,j,k) * betayy(i,j,k) + gyz(i,j,k) * betazy(i,j,k) )
 
-  gyz_rhs = - TWO * alpn1 * Ayz    +  F1o3 * gyz    * div_beta       + &
-                      gxy * betaxz +   gyy * betayz                  + &
-                      gxz * betaxy                  +   gzz * betazy   &
-                                                    -   gyz * betaxx
+    gzz_rhs(i,j,k) = - TWO * alpn1(i,j,k) * Azz(i,j,k) - F2o3 * gzz(i,j,k) * divb_loc + &
+         TWO * ( gxz(i,j,k) * betaxz(i,j,k) + gyz(i,j,k) * betayz(i,j,k) + gzz(i,j,k) * betazz(i,j,k) )
 
-  gxz_rhs = - TWO * alpn1 * Axz    +  F1o3 * gxz    * div_beta       + &
-                      gxx * betaxz +   gxy * betayz                  + &
-                                       gyz * betayx +   gzz * betazx   &
-                                                    -   gxz * betayy     !rhs for gij
+    gxy_rhs(i,j,k) = - TWO * alpn1(i,j,k) * Axy(i,j,k) + F1o3 * gxy(i,j,k) * divb_loc + &
+         gxx(i,j,k) * betaxy(i,j,k) + gxz(i,j,k) * betazy(i,j,k) + gyy(i,j,k) * betayx(i,j,k) + &
+         gyz(i,j,k) * betazx(i,j,k) - gxy(i,j,k) * betazz(i,j,k)
 
-! invert tilted metric
-  gupzz =  gxx * gyy * gzz + gxy * gyz * gxz + gxz * gxy * gyz - &
-           gxz * gyy * gxz - gxy * gxy * gzz - gxx * gyz * gyz
-  gupxx =   ( gyy * gzz - gyz * gyz ) / gupzz
-  gupxy = - ( gxy * gzz - gyz * gxz ) / gupzz
-  gupxz =   ( gxy * gyz - gyy * gxz ) / gupzz
-  gupyy =   ( gxx * gzz - gxz * gxz ) / gupzz
-  gupyz = - ( gxx * gyz - gxy * gxz ) / gupzz
-  gupzz =   ( gxx * gyy - gxy * gxy ) / gupzz
+    gyz_rhs(i,j,k) = - TWO * alpn1(i,j,k) * Ayz(i,j,k) + F1o3 * gyz(i,j,k) * divb_loc + &
+         gxy(i,j,k) * betaxz(i,j,k) + gyy(i,j,k) * betayz(i,j,k) + gxz(i,j,k) * betaxy(i,j,k) + &
+         gzz(i,j,k) * betazy(i,j,k) - gyz(i,j,k) * betaxx(i,j,k)
 
-  if(co == 0)then
-! Gam^i_Res = Gam^i + gup^ij_,j
-  Gmx_Res = Gamx - (gupxx*(gupxx*gxxx+gupxy*gxyx+gupxz*gxzx)&
-                   +gupxy*(gupxx*gxyx+gupxy*gyyx+gupxz*gyzx)&
-                   +gupxz*(gupxx*gxzx+gupxy*gyzx+gupxz*gzzx)&
-                   +gupxx*(gupxy*gxxy+gupyy*gxyy+gupyz*gxzy)&
-                   +gupxy*(gupxy*gxyy+gupyy*gyyy+gupyz*gyzy)&
-                   +gupxz*(gupxy*gxzy+gupyy*gyzy+gupyz*gzzy)&
-                   +gupxx*(gupxz*gxxz+gupyz*gxyz+gupzz*gxzz)&
-                   +gupxy*(gupxz*gxyz+gupyz*gyyz+gupzz*gyzz)&
-                   +gupxz*(gupxz*gxzz+gupyz*gyzz+gupzz*gzzz))
-  Gmy_Res = Gamy - (gupxx*(gupxy*gxxx+gupyy*gxyx+gupyz*gxzx)&
-                   +gupxy*(gupxy*gxyx+gupyy*gyyx+gupyz*gyzx)&
-                   +gupxz*(gupxy*gxzx+gupyy*gyzx+gupyz*gzzx)&
-                   +gupxy*(gupxy*gxxy+gupyy*gxyy+gupyz*gxzy)&
-                   +gupyy*(gupxy*gxyy+gupyy*gyyy+gupyz*gyzy)&
-                   +gupyz*(gupxy*gxzy+gupyy*gyzy+gupyz*gzzy)&
-                   +gupxy*(gupxz*gxxz+gupyz*gxyz+gupzz*gxzz)&
-                   +gupyy*(gupxz*gxyz+gupyz*gyyz+gupzz*gyzz)&
-                   +gupyz*(gupxz*gxzz+gupyz*gyzz+gupzz*gzzz))
-  Gmz_Res = Gamz - (gupxx*(gupxz*gxxx+gupyz*gxyx+gupzz*gxzx)&
-                   +gupxy*(gupxz*gxyx+gupyz*gyyx+gupzz*gyzx)&
-                   +gupxz*(gupxz*gxzx+gupyz*gyzx+gupzz*gzzx)&
-                   +gupxy*(gupxz*gxxy+gupyz*gxyy+gupzz*gxzy)&
-                   +gupyy*(gupxz*gxyy+gupyz*gyyy+gupzz*gyzy)&
-                   +gupyz*(gupxz*gxzy+gupyz*gyzy+gupzz*gzzy)&
-                   +gupxz*(gupxz*gxxz+gupyz*gxyz+gupzz*gxzz)&
-                   +gupyz*(gupxz*gxyz+gupyz*gyyz+gupzz*gyzz)&
-                   +gupzz*(gupxz*gxzz+gupyz*gyzz+gupzz*gzzz))
-  endif
+    gxz_rhs(i,j,k) = - TWO * alpn1(i,j,k) * Axz(i,j,k) + F1o3 * gxz(i,j,k) * divb_loc + &
+         gxx(i,j,k) * betaxz(i,j,k) + gxy(i,j,k) * betayz(i,j,k) + gyz(i,j,k) * betayx(i,j,k) + &
+         gzz(i,j,k) * betazx(i,j,k) - gxz(i,j,k) * betayy(i,j,k)
 
-! second kind of connection
-  Gamxxx =HALF*( gupxx*gxxx + gupxy*(TWO*gxyx - gxxy ) + gupxz*(TWO*gxzx - gxxz ))
-  Gamyxx =HALF*( gupxy*gxxx + gupyy*(TWO*gxyx - gxxy ) + gupyz*(TWO*gxzx - gxxz ))
-  Gamzxx =HALF*( gupxz*gxxx + gupyz*(TWO*gxyx - gxxy ) + gupzz*(TWO*gxzx - gxxz ))
+    det_loc = gxx(i,j,k) * gyy(i,j,k) * gzz(i,j,k) + gxy(i,j,k) * gyz(i,j,k) * gxz(i,j,k) + &
+         gxz(i,j,k) * gxy(i,j,k) * gyz(i,j,k) - gxz(i,j,k) * gyy(i,j,k) * gxz(i,j,k) - &
+         gxy(i,j,k) * gxy(i,j,k) * gzz(i,j,k) - gxx(i,j,k) * gyz(i,j,k) * gyz(i,j,k)
+    gupxx_loc = ( gyy(i,j,k) * gzz(i,j,k) - gyz(i,j,k) * gyz(i,j,k) ) / det_loc
+    gupxy_loc = - ( gxy(i,j,k) * gzz(i,j,k) - gyz(i,j,k) * gxz(i,j,k) ) / det_loc
+    gupxz_loc = ( gxy(i,j,k) * gyz(i,j,k) - gyy(i,j,k) * gxz(i,j,k) ) / det_loc
+    gupyy_loc = ( gxx(i,j,k) * gzz(i,j,k) - gxz(i,j,k) * gxz(i,j,k) ) / det_loc
+    gupyz_loc = - ( gxx(i,j,k) * gyz(i,j,k) - gxy(i,j,k) * gxz(i,j,k) ) / det_loc
+    gupzz_loc = ( gxx(i,j,k) * gyy(i,j,k) - gxy(i,j,k) * gxy(i,j,k) ) / det_loc
+    gupxx(i,j,k) = gupxx_loc
+    gupxy(i,j,k) = gupxy_loc
+    gupxz(i,j,k) = gupxz_loc
+    gupyy(i,j,k) = gupyy_loc
+    gupyz(i,j,k) = gupyz_loc
+    gupzz(i,j,k) = gupzz_loc
 
-  Gamxyy =HALF*( gupxx*(TWO*gxyy - gyyx ) + gupxy*gyyy + gupxz*(TWO*gyzy - gyyz ))
-  Gamyyy =HALF*( gupxy*(TWO*gxyy - gyyx ) + gupyy*gyyy + gupyz*(TWO*gyzy - gyyz ))
-  Gamzyy =HALF*( gupxz*(TWO*gxyy - gyyx ) + gupyz*gyyy + gupzz*(TWO*gyzy - gyyz ))
+    if(co == 0)then
+      Gmx_Res(i,j,k) = Gamx(i,j,k) - ( &
+           gupxx_loc*(gupxx_loc*gxxx(i,j,k)+gupxy_loc*gxyx(i,j,k)+gupxz_loc*gxzx(i,j,k)) + &
+           gupxy_loc*(gupxx_loc*gxyx(i,j,k)+gupxy_loc*gyyx(i,j,k)+gupxz_loc*gyzx(i,j,k)) + &
+           gupxz_loc*(gupxx_loc*gxzx(i,j,k)+gupxy_loc*gyzx(i,j,k)+gupxz_loc*gzzx(i,j,k)) + &
+           gupxx_loc*(gupxy_loc*gxxy(i,j,k)+gupyy_loc*gxyy(i,j,k)+gupyz_loc*gxzy(i,j,k)) + &
+           gupxy_loc*(gupxy_loc*gxyy(i,j,k)+gupyy_loc*gyyy(i,j,k)+gupyz_loc*gyzy(i,j,k)) + &
+           gupxz_loc*(gupxy_loc*gxzy(i,j,k)+gupyy_loc*gyzy(i,j,k)+gupyz_loc*gzzy(i,j,k)) + &
+           gupxx_loc*(gupxz_loc*gxxz(i,j,k)+gupyz_loc*gxyz(i,j,k)+gupzz_loc*gxzz(i,j,k)) + &
+           gupxy_loc*(gupxz_loc*gxyz(i,j,k)+gupyz_loc*gyyz(i,j,k)+gupzz_loc*gyzz(i,j,k)) + &
+           gupxz_loc*(gupxz_loc*gxzz(i,j,k)+gupyz_loc*gyzz(i,j,k)+gupzz_loc*gzzz(i,j,k)))
+      Gmy_Res(i,j,k) = Gamy(i,j,k) - ( &
+           gupxx_loc*(gupxy_loc*gxxx(i,j,k)+gupyy_loc*gxyx(i,j,k)+gupyz_loc*gxzx(i,j,k)) + &
+           gupxy_loc*(gupxy_loc*gxyx(i,j,k)+gupyy_loc*gyyx(i,j,k)+gupyz_loc*gyzx(i,j,k)) + &
+           gupxz_loc*(gupxy_loc*gxzx(i,j,k)+gupyy_loc*gyzx(i,j,k)+gupyz_loc*gzzx(i,j,k)) + &
+           gupxy_loc*(gupxy_loc*gxxy(i,j,k)+gupyy_loc*gxyy(i,j,k)+gupyz_loc*gxzy(i,j,k)) + &
+           gupyy_loc*(gupxy_loc*gxyy(i,j,k)+gupyy_loc*gyyy(i,j,k)+gupyz_loc*gyzy(i,j,k)) + &
+           gupyz_loc*(gupxy_loc*gxzy(i,j,k)+gupyy_loc*gyzy(i,j,k)+gupyz_loc*gzzy(i,j,k)) + &
+           gupxy_loc*(gupxz_loc*gxxz(i,j,k)+gupyz_loc*gxyz(i,j,k)+gupzz_loc*gxzz(i,j,k)) + &
+           gupyy_loc*(gupxz_loc*gxyz(i,j,k)+gupyz_loc*gyyz(i,j,k)+gupzz_loc*gyzz(i,j,k)) + &
+           gupyz_loc*(gupxz_loc*gxzz(i,j,k)+gupyz_loc*gyzz(i,j,k)+gupzz_loc*gzzz(i,j,k)))
+      Gmz_Res(i,j,k) = Gamz(i,j,k) - ( &
+           gupxx_loc*(gupxz_loc*gxxx(i,j,k)+gupyz_loc*gxyx(i,j,k)+gupzz_loc*gxzx(i,j,k)) + &
+           gupxy_loc*(gupxz_loc*gxyx(i,j,k)+gupyz_loc*gyyx(i,j,k)+gupzz_loc*gyzx(i,j,k)) + &
+           gupxz_loc*(gupxz_loc*gxzx(i,j,k)+gupyz_loc*gyzx(i,j,k)+gupzz_loc*gzzx(i,j,k)) + &
+           gupxy_loc*(gupxz_loc*gxxy(i,j,k)+gupyz_loc*gxyy(i,j,k)+gupzz_loc*gxzy(i,j,k)) + &
+           gupyy_loc*(gupxz_loc*gxyy(i,j,k)+gupyz_loc*gyyy(i,j,k)+gupzz_loc*gyzy(i,j,k)) + &
+           gupyz_loc*(gupxz_loc*gxzy(i,j,k)+gupyz_loc*gyzy(i,j,k)+gupzz_loc*gzzy(i,j,k)) + &
+           gupxz_loc*(gupxz_loc*gxxz(i,j,k)+gupyz_loc*gxyz(i,j,k)+gupzz_loc*gxzz(i,j,k)) + &
+           gupyz_loc*(gupxz_loc*gxyz(i,j,k)+gupyz_loc*gyyz(i,j,k)+gupzz_loc*gyzz(i,j,k)) + &
+           gupzz_loc*(gupxz_loc*gxzz(i,j,k)+gupyz_loc*gyzz(i,j,k)+gupzz_loc*gzzz(i,j,k)))
+    endif
 
-  Gamxzz =HALF*( gupxx*(TWO*gxzz - gzzx ) + gupxy*(TWO*gyzz - gzzy ) + gupxz*gzzz)
-  Gamyzz =HALF*( gupxy*(TWO*gxzz - gzzx ) + gupyy*(TWO*gyzz - gzzy ) + gupyz*gzzz)
-  Gamzzz =HALF*( gupxz*(TWO*gxzz - gzzx ) + gupyz*(TWO*gyzz - gzzy ) + gupzz*gzzz)
+    Gamxxx(i,j,k)=HALF*( gupxx_loc*gxxx(i,j,k) + gupxy_loc*(TWO*gxyx(i,j,k) - gxxy(i,j,k)) + gupxz_loc*(TWO*gxzx(i,j,k) - gxxz(i,j,k)))
+    Gamyxx(i,j,k)=HALF*( gupxy_loc*gxxx(i,j,k) + gupyy_loc*(TWO*gxyx(i,j,k) - gxxy(i,j,k)) + gupyz_loc*(TWO*gxzx(i,j,k) - gxxz(i,j,k)))
+    Gamzxx(i,j,k)=HALF*( gupxz_loc*gxxx(i,j,k) + gupyz_loc*(TWO*gxyx(i,j,k) - gxxy(i,j,k)) + gupzz_loc*(TWO*gxzx(i,j,k) - gxxz(i,j,k)))
 
-  Gamxxy =HALF*( gupxx*gxxy + gupxy*gyyx + gupxz*( gxzy + gyzx - gxyz ) )
-  Gamyxy =HALF*( gupxy*gxxy + gupyy*gyyx + gupyz*( gxzy + gyzx - gxyz ) )
-  Gamzxy =HALF*( gupxz*gxxy + gupyz*gyyx + gupzz*( gxzy + gyzx - gxyz ) )
+    Gamxyy(i,j,k)=HALF*( gupxx_loc*(TWO*gxyy(i,j,k) - gyyx(i,j,k)) + gupxy_loc*gyyy(i,j,k) + gupxz_loc*(TWO*gyzy(i,j,k) - gyyz(i,j,k)))
+    Gamyyy(i,j,k)=HALF*( gupxy_loc*(TWO*gxyy(i,j,k) - gyyx(i,j,k)) + gupyy_loc*gyyy(i,j,k) + gupyz_loc*(TWO*gyzy(i,j,k) - gyyz(i,j,k)))
+    Gamzyy(i,j,k)=HALF*( gupxz_loc*(TWO*gxyy(i,j,k) - gyyx(i,j,k)) + gupyz_loc*gyyy(i,j,k) + gupzz_loc*(TWO*gyzy(i,j,k) - gyyz(i,j,k)))
 
-  Gamxxz =HALF*( gupxx*gxxz + gupxy*( gxyz + gyzx - gxzy ) + gupxz*gzzx )
-  Gamyxz =HALF*( gupxy*gxxz + gupyy*( gxyz + gyzx - gxzy ) + gupyz*gzzx )
-  Gamzxz =HALF*( gupxz*gxxz + gupyz*( gxyz + gyzx - gxzy ) + gupzz*gzzx )
+    Gamxzz(i,j,k)=HALF*( gupxx_loc*(TWO*gxzz(i,j,k) - gzzx(i,j,k)) + gupxy_loc*(TWO*gyzz(i,j,k) - gzzy(i,j,k)) + gupxz_loc*gzzz(i,j,k))
+    Gamyzz(i,j,k)=HALF*( gupxy_loc*(TWO*gxzz(i,j,k) - gzzx(i,j,k)) + gupyy_loc*(TWO*gyzz(i,j,k) - gzzy(i,j,k)) + gupyz_loc*gzzz(i,j,k))
+    Gamzzz(i,j,k)=HALF*( gupxz_loc*(TWO*gxzz(i,j,k) - gzzx(i,j,k)) + gupyz_loc*(TWO*gyzz(i,j,k) - gzzy(i,j,k)) + gupzz_loc*gzzz(i,j,k))
 
-  Gamxyz =HALF*( gupxx*( gxyz + gxzy - gyzx ) + gupxy*gyyz + gupxz*gzzy )
-  Gamyyz =HALF*( gupxy*( gxyz + gxzy - gyzx ) + gupyy*gyyz + gupyz*gzzy )
-  Gamzyz =HALF*( gupxz*( gxyz + gxzy - gyzx ) + gupyz*gyyz + gupzz*gzzy )
+    Gamxxy(i,j,k)=HALF*( gupxx_loc*gxxy(i,j,k) + gupxy_loc*gyyx(i,j,k) + gupxz_loc*(gxzy(i,j,k) + gyzx(i,j,k) - gxyz(i,j,k)) )
+    Gamyxy(i,j,k)=HALF*( gupxy_loc*gxxy(i,j,k) + gupyy_loc*gyyx(i,j,k) + gupyz_loc*(gxzy(i,j,k) + gyzx(i,j,k) - gxyz(i,j,k)) )
+    Gamzxy(i,j,k)=HALF*( gupxz_loc*gxxy(i,j,k) + gupyz_loc*gyyx(i,j,k) + gupzz_loc*(gxzy(i,j,k) + gyzx(i,j,k) - gxyz(i,j,k)) )
+
+    Gamxxz(i,j,k)=HALF*( gupxx_loc*gxxz(i,j,k) + gupxy_loc*(gxyz(i,j,k) + gyzx(i,j,k) - gxzy(i,j,k)) + gupxz_loc*gzzx(i,j,k) )
+    Gamyxz(i,j,k)=HALF*( gupxy_loc*gxxz(i,j,k) + gupyy_loc*(gxyz(i,j,k) + gyzx(i,j,k) - gxzy(i,j,k)) + gupyz_loc*gzzx(i,j,k) )
+    Gamzxz(i,j,k)=HALF*( gupxz_loc*gxxz(i,j,k) + gupyz_loc*(gxyz(i,j,k) + gyzx(i,j,k) - gxzy(i,j,k)) + gupzz_loc*gzzx(i,j,k) )
+
+    Gamxyz(i,j,k)=HALF*( gupxx_loc*(gxyz(i,j,k) + gxzy(i,j,k) - gyzx(i,j,k)) + gupxy_loc*gyyz(i,j,k) + gupxz_loc*gzzy(i,j,k) )
+    Gamyyz(i,j,k)=HALF*( gupxy_loc*(gxyz(i,j,k) + gxzy(i,j,k) - gyzx(i,j,k)) + gupyy_loc*gyyz(i,j,k) + gupyz_loc*gzzy(i,j,k) )
+    Gamzyz(i,j,k)=HALF*( gupxz_loc*(gxyz(i,j,k) + gxzy(i,j,k) - gyzx(i,j,k)) + gupyz_loc*gyyz(i,j,k) + gupzz_loc*gzzy(i,j,k) )
+  enddo
+  enddo
+  enddo
 ! Raise indices of \tilde A_{ij} and store in R_ij
 
-  Rxx =    gupxx * gupxx * Axx + gupxy * gupxy * Ayy + gupxz * gupxz * Azz + &
-      TWO*(gupxx * gupxy * Axy + gupxx * gupxz * Axz + gupxy * gupxz * Ayz)
-
-  Ryy =    gupxy * gupxy * Axx + gupyy * gupyy * Ayy + gupyz * gupyz * Azz + &
-      TWO*(gupxy * gupyy * Axy + gupxy * gupyz * Axz + gupyy * gupyz * Ayz)
-
-  Rzz =    gupxz * gupxz * Axx + gupyz * gupyz * Ayy + gupzz * gupzz * Azz + &
-      TWO*(gupxz * gupyz * Axy + gupxz * gupzz * Axz + gupyz * gupzz * Ayz)
-
-  Rxy =    gupxx * gupxy * Axx + gupxy * gupyy * Ayy + gupxz * gupyz * Azz + &
-          (gupxx * gupyy       + gupxy * gupxy)* Axy                       + &
-          (gupxx * gupyz       + gupxz * gupxy)* Axz                       + &
-          (gupxy * gupyz       + gupxz * gupyy)* Ayz
-
-  Rxz =    gupxx * gupxz * Axx + gupxy * gupyz * Ayy + gupxz * gupzz * Azz + &
-          (gupxx * gupyz       + gupxy * gupxz)* Axy                       + &
-          (gupxx * gupzz       + gupxz * gupxz)* Axz                       + &
-          (gupxy * gupzz       + gupxz * gupyz)* Ayz
-
-  Ryz =    gupxy * gupxz * Axx + gupyy * gupyz * Ayy + gupyz * gupzz * Azz + &
-          (gupxy * gupyz       + gupyy * gupxz)* Axy                       + &
-          (gupxy * gupzz       + gupyz * gupxz)* Axz                       + &
-          (gupyy * gupzz       + gupyz * gupyz)* Ayz
-
 ! Right hand side for Gam^i without shift terms...
   call fderivs(ex,Lap,Lapx,Lapy,Lapz,X,Y,Z,SYM,SYM,SYM,Symmetry,Lev)
   call fderivs(ex,trK,Kx,Ky,Kz,X,Y,Z,SYM,SYM,SYM,symmetry,Lev)
+  do k=1,ex(3)
+  do j=1,ex(2)
+  do i=1,ex(1)
+    gupxx_loc = gupxx(i,j,k)
+    gupxy_loc = gupxy(i,j,k)
+    gupxz_loc = gupxz(i,j,k)
+    gupyy_loc = gupyy(i,j,k)
+    gupyz_loc = gupyz(i,j,k)
+    gupzz_loc = gupzz(i,j,k)
 
-   Gamx_rhs = - TWO * (   Lapx * Rxx +   Lapy * Rxy +   Lapz * Rxz ) + &
-        TWO * alpn1 * (                                                &
-        -F3o2/chin1 * (   chix * Rxx +   chiy * Rxy +   chiz * Rxz ) - &
-              gupxx * (   F2o3 * Kx  +  EIGHT * PI * Sx            ) - &
-              gupxy * (   F2o3 * Ky  +  EIGHT * PI * Sy            ) - &
-              gupxz * (   F2o3 * Kz  +  EIGHT * PI * Sz            ) + &
-                        Gamxxx * Rxx + Gamxyy * Ryy + Gamxzz * Rzz   + &
-                TWO * ( Gamxxy * Rxy + Gamxxz * Rxz + Gamxyz * Ryz ) )
+    Rxx_loc = gupxx_loc * gupxx_loc * Axx(i,j,k) + gupxy_loc * gupxy_loc * Ayy(i,j,k) + gupxz_loc * gupxz_loc * Azz(i,j,k) + &
+         TWO * (gupxx_loc * gupxy_loc * Axy(i,j,k) + gupxx_loc * gupxz_loc * Axz(i,j,k) + gupxy_loc * gupxz_loc * Ayz(i,j,k))
+    Ryy_loc = gupxy_loc * gupxy_loc * Axx(i,j,k) + gupyy_loc * gupyy_loc * Ayy(i,j,k) + gupyz_loc * gupyz_loc * Azz(i,j,k) + &
+         TWO * (gupxy_loc * gupyy_loc * Axy(i,j,k) + gupxy_loc * gupyz_loc * Axz(i,j,k) + gupyy_loc * gupyz_loc * Ayz(i,j,k))
+    Rzz_loc = gupxz_loc * gupxz_loc * Axx(i,j,k) + gupyz_loc * gupyz_loc * Ayy(i,j,k) + gupzz_loc * gupzz_loc * Azz(i,j,k) + &
+         TWO * (gupxz_loc * gupyz_loc * Axy(i,j,k) + gupxz_loc * gupzz_loc * Axz(i,j,k) + gupyz_loc * gupzz_loc * Ayz(i,j,k))
+    Rxy_loc = gupxx_loc * gupxy_loc * Axx(i,j,k) + gupxy_loc * gupyy_loc * Ayy(i,j,k) + gupxz_loc * gupyz_loc * Azz(i,j,k) + &
+         (gupxx_loc * gupyy_loc + gupxy_loc * gupxy_loc) * Axy(i,j,k) + &
+         (gupxx_loc * gupyz_loc + gupxz_loc * gupxy_loc) * Axz(i,j,k) + &
+         (gupxy_loc * gupyz_loc + gupxz_loc * gupyy_loc) * Ayz(i,j,k)
+    Rxz_loc = gupxx_loc * gupxz_loc * Axx(i,j,k) + gupxy_loc * gupyz_loc * Ayy(i,j,k) + gupxz_loc * gupzz_loc * Azz(i,j,k) + &
+         (gupxx_loc * gupyz_loc + gupxy_loc * gupxz_loc) * Axy(i,j,k) + &
+         (gupxx_loc * gupzz_loc + gupxz_loc * gupxz_loc) * Axz(i,j,k) + &
+         (gupxy_loc * gupzz_loc + gupxz_loc * gupyz_loc) * Ayz(i,j,k)
+    Ryz_loc = gupxy_loc * gupxz_loc * Axx(i,j,k) + gupyy_loc * gupyz_loc * Ayy(i,j,k) + gupyz_loc * gupzz_loc * Azz(i,j,k) + &
+         (gupxy_loc * gupyz_loc + gupyy_loc * gupxz_loc) * Axy(i,j,k) + &
+         (gupxy_loc * gupzz_loc + gupyz_loc * gupxz_loc) * Axz(i,j,k) + &
+         (gupyy_loc * gupzz_loc + gupyz_loc * gupyz_loc) * Ayz(i,j,k)
+    Rxx(i,j,k) = Rxx_loc
+    Ryy(i,j,k) = Ryy_loc
+    Rzz(i,j,k) = Rzz_loc
+    Rxy(i,j,k) = Rxy_loc
+    Rxz(i,j,k) = Rxz_loc
+    Ryz(i,j,k) = Ryz_loc
 
-   Gamy_rhs = - TWO * (   Lapx * Rxy +   Lapy * Ryy +   Lapz * Ryz ) + &
-        TWO * alpn1 * (                                                &
-        -F3o2/chin1 * (   chix * Rxy +  chiy * Ryy +    chiz * Ryz ) - &
-              gupxy * (   F2o3 * Kx  +  EIGHT * PI * Sx            ) - &
-              gupyy * (   F2o3 * Ky  +  EIGHT * PI * Sy            ) - &
-              gupyz * (   F2o3 * Kz  +  EIGHT * PI * Sz            ) + &
-                        Gamyxx * Rxx + Gamyyy * Ryy + Gamyzz * Rzz   + &
-                TWO * ( Gamyxy * Rxy + Gamyxz * Rxz + Gamyyz * Ryz ) )
+    Gamx_rhs(i,j,k) = - TWO * (Lapx(i,j,k) * Rxx_loc + Lapy(i,j,k) * Rxy_loc + Lapz(i,j,k) * Rxz_loc) + &
+         TWO * alpn1(i,j,k) * ( &
+         -F3o2/chin1(i,j,k) * (chix(i,j,k) * Rxx_loc + chiy(i,j,k) * Rxy_loc + chiz(i,j,k) * Rxz_loc) - &
+         gupxx_loc * (F2o3 * Kx(i,j,k) + EIGHT * PI * Sx(i,j,k)) - &
+         gupxy_loc * (F2o3 * Ky(i,j,k) + EIGHT * PI * Sy(i,j,k)) - &
+         gupxz_loc * (F2o3 * Kz(i,j,k) + EIGHT * PI * Sz(i,j,k)) + &
+         Gamxxx(i,j,k) * Rxx_loc + Gamxyy(i,j,k) * Ryy_loc + Gamxzz(i,j,k) * Rzz_loc + &
+         TWO * (Gamxxy(i,j,k) * Rxy_loc + Gamxxz(i,j,k) * Rxz_loc + Gamxyz(i,j,k) * Ryz_loc))
 
-   Gamz_rhs = - TWO * (   Lapx * Rxz +   Lapy * Ryz +   Lapz * Rzz ) + &
-        TWO * alpn1 * (                                                &
-        -F3o2/chin1 * (   chix * Rxz +  chiy * Ryz +    chiz * Rzz ) - &
-              gupxz * (   F2o3 * Kx  +  EIGHT * PI * Sx            ) - &
-              gupyz * (   F2o3 * Ky  +  EIGHT * PI * Sy            ) - &
-              gupzz * (   F2o3 * Kz  +  EIGHT * PI * Sz            ) + &
-                        Gamzxx * Rxx + Gamzyy * Ryy + Gamzzz * Rzz   + &
-                TWO * ( Gamzxy * Rxy + Gamzxz * Rxz + Gamzyz * Ryz ) )
+    Gamy_rhs(i,j,k) = - TWO * (Lapx(i,j,k) * Rxy_loc + Lapy(i,j,k) * Ryy_loc + Lapz(i,j,k) * Ryz_loc) + &
+         TWO * alpn1(i,j,k) * ( &
+         -F3o2/chin1(i,j,k) * (chix(i,j,k) * Rxy_loc + chiy(i,j,k) * Ryy_loc + chiz(i,j,k) * Ryz_loc) - &
+         gupxy_loc * (F2o3 * Kx(i,j,k) + EIGHT * PI * Sx(i,j,k)) - &
+         gupyy_loc * (F2o3 * Ky(i,j,k) + EIGHT * PI * Sy(i,j,k)) - &
+         gupyz_loc * (F2o3 * Kz(i,j,k) + EIGHT * PI * Sz(i,j,k)) + &
+         Gamyxx(i,j,k) * Rxx_loc + Gamyyy(i,j,k) * Ryy_loc + Gamyzz(i,j,k) * Rzz_loc + &
+         TWO * (Gamyxy(i,j,k) * Rxy_loc + Gamyxz(i,j,k) * Rxz_loc + Gamyyz(i,j,k) * Ryz_loc))
+
+    Gamz_rhs(i,j,k) = - TWO * (Lapx(i,j,k) * Rxz_loc + Lapy(i,j,k) * Ryz_loc + Lapz(i,j,k) * Rzz_loc) + &
+         TWO * alpn1(i,j,k) * ( &
+         -F3o2/chin1(i,j,k) * (chix(i,j,k) * Rxz_loc + chiy(i,j,k) * Ryz_loc + chiz(i,j,k) * Rzz_loc) - &
+         gupxz_loc * (F2o3 * Kx(i,j,k) + EIGHT * PI * Sx(i,j,k)) - &
+         gupyz_loc * (F2o3 * Ky(i,j,k) + EIGHT * PI * Sy(i,j,k)) - &
+         gupzz_loc * (F2o3 * Kz(i,j,k) + EIGHT * PI * Sz(i,j,k)) + &
+         Gamzxx(i,j,k) * Rxx_loc + Gamzyy(i,j,k) * Ryy_loc + Gamzzz(i,j,k) * Rzz_loc + &
+         TWO * (Gamzxy(i,j,k) * Rxy_loc + Gamzxz(i,j,k) * Rxz_loc + Gamzyz(i,j,k) * Ryz_loc))
+  enddo
+  enddo
+  enddo
 
   call fdderivs(ex,betax,gxxx,gxyx,gxzx,gyyx,gyzx,gzzx,&
                 X,Y,Z,ANTI,SYM, SYM ,Symmetry,Lev)
@@ -321,38 +359,54 @@
   call fdderivs(ex,betaz,gxxz,gxyz,gxzz,gyyz,gyzz,gzzz,&
                 X,Y,Z,SYM ,SYM, ANTI,Symmetry,Lev)
 
-  fxx = gxxx + gxyy + gxzz
-  fxy = gxyx + gyyy + gyzz
-  fxz = gxzx + gyzy + gzzz
-
-  Gamxa =       gupxx * Gamxxx + gupyy * Gamxyy + gupzz * Gamxzz + &
-          TWO*( gupxy * Gamxxy + gupxz * Gamxxz + gupyz * Gamxyz )
-  Gamya =       gupxx * Gamyxx + gupyy * Gamyyy + gupzz * Gamyzz + &
-          TWO*( gupxy * Gamyxy + gupxz * Gamyxz + gupyz * Gamyyz )
-  Gamza =       gupxx * Gamzxx + gupyy * Gamzyy + gupzz * Gamzzz + &
-          TWO*( gupxy * Gamzxy + gupxz * Gamzxz + gupyz * Gamzyz )
-
   call fderivs(ex,Gamx,Gamxx,Gamxy,Gamxz,X,Y,Z,ANTI,SYM ,SYM ,Symmetry,Lev)
   call fderivs(ex,Gamy,Gamyx,Gamyy,Gamyz,X,Y,Z,SYM ,ANTI,SYM ,Symmetry,Lev)
   call fderivs(ex,Gamz,Gamzx,Gamzy,Gamzz,X,Y,Z,SYM ,SYM ,ANTI,Symmetry,Lev)
+  do k=1,ex(3)
+  do j=1,ex(2)
+  do i=1,ex(1)
+    divb_loc = div_beta(i,j,k)
+    fxx_loc = gxxx(i,j,k) + gxyy(i,j,k) + gxzz(i,j,k)
+    fxy_loc = gxyx(i,j,k) + gyyy(i,j,k) + gyzz(i,j,k)
+    fxz_loc = gxzx(i,j,k) + gyzy(i,j,k) + gzzz(i,j,k)
 
-  Gamx_rhs =               Gamx_rhs +  F2o3 *  Gamxa * div_beta        - &
-                     Gamxa * betaxx - Gamya * betaxy - Gamza * betaxz  + &
-             F1o3 * (gupxx * fxx    + gupxy * fxy    + gupxz * fxz    ) + &
-                     gupxx * gxxx   + gupyy * gyyx   + gupzz * gzzx    + &
-              TWO * (gupxy * gxyx   + gupxz * gxzx   + gupyz * gyzx  )
+    gupxx_loc = gupxx(i,j,k)
+    gupxy_loc = gupxy(i,j,k)
+    gupxz_loc = gupxz(i,j,k)
+    gupyy_loc = gupyy(i,j,k)
+    gupyz_loc = gupyz(i,j,k)
+    gupzz_loc = gupzz(i,j,k)
 
-  Gamy_rhs =               Gamy_rhs +  F2o3 *  Gamya * div_beta        - &
-                     Gamxa * betayx - Gamya * betayy - Gamza * betayz  + &
-             F1o3 * (gupxy * fxx    + gupyy * fxy    + gupyz * fxz    ) + &
-                     gupxx * gxxy   + gupyy * gyyy   + gupzz * gzzy    + &
-              TWO * (gupxy * gxyy   + gupxz * gxzy   + gupyz * gyzy  )
+    Gamxa_loc = gupxx_loc * Gamxxx(i,j,k) + gupyy_loc * Gamxyy(i,j,k) + gupzz_loc * Gamxzz(i,j,k) + &
+         TWO * (gupxy_loc * Gamxxy(i,j,k) + gupxz_loc * Gamxxz(i,j,k) + gupyz_loc * Gamxyz(i,j,k))
+    Gamya_loc = gupxx_loc * Gamyxx(i,j,k) + gupyy_loc * Gamyyy(i,j,k) + gupzz_loc * Gamyzz(i,j,k) + &
+         TWO * (gupxy_loc * Gamyxy(i,j,k) + gupxz_loc * Gamyxz(i,j,k) + gupyz_loc * Gamyyz(i,j,k))
+    Gamza_loc = gupxx_loc * Gamzxx(i,j,k) + gupyy_loc * Gamzyy(i,j,k) + gupzz_loc * Gamzzz(i,j,k) + &
+         TWO * (gupxy_loc * Gamzxy(i,j,k) + gupxz_loc * Gamzxz(i,j,k) + gupyz_loc * Gamzyz(i,j,k))
+    Gamxa(i,j,k) = Gamxa_loc
+    Gamya(i,j,k) = Gamya_loc
+    Gamza(i,j,k) = Gamza_loc
 
-  Gamz_rhs =               Gamz_rhs +  F2o3 *  Gamza * div_beta        - &
-                     Gamxa * betazx - Gamya * betazy - Gamza * betazz  + &
-             F1o3 * (gupxz * fxx    + gupyz * fxy    + gupzz * fxz    ) + &
-                     gupxx * gxxz   + gupyy * gyyz   + gupzz * gzzz    + &
-              TWO * (gupxy * gxyz   + gupxz * gxzz   + gupyz * gyzz  )    !rhs for Gam^i
+    Gamx_rhs(i,j,k) = Gamx_rhs(i,j,k) + F2o3 * Gamxa_loc * divb_loc - &
+         Gamxa_loc * betaxx(i,j,k) - Gamya_loc * betaxy(i,j,k) - Gamza_loc * betaxz(i,j,k) + &
+         F1o3 * (gupxx_loc * fxx_loc + gupxy_loc * fxy_loc + gupxz_loc * fxz_loc) + &
+         gupxx_loc * gxxx(i,j,k) + gupyy_loc * gyyx(i,j,k) + gupzz_loc * gzzx(i,j,k) + &
+         TWO * (gupxy_loc * gxyx(i,j,k) + gupxz_loc * gxzx(i,j,k) + gupyz_loc * gyzx(i,j,k))
+
+    Gamy_rhs(i,j,k) = Gamy_rhs(i,j,k) + F2o3 * Gamya_loc * divb_loc - &
+         Gamxa_loc * betayx(i,j,k) - Gamya_loc * betayy(i,j,k) - Gamza_loc * betayz(i,j,k) + &
+         F1o3 * (gupxy_loc * fxx_loc + gupyy_loc * fxy_loc + gupyz_loc * fxz_loc) + &
+         gupxx_loc * gxxy(i,j,k) + gupyy_loc * gyyy(i,j,k) + gupzz_loc * gzzy(i,j,k) + &
+         TWO * (gupxy_loc * gxyy(i,j,k) + gupxz_loc * gxzy(i,j,k) + gupyz_loc * gyzy(i,j,k))
+
+    Gamz_rhs(i,j,k) = Gamz_rhs(i,j,k) + F2o3 * Gamza_loc * divb_loc - &
+         Gamxa_loc * betazx(i,j,k) - Gamya_loc * betazy(i,j,k) - Gamza_loc * betazz(i,j,k) + &
+         F1o3 * (gupxz_loc * fxx_loc + gupyz_loc * fxy_loc + gupzz_loc * fxz_loc) + &
+         gupxx_loc * gxxz(i,j,k) + gupyy_loc * gyyz(i,j,k) + gupzz_loc * gzzz(i,j,k) + &
+         TWO * (gupxy_loc * gxyz(i,j,k) + gupxz_loc * gxzz(i,j,k) + gupyz_loc * gyzz(i,j,k))
+  enddo
+  enddo
+  enddo
 
 !first kind of connection stored in gij,k
   gxxx = gxx * Gamxxx + gxy * Gamyxx + gxz * Gamzxx
@@ -604,189 +658,187 @@
 !covariant second derivative of chi respect to tilted metric
   call fdderivs(ex,chi,fxx,fxy,fxz,fyy,fyz,fzz,X,Y,Z,SYM,SYM,SYM,Symmetry,Lev)
 
-  fxx = fxx - Gamxxx * chix - Gamyxx * chiy - Gamzxx * chiz
-  fxy = fxy - Gamxxy * chix - Gamyxy * chiy - Gamzxy * chiz
-  fxz = fxz - Gamxxz * chix - Gamyxz * chiy - Gamzxz * chiz
-  fyy = fyy - Gamxyy * chix - Gamyyy * chiy - Gamzyy * chiz
-  fyz = fyz - Gamxyz * chix - Gamyyz * chiy - Gamzyz * chiz
-  fzz = fzz - Gamxzz * chix - Gamyzz * chiy - Gamzzz * chiz
-! Store D^l D_l chi - 3/(2*chi) D^l chi D_l chi in f
+  do k=1,ex(3)
+  do j=1,ex(2)
+  do i=1,ex(1)
+    fxx(i,j,k) = fxx(i,j,k) - Gamxxx(i,j,k) * chix(i,j,k) - Gamyxx(i,j,k) * chiy(i,j,k) - Gamzxx(i,j,k) * chiz(i,j,k)
+    fxy(i,j,k) = fxy(i,j,k) - Gamxxy(i,j,k) * chix(i,j,k) - Gamyxy(i,j,k) * chiy(i,j,k) - Gamzxy(i,j,k) * chiz(i,j,k)
+    fxz(i,j,k) = fxz(i,j,k) - Gamxxz(i,j,k) * chix(i,j,k) - Gamyxz(i,j,k) * chiy(i,j,k) - Gamzxz(i,j,k) * chiz(i,j,k)
+    fyy(i,j,k) = fyy(i,j,k) - Gamxyy(i,j,k) * chix(i,j,k) - Gamyyy(i,j,k) * chiy(i,j,k) - Gamzyy(i,j,k) * chiz(i,j,k)
+    fyz(i,j,k) = fyz(i,j,k) - Gamxyz(i,j,k) * chix(i,j,k) - Gamyyz(i,j,k) * chiy(i,j,k) - Gamzyz(i,j,k) * chiz(i,j,k)
+    fzz(i,j,k) = fzz(i,j,k) - Gamxzz(i,j,k) * chix(i,j,k) - Gamyzz(i,j,k) * chiy(i,j,k) - Gamzzz(i,j,k) * chiz(i,j,k)
 
-  f =        gupxx * ( fxx - F3o2/chin1 * chix * chix ) + &
-             gupyy * ( fyy - F3o2/chin1 * chiy * chiy ) + &
-             gupzz * ( fzz - F3o2/chin1 * chiz * chiz ) + &
-       TWO * gupxy * ( fxy - F3o2/chin1 * chix * chiy ) + &
-       TWO * gupxz * ( fxz - F3o2/chin1 * chix * chiz ) + &
-       TWO * gupyz * ( fyz - F3o2/chin1 * chiy * chiz ) 
-! Add chi part to Ricci tensor:
+    chin_loc = chin1(i,j,k)
+    f_loc = gupxx(i,j,k) * (fxx(i,j,k) - F3o2/chin_loc * chix(i,j,k) * chix(i,j,k)) + &
+            gupyy(i,j,k) * (fyy(i,j,k) - F3o2/chin_loc * chiy(i,j,k) * chiy(i,j,k)) + &
+            gupzz(i,j,k) * (fzz(i,j,k) - F3o2/chin_loc * chiz(i,j,k) * chiz(i,j,k)) + &
+            TWO * gupxy(i,j,k) * (fxy(i,j,k) - F3o2/chin_loc * chix(i,j,k) * chiy(i,j,k)) + &
+            TWO * gupxz(i,j,k) * (fxz(i,j,k) - F3o2/chin_loc * chix(i,j,k) * chiz(i,j,k)) + &
+            TWO * gupyz(i,j,k) * (fyz(i,j,k) - F3o2/chin_loc * chiy(i,j,k) * chiz(i,j,k))
+    f(i,j,k) = f_loc
 
-  Rxx = Rxx + (fxx - chix*chix/chin1/TWO + gxx * f)/chin1/TWO
-  Ryy = Ryy + (fyy - chiy*chiy/chin1/TWO + gyy * f)/chin1/TWO
-  Rzz = Rzz + (fzz - chiz*chiz/chin1/TWO + gzz * f)/chin1/TWO
-  Rxy = Rxy + (fxy - chix*chiy/chin1/TWO + gxy * f)/chin1/TWO
-  Rxz = Rxz + (fxz - chix*chiz/chin1/TWO + gxz * f)/chin1/TWO
-  Ryz = Ryz + (fyz - chiy*chiz/chin1/TWO + gyz * f)/chin1/TWO
+    Rxx(i,j,k) = Rxx(i,j,k) + (fxx(i,j,k) - chix(i,j,k)*chix(i,j,k)/chin_loc/TWO + gxx(i,j,k) * f_loc)/chin_loc/TWO
+    Ryy(i,j,k) = Ryy(i,j,k) + (fyy(i,j,k) - chiy(i,j,k)*chiy(i,j,k)/chin_loc/TWO + gyy(i,j,k) * f_loc)/chin_loc/TWO
+    Rzz(i,j,k) = Rzz(i,j,k) + (fzz(i,j,k) - chiz(i,j,k)*chiz(i,j,k)/chin_loc/TWO + gzz(i,j,k) * f_loc)/chin_loc/TWO
+    Rxy(i,j,k) = Rxy(i,j,k) + (fxy(i,j,k) - chix(i,j,k)*chiy(i,j,k)/chin_loc/TWO + gxy(i,j,k) * f_loc)/chin_loc/TWO
+    Rxz(i,j,k) = Rxz(i,j,k) + (fxz(i,j,k) - chix(i,j,k)*chiz(i,j,k)/chin_loc/TWO + gxz(i,j,k) * f_loc)/chin_loc/TWO
+    Ryz(i,j,k) = Ryz(i,j,k) + (fyz(i,j,k) - chiy(i,j,k)*chiz(i,j,k)/chin_loc/TWO + gyz(i,j,k) * f_loc)/chin_loc/TWO
+  enddo
+  enddo
+  enddo
 
 ! covariant second derivatives of the lapse respect to physical metric
   call fdderivs(ex,Lap,fxx,fxy,fxz,fyy,fyz,fzz,X,Y,Z, &
                 SYM,SYM,SYM,symmetry,Lev)
 
-  gxxx = (gupxx * chix + gupxy * chiy + gupxz * chiz)/chin1
-  gxxy = (gupxy * chix + gupyy * chiy + gupyz * chiz)/chin1
-  gxxz = (gupxz * chix + gupyz * chiy + gupzz * chiz)/chin1
-! now get physical second kind of connection
-  Gamxxx = Gamxxx - ( (chix + chix)/chin1 - gxx * gxxx )*HALF
-  Gamyxx = Gamyxx - (                     - gxx * gxxy )*HALF
-  Gamzxx = Gamzxx - (                     - gxx * gxxz )*HALF
-  Gamxyy = Gamxyy - (                     - gyy * gxxx )*HALF
-  Gamyyy = Gamyyy - ( (chiy + chiy)/chin1 - gyy * gxxy )*HALF
-  Gamzyy = Gamzyy - (                     - gyy * gxxz )*HALF
-  Gamxzz = Gamxzz - (                     - gzz * gxxx )*HALF
-  Gamyzz = Gamyzz - (                     - gzz * gxxy )*HALF
-  Gamzzz = Gamzzz - ( (chiz + chiz)/chin1 - gzz * gxxz )*HALF
-  Gamxxy = Gamxxy - (  chiy        /chin1 - gxy * gxxx )*HALF
-  Gamyxy = Gamyxy - (         chix /chin1 - gxy * gxxy )*HALF
-  Gamzxy = Gamzxy - (                     - gxy * gxxz )*HALF
-  Gamxxz = Gamxxz - (  chiz        /chin1 - gxz * gxxx )*HALF
-  Gamyxz = Gamyxz - (                     - gxz * gxxy )*HALF
-  Gamzxz = Gamzxz - (         chix /chin1 - gxz * gxxz )*HALF
-  Gamxyz = Gamxyz - (                     - gyz * gxxx )*HALF
-  Gamyyz = Gamyyz - (  chiz        /chin1 - gyz * gxxy )*HALF
-  Gamzyz = Gamzyz - (         chiy /chin1 - gyz * gxxz )*HALF
+  do k=1,ex(3)
+  do j=1,ex(2)
+  do i=1,ex(1)
+    chin_loc = chin1(i,j,k)
+    gxxx(i,j,k) = (gupxx(i,j,k) * chix(i,j,k) + gupxy(i,j,k) * chiy(i,j,k) + gupxz(i,j,k) * chiz(i,j,k)) / chin_loc
+    gxxy(i,j,k) = (gupxy(i,j,k) * chix(i,j,k) + gupyy(i,j,k) * chiy(i,j,k) + gupyz(i,j,k) * chiz(i,j,k)) / chin_loc
+    gxxz(i,j,k) = (gupxz(i,j,k) * chix(i,j,k) + gupyz(i,j,k) * chiy(i,j,k) + gupzz(i,j,k) * chiz(i,j,k)) / chin_loc
 
-  fxx = fxx - Gamxxx*Lapx - Gamyxx*Lapy - Gamzxx*Lapz
-  fyy = fyy - Gamxyy*Lapx - Gamyyy*Lapy - Gamzyy*Lapz
-  fzz = fzz - Gamxzz*Lapx - Gamyzz*Lapy - Gamzzz*Lapz
-  fxy = fxy - Gamxxy*Lapx - Gamyxy*Lapy - Gamzxy*Lapz
-  fxz = fxz - Gamxxz*Lapx - Gamyxz*Lapy - Gamzxz*Lapz
-  fyz = fyz - Gamxyz*Lapx - Gamyyz*Lapy - Gamzyz*Lapz
+    Gamxxx(i,j,k) = Gamxxx(i,j,k) - ( (chix(i,j,k) + chix(i,j,k))/chin_loc - gxx(i,j,k) * gxxx(i,j,k) )*HALF
+    Gamyxx(i,j,k) = Gamyxx(i,j,k) - (                                   - gxx(i,j,k) * gxxy(i,j,k) )*HALF
+    Gamzxx(i,j,k) = Gamzxx(i,j,k) - (                                   - gxx(i,j,k) * gxxz(i,j,k) )*HALF
+    Gamxyy(i,j,k) = Gamxyy(i,j,k) - (                                   - gyy(i,j,k) * gxxx(i,j,k) )*HALF
+    Gamyyy(i,j,k) = Gamyyy(i,j,k) - ( (chiy(i,j,k) + chiy(i,j,k))/chin_loc - gyy(i,j,k) * gxxy(i,j,k) )*HALF
+    Gamzyy(i,j,k) = Gamzyy(i,j,k) - (                                   - gyy(i,j,k) * gxxz(i,j,k) )*HALF
+    Gamxzz(i,j,k) = Gamxzz(i,j,k) - (                                   - gzz(i,j,k) * gxxx(i,j,k) )*HALF
+    Gamyzz(i,j,k) = Gamyzz(i,j,k) - (                                   - gzz(i,j,k) * gxxy(i,j,k) )*HALF
+    Gamzzz(i,j,k) = Gamzzz(i,j,k) - ( (chiz(i,j,k) + chiz(i,j,k))/chin_loc - gzz(i,j,k) * gxxz(i,j,k) )*HALF
+    Gamxxy(i,j,k) = Gamxxy(i,j,k) - ( chiy(i,j,k) /chin_loc - gxy(i,j,k) * gxxx(i,j,k) )*HALF
+    Gamyxy(i,j,k) = Gamyxy(i,j,k) - ( chix(i,j,k) /chin_loc - gxy(i,j,k) * gxxy(i,j,k) )*HALF
+    Gamzxy(i,j,k) = Gamzxy(i,j,k) - (                     - gxy(i,j,k) * gxxz(i,j,k) )*HALF
+    Gamxxz(i,j,k) = Gamxxz(i,j,k) - ( chiz(i,j,k) /chin_loc - gxz(i,j,k) * gxxx(i,j,k) )*HALF
+    Gamyxz(i,j,k) = Gamyxz(i,j,k) - (                     - gxz(i,j,k) * gxxy(i,j,k) )*HALF
+    Gamzxz(i,j,k) = Gamzxz(i,j,k) - ( chix(i,j,k) /chin_loc - gxz(i,j,k) * gxxz(i,j,k) )*HALF
+    Gamxyz(i,j,k) = Gamxyz(i,j,k) - (                     - gyz(i,j,k) * gxxx(i,j,k) )*HALF
+    Gamyyz(i,j,k) = Gamyyz(i,j,k) - ( chiz(i,j,k) /chin_loc - gyz(i,j,k) * gxxy(i,j,k) )*HALF
+    Gamzyz(i,j,k) = Gamzyz(i,j,k) - ( chiy(i,j,k) /chin_loc - gyz(i,j,k) * gxxz(i,j,k) )*HALF
 
-! store D^i D_i Lap in trK_rhs upto chi
-  trK_rhs =    gupxx * fxx + gupyy * fyy + gupzz * fzz + &
-        TWO* ( gupxy * fxy + gupxz * fxz + gupyz * fyz )
-#if 1        
-!! follow bam code
-  S =  chin1 * ( gupxx * Sxx + gupyy * Syy + gupzz * Szz + &
-     TWO * ( gupxy * Sxy + gupxz * Sxz + gupyz * Syz ) )
-  f = F2o3 * trK * trK -(&
-       gupxx * ( &
-       gupxx * Axx * Axx + gupyy * Axy * Axy + gupzz * Axz * Axz + &
-       TWO * (gupxy * Axx * Axy + gupxz * Axx * Axz + gupyz * Axy * Axz) ) + &
-       gupyy * ( &
-       gupxx * Axy * Axy + gupyy * Ayy * Ayy + gupzz * Ayz * Ayz + &
-       TWO * (gupxy * Axy * Ayy + gupxz * Axy * Ayz + gupyz * Ayy * Ayz) ) + &
-       gupzz * ( &
-       gupxx * Axz * Axz + gupyy * Ayz * Ayz + gupzz * Azz * Azz + &
-       TWO * (gupxy * Axz * Ayz + gupxz * Axz * Azz + gupyz * Ayz * Azz) ) + &
-       TWO * ( &
-       gupxy * ( &
-       gupxx * Axx * Axy + gupyy * Axy * Ayy + gupzz * Axz * Ayz + &
-       gupxy * (Axx * Ayy + Axy * Axy) + &
-       gupxz * (Axx * Ayz + Axz * Axy) + &
-       gupyz * (Axy * Ayz + Axz * Ayy) ) + &
-       gupxz * ( &
-       gupxx * Axx * Axz + gupyy * Axy * Ayz + gupzz * Axz * Azz + &
-       gupxy * (Axx * Ayz + Axy * Axz) + &
-       gupxz * (Axx * Azz + Axz * Axz) + &
-       gupyz * (Axy * Azz + Axz * Ayz) ) + &
-       gupyz * ( &
-       gupxx * Axy * Axz + gupyy * Ayy * Ayz + gupzz * Ayz * Azz + &
-       gupxy * (Axy * Ayz + Ayy * Axz) + &
-       gupxz * (Axy * Azz + Ayz * Axz) + &
-       gupyz * (Ayy * Azz + Ayz * Ayz) ) )) -1.6d1*PI*rho + EIGHT * PI * S
-  f = - F1o3 *(  gupxx * fxx + gupyy * fyy + gupzz * fzz + &
-        TWO* ( gupxy * fxy + gupxz * fxz + gupyz * fyz ) + alpn1/chin1*f)
+    fxx(i,j,k) = fxx(i,j,k) - Gamxxx(i,j,k)*Lapx(i,j,k) - Gamyxx(i,j,k)*Lapy(i,j,k) - Gamzxx(i,j,k)*Lapz(i,j,k)
+    fyy(i,j,k) = fyy(i,j,k) - Gamxyy(i,j,k)*Lapx(i,j,k) - Gamyyy(i,j,k)*Lapy(i,j,k) - Gamzyy(i,j,k)*Lapz(i,j,k)
+    fzz(i,j,k) = fzz(i,j,k) - Gamxzz(i,j,k)*Lapx(i,j,k) - Gamyzz(i,j,k)*Lapy(i,j,k) - Gamzzz(i,j,k)*Lapz(i,j,k)
+    fxy(i,j,k) = fxy(i,j,k) - Gamxxy(i,j,k)*Lapx(i,j,k) - Gamyxy(i,j,k)*Lapy(i,j,k) - Gamzxy(i,j,k)*Lapz(i,j,k)
+    fxz(i,j,k) = fxz(i,j,k) - Gamxxz(i,j,k)*Lapx(i,j,k) - Gamyxz(i,j,k)*Lapy(i,j,k) - Gamzxz(i,j,k)*Lapz(i,j,k)
+    fyz(i,j,k) = fyz(i,j,k) - Gamxyz(i,j,k)*Lapx(i,j,k) - Gamyyz(i,j,k)*Lapy(i,j,k) - Gamzyz(i,j,k)*Lapz(i,j,k)
 
-  fxx = alpn1 * (Rxx - EIGHT * PI * Sxx) - fxx
-  fxy = alpn1 * (Rxy - EIGHT * PI * Sxy) - fxy
-  fxz = alpn1 * (Rxz - EIGHT * PI * Sxz) - fxz
-  fyy = alpn1 * (Ryy - EIGHT * PI * Syy) - fyy
-  fyz = alpn1 * (Ryz - EIGHT * PI * Syz) - fyz
-  fzz = alpn1 * (Rzz - EIGHT * PI * Szz) - fzz
-#else        
-! Add lapse and S_ij parts to Ricci tensor:
+    trK_rhs(i,j,k) = gupxx(i,j,k) * fxx(i,j,k) + gupyy(i,j,k) * fyy(i,j,k) + gupzz(i,j,k) * fzz(i,j,k) + &
+                     TWO * (gupxy(i,j,k) * fxy(i,j,k) + gupxz(i,j,k) * fxz(i,j,k) + gupyz(i,j,k) * fyz(i,j,k))
+  enddo
+  enddo
+  enddo
+  do k=1,ex(3)
+  do j=1,ex(2)
+  do i=1,ex(1)
+    divb_loc = div_beta(i,j,k)
+    chin_loc = chin1(i,j,k)
 
-  fxx = alpn1 * (Rxx - EIGHT * PI * Sxx) - fxx
-  fxy = alpn1 * (Rxy - EIGHT * PI * Sxy) - fxy
-  fxz = alpn1 * (Rxz - EIGHT * PI * Sxz) - fxz
-  fyy = alpn1 * (Ryy - EIGHT * PI * Syy) - fyy
-  fyz = alpn1 * (Ryz - EIGHT * PI * Syz) - fyz
-  fzz = alpn1 * (Rzz - EIGHT * PI * Szz) - fzz
+    S_loc = chin_loc * ( gupxx(i,j,k) * Sxx(i,j,k) + gupyy(i,j,k) * Syy(i,j,k) + gupzz(i,j,k) * Szz(i,j,k) + &
+           TWO * (gupxy(i,j,k) * Sxy(i,j,k) + gupxz(i,j,k) * Sxz(i,j,k) + gupyz(i,j,k) * Syz(i,j,k)) )
+    S(i,j,k) = S_loc
 
-! Compute trace-free part (note: chi^-1 and chi cancel!):
+    f_loc = F2o3 * trK(i,j,k) * trK(i,j,k) - ( &
+            gupxx(i,j,k) * ( gupxx(i,j,k) * Axx(i,j,k) * Axx(i,j,k) + gupyy(i,j,k) * Axy(i,j,k) * Axy(i,j,k) + &
+                             gupzz(i,j,k) * Axz(i,j,k) * Axz(i,j,k) + &
+                             TWO * (gupxy(i,j,k) * Axx(i,j,k) * Axy(i,j,k) + gupxz(i,j,k) * Axx(i,j,k) * Axz(i,j,k) + &
+                                    gupyz(i,j,k) * Axy(i,j,k) * Axz(i,j,k)) ) + &
+            gupyy(i,j,k) * ( gupxx(i,j,k) * Axy(i,j,k) * Axy(i,j,k) + gupyy(i,j,k) * Ayy(i,j,k) * Ayy(i,j,k) + &
+                             gupzz(i,j,k) * Ayz(i,j,k) * Ayz(i,j,k) + &
+                             TWO * (gupxy(i,j,k) * Axy(i,j,k) * Ayy(i,j,k) + gupxz(i,j,k) * Axy(i,j,k) * Ayz(i,j,k) + &
+                                    gupyz(i,j,k) * Ayy(i,j,k) * Ayz(i,j,k)) ) + &
+            gupzz(i,j,k) * ( gupxx(i,j,k) * Axz(i,j,k) * Axz(i,j,k) + gupyy(i,j,k) * Ayz(i,j,k) * Ayz(i,j,k) + &
+                             gupzz(i,j,k) * Azz(i,j,k) * Azz(i,j,k) + &
+                             TWO * (gupxy(i,j,k) * Axz(i,j,k) * Ayz(i,j,k) + gupxz(i,j,k) * Axz(i,j,k) * Azz(i,j,k) + &
+                                    gupyz(i,j,k) * Ayz(i,j,k) * Azz(i,j,k)) ) + &
+            TWO * ( gupxy(i,j,k) * ( gupxx(i,j,k) * Axx(i,j,k) * Axy(i,j,k) + gupyy(i,j,k) * Axy(i,j,k) * Ayy(i,j,k) + &
+                                     gupzz(i,j,k) * Axz(i,j,k) * Ayz(i,j,k) + &
+                                     gupxy(i,j,k) * (Axx(i,j,k) * Ayy(i,j,k) + Axy(i,j,k) * Axy(i,j,k)) + &
+                                     gupxz(i,j,k) * (Axx(i,j,k) * Ayz(i,j,k) + Axz(i,j,k) * Axy(i,j,k)) + &
+                                     gupyz(i,j,k) * (Axy(i,j,k) * Ayz(i,j,k) + Axz(i,j,k) * Ayy(i,j,k)) ) + &
+                    gupxz(i,j,k) * ( gupxx(i,j,k) * Axx(i,j,k) * Axz(i,j,k) + gupyy(i,j,k) * Axy(i,j,k) * Ayz(i,j,k) + &
+                                     gupzz(i,j,k) * Axz(i,j,k) * Azz(i,j,k) + &
+                                     gupxy(i,j,k) * (Axx(i,j,k) * Ayz(i,j,k) + Axy(i,j,k) * Axz(i,j,k)) + &
+                                     gupxz(i,j,k) * (Axx(i,j,k) * Azz(i,j,k) + Axz(i,j,k) * Axz(i,j,k)) + &
+                                     gupyz(i,j,k) * (Axy(i,j,k) * Azz(i,j,k) + Axz(i,j,k) * Ayz(i,j,k)) ) + &
+                    gupyz(i,j,k) * ( gupxx(i,j,k) * Axy(i,j,k) * Axz(i,j,k) + gupyy(i,j,k) * Ayy(i,j,k) * Ayz(i,j,k) + &
+                                     gupzz(i,j,k) * Ayz(i,j,k) * Azz(i,j,k) + &
+                                     gupxy(i,j,k) * (Axy(i,j,k) * Ayz(i,j,k) + Ayy(i,j,k) * Axz(i,j,k)) + &
+                                     gupxz(i,j,k) * (Axy(i,j,k) * Azz(i,j,k) + Ayz(i,j,k) * Axz(i,j,k)) + &
+                                     gupyz(i,j,k) * (Ayy(i,j,k) * Azz(i,j,k) + Ayz(i,j,k) * Ayz(i,j,k)) ) ) ) - &
+            F16 * PI * rho(i,j,k) + EIGHT * PI * S_loc
 
-  f = F1o3 *(  gupxx * fxx + gupyy * fyy + gupzz * fzz + &
-        TWO* ( gupxy * fxy + gupxz * fxz + gupyz * fyz ) )
-#endif
+    f_loc = -F1o3 * ( gupxx(i,j,k) * fxx(i,j,k) + gupyy(i,j,k) * fyy(i,j,k) + gupzz(i,j,k) * fzz(i,j,k) + &
+            TWO * (gupxy(i,j,k) * fxy(i,j,k) + gupxz(i,j,k) * fxz(i,j,k) + gupyz(i,j,k) * fyz(i,j,k)) + &
+            alpn1(i,j,k)/chin_loc * f_loc )
+    f(i,j,k) = f_loc
 
-  Axx_rhs = fxx - gxx * f
-  Ayy_rhs = fyy - gyy * f
-  Azz_rhs = fzz - gzz * f
-  Axy_rhs = fxy - gxy * f
-  Axz_rhs = fxz - gxz * f
-  Ayz_rhs = fyz - gyz * f
+    l_fxx = alpn1(i,j,k) * (Rxx(i,j,k) - EIGHT * PI * Sxx(i,j,k)) - fxx(i,j,k)
+    l_fxy = alpn1(i,j,k) * (Rxy(i,j,k) - EIGHT * PI * Sxy(i,j,k)) - fxy(i,j,k)
+    l_fxz = alpn1(i,j,k) * (Rxz(i,j,k) - EIGHT * PI * Sxz(i,j,k)) - fxz(i,j,k)
+    l_fyy = alpn1(i,j,k) * (Ryy(i,j,k) - EIGHT * PI * Syy(i,j,k)) - fyy(i,j,k)
+    l_fyz = alpn1(i,j,k) * (Ryz(i,j,k) - EIGHT * PI * Syz(i,j,k)) - fyz(i,j,k)
+    l_fzz = alpn1(i,j,k) * (Rzz(i,j,k) - EIGHT * PI * Szz(i,j,k)) - fzz(i,j,k)
 
-! Now: store A_il A^l_j into fij:
+    Axx_rhs(i,j,k) = l_fxx - gxx(i,j,k) * f_loc
+    Ayy_rhs(i,j,k) = l_fyy - gyy(i,j,k) * f_loc
+    Azz_rhs(i,j,k) = l_fzz - gzz(i,j,k) * f_loc
+    Axy_rhs(i,j,k) = l_fxy - gxy(i,j,k) * f_loc
+    Axz_rhs(i,j,k) = l_fxz - gxz(i,j,k) * f_loc
+    Ayz_rhs(i,j,k) = l_fyz - gyz(i,j,k) * f_loc
 
-  fxx =       gupxx * Axx * Axx + gupyy * Axy * Axy + gupzz * Axz * Axz + &
-       TWO * (gupxy * Axx * Axy + gupxz * Axx * Axz + gupyz * Axy * Axz)
-  fyy =       gupxx * Axy * Axy + gupyy * Ayy * Ayy + gupzz * Ayz * Ayz + &
-       TWO * (gupxy * Axy * Ayy + gupxz * Axy * Ayz + gupyz * Ayy * Ayz)
-  fzz =       gupxx * Axz * Axz + gupyy * Ayz * Ayz + gupzz * Azz * Azz + &
-       TWO * (gupxy * Axz * Ayz + gupxz * Axz * Azz + gupyz * Ayz * Azz)
-  fxy =       gupxx * Axx * Axy + gupyy * Axy * Ayy + gupzz * Axz * Ayz + &
-              gupxy *(Axx * Ayy + Axy * Axy)                            + &
-              gupxz *(Axx * Ayz + Axz * Axy)                            + &
-              gupyz *(Axy * Ayz + Axz * Ayy)
-  fxz =       gupxx * Axx * Axz + gupyy * Axy * Ayz + gupzz * Axz * Azz + &
-              gupxy *(Axx * Ayz + Axy * Axz)                            + &
-              gupxz *(Axx * Azz + Axz * Axz)                            + &
-              gupyz *(Axy * Azz + Axz * Ayz)
-  fyz =       gupxx * Axy * Axz + gupyy * Ayy * Ayz + gupzz * Ayz * Azz + &
-              gupxy *(Axy * Ayz + Ayy * Axz)                            + &
-              gupxz *(Axy * Azz + Ayz * Axz)                            + &
-              gupyz *(Ayy * Azz + Ayz * Ayz)
+    fxx(i,j,k) = gupxx(i,j,k) * Axx(i,j,k) * Axx(i,j,k) + gupyy(i,j,k) * Axy(i,j,k) * Axy(i,j,k) + &
+                 gupzz(i,j,k) * Axz(i,j,k) * Axz(i,j,k) + TWO * (gupxy(i,j,k) * Axx(i,j,k) * Axy(i,j,k) + &
+                 gupxz(i,j,k) * Axx(i,j,k) * Axz(i,j,k) + gupyz(i,j,k) * Axy(i,j,k) * Axz(i,j,k))
+    fyy(i,j,k) = gupxx(i,j,k) * Axy(i,j,k) * Axy(i,j,k) + gupyy(i,j,k) * Ayy(i,j,k) * Ayy(i,j,k) + &
+                 gupzz(i,j,k) * Ayz(i,j,k) * Ayz(i,j,k) + TWO * (gupxy(i,j,k) * Axy(i,j,k) * Ayy(i,j,k) + &
+                 gupxz(i,j,k) * Axy(i,j,k) * Ayz(i,j,k) + gupyz(i,j,k) * Ayy(i,j,k) * Ayz(i,j,k))
+    fzz(i,j,k) = gupxx(i,j,k) * Axz(i,j,k) * Axz(i,j,k) + gupyy(i,j,k) * Ayz(i,j,k) * Ayz(i,j,k) + &
+                 gupzz(i,j,k) * Azz(i,j,k) * Azz(i,j,k) + TWO * (gupxy(i,j,k) * Axz(i,j,k) * Ayz(i,j,k) + &
+                 gupxz(i,j,k) * Axz(i,j,k) * Azz(i,j,k) + gupyz(i,j,k) * Ayz(i,j,k) * Azz(i,j,k))
+    fxy(i,j,k) = gupxx(i,j,k) * Axx(i,j,k) * Axy(i,j,k) + gupyy(i,j,k) * Axy(i,j,k) * Ayy(i,j,k) + &
+                 gupzz(i,j,k) * Axz(i,j,k) * Ayz(i,j,k) + gupxy(i,j,k) * (Axx(i,j,k) * Ayy(i,j,k) + Axy(i,j,k) * Axy(i,j,k)) + &
+                 gupxz(i,j,k) * (Axx(i,j,k) * Ayz(i,j,k) + Axz(i,j,k) * Axy(i,j,k)) + &
+                 gupyz(i,j,k) * (Axy(i,j,k) * Ayz(i,j,k) + Axz(i,j,k) * Ayy(i,j,k))
+    fxz(i,j,k) = gupxx(i,j,k) * Axx(i,j,k) * Axz(i,j,k) + gupyy(i,j,k) * Axy(i,j,k) * Ayz(i,j,k) + &
+                 gupzz(i,j,k) * Axz(i,j,k) * Azz(i,j,k) + gupxy(i,j,k) * (Axx(i,j,k) * Ayz(i,j,k) + Axy(i,j,k) * Axz(i,j,k)) + &
+                 gupxz(i,j,k) * (Axx(i,j,k) * Azz(i,j,k) + Axz(i,j,k) * Axz(i,j,k)) + &
+                 gupyz(i,j,k) * (Axy(i,j,k) * Azz(i,j,k) + Axz(i,j,k) * Ayz(i,j,k))
+    fyz(i,j,k) = gupxx(i,j,k) * Axy(i,j,k) * Axz(i,j,k) + gupyy(i,j,k) * Ayy(i,j,k) * Ayz(i,j,k) + &
+                 gupzz(i,j,k) * Ayz(i,j,k) * Azz(i,j,k) + gupxy(i,j,k) * (Axy(i,j,k) * Ayz(i,j,k) + Ayy(i,j,k) * Axz(i,j,k)) + &
+                 gupxz(i,j,k) * (Axy(i,j,k) * Azz(i,j,k) + Ayz(i,j,k) * Axz(i,j,k)) + &
+                 gupyz(i,j,k) * (Ayy(i,j,k) * Azz(i,j,k) + Ayz(i,j,k) * Ayz(i,j,k))
 
-  f = chin1
-! store D^i D_i Lap in trK_rhs
-  trK_rhs = f*trK_rhs
+    trK_rhs(i,j,k) = chin_loc * trK_rhs(i,j,k)
 
-  Axx_rhs =           f * Axx_rhs+ alpn1 * (trK * Axx - TWO * fxx)  + &
-           TWO * (  Axx * betaxx +   Axy * betayx +   Axz * betazx )- &
-             F2o3 * Axx * div_beta
+    Axx_rhs(i,j,k) = chin_loc * Axx_rhs(i,j,k) + alpn1(i,j,k) * (trK(i,j,k) * Axx(i,j,k) - TWO * fxx(i,j,k)) + &
+                     TWO * (Axx(i,j,k) * betaxx(i,j,k) + Axy(i,j,k) * betayx(i,j,k) + Axz(i,j,k) * betazx(i,j,k)) - &
+                     F2o3 * Axx(i,j,k) * divb_loc
+    Ayy_rhs(i,j,k) = chin_loc * Ayy_rhs(i,j,k) + alpn1(i,j,k) * (trK(i,j,k) * Ayy(i,j,k) - TWO * fyy(i,j,k)) + &
+                     TWO * (Axy(i,j,k) * betaxy(i,j,k) + Ayy(i,j,k) * betayy(i,j,k) + Ayz(i,j,k) * betazy(i,j,k)) - &
+                     F2o3 * Ayy(i,j,k) * divb_loc
+    Azz_rhs(i,j,k) = chin_loc * Azz_rhs(i,j,k) + alpn1(i,j,k) * (trK(i,j,k) * Azz(i,j,k) - TWO * fzz(i,j,k)) + &
+                     TWO * (Axz(i,j,k) * betaxz(i,j,k) + Ayz(i,j,k) * betayz(i,j,k) + Azz(i,j,k) * betazz(i,j,k)) - &
+                     F2o3 * Azz(i,j,k) * divb_loc
+    Axy_rhs(i,j,k) = chin_loc * Axy_rhs(i,j,k) + alpn1(i,j,k) * (trK(i,j,k) * Axy(i,j,k) - TWO * fxy(i,j,k)) + &
+                     Axx(i,j,k) * betaxy(i,j,k) + Axz(i,j,k) * betazy(i,j,k) + Ayy(i,j,k) * betayx(i,j,k) + &
+                     Ayz(i,j,k) * betazx(i,j,k) + F1o3 * Axy(i,j,k) * divb_loc - Axy(i,j,k) * betazz(i,j,k)
+    Ayz_rhs(i,j,k) = chin_loc * Ayz_rhs(i,j,k) + alpn1(i,j,k) * (trK(i,j,k) * Ayz(i,j,k) - TWO * fyz(i,j,k)) + &
+                     Axy(i,j,k) * betaxz(i,j,k) + Ayy(i,j,k) * betayz(i,j,k) + Axz(i,j,k) * betaxy(i,j,k) + &
+                     Azz(i,j,k) * betazy(i,j,k) + F1o3 * Ayz(i,j,k) * divb_loc - Ayz(i,j,k) * betaxx(i,j,k)
+    Axz_rhs(i,j,k) = chin_loc * Axz_rhs(i,j,k) + alpn1(i,j,k) * (trK(i,j,k) * Axz(i,j,k) - TWO * fxz(i,j,k)) + &
+                     Axx(i,j,k) * betaxz(i,j,k) + Axy(i,j,k) * betayz(i,j,k) + Ayz(i,j,k) * betayx(i,j,k) + &
+                     Azz(i,j,k) * betazx(i,j,k) + F1o3 * Axz(i,j,k) * divb_loc - Axz(i,j,k) * betayy(i,j,k)
 
-  Ayy_rhs =           f * Ayy_rhs+ alpn1 * (trK * Ayy - TWO * fyy)  + &
-           TWO * (  Axy * betaxy +   Ayy * betayy +   Ayz * betazy )- &
-             F2o3 * Ayy * div_beta
-
-  Azz_rhs =           f * Azz_rhs+ alpn1 * (trK * Azz - TWO * fzz)  + &
-           TWO * (  Axz * betaxz +   Ayz * betayz +   Azz * betazz )- &
-             F2o3 * Azz * div_beta
-
-  Axy_rhs =           f * Axy_rhs+ alpn1 *( trK * Axy  - TWO * fxy )+ &
-                    Axx * betaxy                  +   Axz * betazy  + &
-                                     Ayy * betayx +   Ayz * betazx  + &
-             F1o3 * Axy * div_beta                -   Axy * betazz
-
-  Ayz_rhs =           f * Ayz_rhs+ alpn1 *( trK * Ayz  - TWO * fyz )+ &
-                    Axy * betaxz +   Ayy * betayz                   + &
-                    Axz * betaxy                  +   Azz * betazy  + &
-             F1o3 * Ayz * div_beta                -   Ayz * betaxx
- 
-  Axz_rhs =           f * Axz_rhs+ alpn1 *( trK * Axz  - TWO * fxz )+ &
-                    Axx * betaxz +   Axy * betayz                   + &
-                                     Ayz * betayx +   Azz * betazx  + &
-             F1o3 * Axz * div_beta                -   Axz * betayy      !rhs for Aij
-
-! Compute trace of S_ij
-
-  S =  f * ( gupxx * Sxx + gupyy * Syy + gupzz * Szz + &
-     TWO * ( gupxy * Sxy + gupxz * Sxz + gupyz * Syz ) )
-
-  trK_rhs = - trK_rhs + alpn1 *( F1o3 * trK * trK         + &
-                gupxx * fxx + gupyy * fyy + gupzz * fzz   + &
-        TWO * ( gupxy * fxy + gupxz * fxz + gupyz * fyz ) + &
-       FOUR * PI * ( rho + S ))                                !rhs for trK
+    trK_rhs(i,j,k) = - trK_rhs(i,j,k) + alpn1(i,j,k) * ( F1o3 * trK(i,j,k) * trK(i,j,k) + &
+                    gupxx(i,j,k) * fxx(i,j,k) + gupyy(i,j,k) * fyy(i,j,k) + gupzz(i,j,k) * fzz(i,j,k) + &
+                    TWO * (gupxy(i,j,k) * fxy(i,j,k) + gupxz(i,j,k) * fxz(i,j,k) + gupyz(i,j,k) * fyz(i,j,k)) + &
+                    FOUR * PI * (rho(i,j,k) + S_loc) )
+  enddo
+  enddo
+  enddo
   
 !!!! gauge variable part
 
@@ -948,15 +1000,15 @@
 !!!!!!!!!advection term + Kreiss-Oliger dissipation (merged for cache efficiency)
 ! lopsided_kodis shares the symmetry_bd buffer between advection and
 ! dissipation, eliminating redundant full-grid copies. For metric variables
-! gxx/gyy/gzz (=dxx/dyy/dzz+1): kodis stencil coefficients sum to zero,
+! gxx/gyy/gzz (=dxx/dyy/dzz+1): stencil coefficients sum to zero,
 ! so the constant offset has no effect on dissipation.
 
-  call lopsided_kodis(ex,X,Y,Z,gxx,gxx_rhs,betax,betay,betaz,Symmetry,SSS,eps)
+  call lopsided_kodis(ex,X,Y,Z,dxx,gxx_rhs,betax,betay,betaz,Symmetry,SSS,eps)
   call lopsided_kodis(ex,X,Y,Z,gxy,gxy_rhs,betax,betay,betaz,Symmetry,AAS,eps)
   call lopsided_kodis(ex,X,Y,Z,gxz,gxz_rhs,betax,betay,betaz,Symmetry,ASA,eps)
-  call lopsided_kodis(ex,X,Y,Z,gyy,gyy_rhs,betax,betay,betaz,Symmetry,SSS,eps)
+  call lopsided_kodis(ex,X,Y,Z,dyy,gyy_rhs,betax,betay,betaz,Symmetry,SSS,eps)
   call lopsided_kodis(ex,X,Y,Z,gyz,gyz_rhs,betax,betay,betaz,Symmetry,SAA,eps)
-  call lopsided_kodis(ex,X,Y,Z,gzz,gzz_rhs,betax,betay,betaz,Symmetry,SSS,eps)
+  call lopsided_kodis(ex,X,Y,Z,dzz,gzz_rhs,betax,betay,betaz,Symmetry,SSS,eps)
 
   call lopsided_kodis(ex,X,Y,Z,Axx,Axx_rhs,betax,betay,betaz,Symmetry,SSS,eps)
   call lopsided_kodis(ex,X,Y,Z,Axy,Axy_rhs,betax,betay,betaz,Symmetry,AAS,eps)

AMSS_NCKU_source/bssn_rhs.h

@@ -32,6 +32,19 @@
 #define f_compute_rhs_Z4c_ss compute_rhs_z4c_ss_
 #define f_compute_constraint_fr compute_constraint_fr_
 #endif
+
+#ifdef __cplusplus
+extern "C"
+{
+#endif
+        void f_bssn_rhs_kernel_timing_reset();
+        int f_bssn_rhs_kernel_timing_bucket_count();
+        const double *f_bssn_rhs_kernel_timing_local_seconds();
+        const char *f_bssn_rhs_kernel_timing_label(int);
+#ifdef __cplusplus
+}
+#endif
+
 extern "C"
 {
         int f_compute_rhs_bssn(int *, double &, double *, double *, double *,                                                      // ex,T,X,Y,Z

AMSS_NCKU_source/bssn_rhs_c.C

@@ -2,12 +2,88 @@
 #include "bssn_rhs.h"
 #include "share_func.h"
 #include "tool.h"
+#include <time.h>
 // 0-based i,j,k
 // #define IDX_F(i,j,k,nx,ny) ((i) + (j)*(nx) + (k)*(nx)*(ny))
 // ex(1)=nx, ex(2)=ny, ex(3)=nz
 
 // 用法：a[ IDX_F(i,j,k,nx,ny) ]
 
+#ifndef BSSN_KERNEL_FINE_TIMING
+#define BSSN_KERNEL_FINE_TIMING 0
+#endif
+
+#if BSSN_KERNEL_FINE_TIMING
+namespace rhs_kernel_timing
+{
+    enum Bucket
+    {
+        KB_SETUP_DERIVS = 0,
+        KB_GEOM_GAMMA,
+        KB_RICCI_METRIC,
+        KB_CHI_LAPSE,
+        KB_AIJ_TRK_GAUGE,
+        KB_KO_CONSTRAINT,
+        KB_COUNT
+    };
+
+    static double local_bucket_seconds[KB_COUNT];
+
+    static const char *bucket_labels[KB_COUNT] =
+    {
+        "setup_derivs",
+        "geom_gamma",
+        "ricci_metric",
+        "chi_lapse",
+        "aij_trk_gauge",
+        "ko_constraint"
+    };
+
+    static inline double now_seconds()
+    {
+        struct timespec ts;
+        clock_gettime(CLOCK_MONOTONIC, &ts);
+        return double(ts.tv_sec) + 1.0e-9 * double(ts.tv_nsec);
+    }
+}
+
+extern "C" void f_bssn_rhs_kernel_timing_reset()
+{
+    for (int i = 0; i < rhs_kernel_timing::KB_COUNT; ++i)
+        rhs_kernel_timing::local_bucket_seconds[i] = 0.0;
+}
+
+extern "C" int f_bssn_rhs_kernel_timing_bucket_count()
+{
+    return rhs_kernel_timing::KB_COUNT;
+}
+
+extern "C" const double *f_bssn_rhs_kernel_timing_local_seconds()
+{
+    return rhs_kernel_timing::local_bucket_seconds;
+}
+
+extern "C" const char *f_bssn_rhs_kernel_timing_label(int bucket_index)
+{
+    if (bucket_index < 0 || bucket_index >= rhs_kernel_timing::KB_COUNT)
+        return "unknown";
+    return rhs_kernel_timing::bucket_labels[bucket_index];
+}
+
+#define RHS_KERNEL_TIMER_DECL(var_name) const double var_name = rhs_kernel_timing::now_seconds()
+#define RHS_KERNEL_TIMER_ADD(bucket_name, var_name) \
+    rhs_kernel_timing::local_bucket_seconds[int(rhs_kernel_timing::bucket_name)] += \
+        rhs_kernel_timing::now_seconds() - (var_name)
+#else
+extern "C" void f_bssn_rhs_kernel_timing_reset() {}
+extern "C" int f_bssn_rhs_kernel_timing_bucket_count() { return 0; }
+extern "C" const double *f_bssn_rhs_kernel_timing_local_seconds() { return 0; }
+extern "C" const char *f_bssn_rhs_kernel_timing_label(int) { return "disabled"; }
+
+#define RHS_KERNEL_TIMER_DECL(var_name)
+#define RHS_KERNEL_TIMER_ADD(bucket_name, var_name)
+#endif
+
 // C function that calculates the right-hand side for BSSN equations
 int f_compute_rhs_bssn(int *ex, double &T, 
                        double *X, double *Y, double *Z,
@@ -102,6 +178,7 @@ int f_compute_rhs_bssn(int *ex, double &T,
     dY = Y[1] - Y[0];
     dZ = Z[1] - Z[0];
 
+        RHS_KERNEL_TIMER_DECL(timer_setup_derivs);
         // 1ms //
         for(int i=0;i<all;i+=1){
             alpn1[i] = Lap[i] + 1.0;
@@ -141,6 +218,8 @@ int f_compute_rhs_bssn(int *ex, double &T,
                         (dxx[i] + ONE) * betaxz[i] + gxy[i] * betayz[i] + gyz[i] * betayx[i] 
                         + (dzz[i] + ONE) * betazx[i] - gxz[i] * betayy[i];                  
         }
+        RHS_KERNEL_TIMER_ADD(KB_SETUP_DERIVS, timer_setup_derivs);
+        RHS_KERNEL_TIMER_DECL(timer_geom_gamma);
         // Fused: inverse metric + Gamma constraint + Christoffel (3 loops -> 1)
         for(int i=0;i<all;i+=1){
             double det = (dxx[i] + ONE) * (dyy[i] + ONE) * (dzz[i] + ONE) + gxy[i] * gyz[i] * gxz[i] + gxz[i] * gxy[i] * gyz[i] -
@@ -283,9 +362,6 @@ int f_compute_rhs_bssn(int *ex, double &T,
                 + ( gupxy[i]*gupyz[i] + gupyy[i]*gupxz[i] ) * Axy[i]
                 + ( gupxy[i]*gupzz[i] + gupyz[i]*gupxz[i] ) * Axz[i]
                 + ( gupyy[i]*gupzz[i] + gupyz[i]*gupyz[i] ) * Ayz[i];
-            Rxx[i] = axx; Ryy[i] = ayy; Rzz[i] = azz;
-            Rxy[i] = axy; Rxz[i] = axz; Ryz[i] = ayz;
-
             Gamx_rhs[i] = - TWO * ( Lapx[i]*axx + Lapy[i]*axy + Lapz[i]*axz ) +
                 TWO * alpn1[i] * (
                 -F3o2/chin1[i] * ( chix[i]*axx + chiy[i]*axy + chiz[i]*axz ) -
@@ -315,6 +391,8 @@ int f_compute_rhs_bssn(int *ex, double &T,
                             + TWO * ( Gamzxy[i]*axy + Gamzxz[i]*axz + Gamzyz[i]*ayz )
                         );
         }
+        RHS_KERNEL_TIMER_ADD(KB_GEOM_GAMMA, timer_geom_gamma);
+        RHS_KERNEL_TIMER_DECL(timer_ricci_metric);
         // 22.3ms //
         fdderivs(ex,betax,gxxx,gxyx,gxzx,gyyx,gyzx,gzzx,
         X,Y,Z,ANTI,SYM, SYM ,Symmetry,Lev);
@@ -332,7 +410,6 @@ int f_compute_rhs_bssn(int *ex, double &T,
             double lfxx = gxxx[i] + gxyy[i] + gxzz[i];
             double lfxy = gxyx[i] + gyyy[i] + gyzz[i];
             double lfxz = gxzx[i] + gyzy[i] + gzzz[i];
-            fxx[i] = lfxx; fxy[i] = lfxy; fxz[i] = lfxz;
 
             double gxa = gupxx[i]*Gamxxx[i] + gupyy[i]*Gamxyy[i] + gupzz[i]*Gamxzz[i]
                     + TWO * ( gupxy[i]*Gamxxy[i] + gupxz[i]*Gamxxz[i] + gupyz[i]*Gamxyz[i] );
@@ -686,69 +763,74 @@ int f_compute_rhs_bssn(int *ex, double &T,
                     + Gamxyz[i] * gzzx[i] + Gamyyz[i] * gzzy[i] + Gamzyz[i] * gzzz[i]
                 );
         }
+        RHS_KERNEL_TIMER_ADD(KB_RICCI_METRIC, timer_ricci_metric);
 
+        RHS_KERNEL_TIMER_DECL(timer_chi_lapse);
         // 22.3ms //
             fdderivs(ex,chi,fxx,fxy,fxz,fyy,fyz,fzz,X,Y,Z,SYM,SYM,SYM,Symmetry,Lev);
 
         // 7ms //
         for (int i=0;i<all;i+=1) {
-            fxx[i] = fxx[i] - Gamxxx[i] * chix[i] - Gamyxx[i] * chiy[i] - Gamzxx[i] * chiz[i];
-            fxy[i] = fxy[i] - Gamxxy[i] * chix[i] - Gamyxy[i] * chiy[i] - Gamzxy[i] * chiz[i];
-            fxz[i] = fxz[i] - Gamxxz[i] * chix[i] - Gamyxz[i] * chiy[i] - Gamzxz[i] * chiz[i];
-            fyy[i] = fyy[i] - Gamxyy[i] * chix[i] - Gamyyy[i] * chiy[i] - Gamzyy[i] * chiz[i];
-            fyz[i] = fyz[i] - Gamxyz[i] * chix[i] - Gamyyz[i] * chiy[i] - Gamzyz[i] * chiz[i];
-            fzz[i] = fzz[i] - Gamxzz[i] * chix[i] - Gamyzz[i] * chiy[i] - Gamzzz[i] * chiz[i];
-            f[i] =
-                gupxx[i] * (fxx[i] - (F3o2 / chin1[i]) * chix[i] * chix[i])
-                + gupyy[i] * (fyy[i] - (F3o2 / chin1[i]) * chiy[i] * chiy[i])
-                + gupzz[i] * (fzz[i] - (F3o2 / chin1[i]) * chiz[i] * chiz[i])
-                + TWO * gupxy[i] * (fxy[i] - (F3o2 / chin1[i]) * chix[i] * chiy[i])
-                + TWO * gupxz[i] * (fxz[i] - (F3o2 / chin1[i]) * chix[i] * chiz[i])
-                + TWO * gupyz[i] * (fyz[i] - (F3o2 / chin1[i]) * chiy[i] * chiz[i]);
-            Rxx[i] = Rxx[i] + ( fxx[i] - (chix[i] * chix[i]) / (chin1[i] * TWO) + (dxx[i] + ONE) * f[i] ) / (chin1[i] * TWO);
-            Ryy[i] = Ryy[i] + ( fyy[i] - (chiy[i] * chiy[i]) / (chin1[i] * TWO) + (dyy[i] + ONE) * f[i] ) / (chin1[i] * TWO);
-            Rzz[i] = Rzz[i] + ( fzz[i] - (chiz[i] * chiz[i]) / (chin1[i] * TWO) + (dzz[i] + ONE) * f[i] ) / (chin1[i] * TWO);
+            const double inv_chin1 = ONE / chin1[i];
+            const double half_inv_chin1 = HALF * inv_chin1;
+            const double scaled_inv = F3o2 * inv_chin1;
+            const double cxx = fxx[i] - Gamxxx[i] * chix[i] - Gamyxx[i] * chiy[i] - Gamzxx[i] * chiz[i];
+            const double cxy = fxy[i] - Gamxxy[i] * chix[i] - Gamyxy[i] * chiy[i] - Gamzxy[i] * chiz[i];
+            const double cxz = fxz[i] - Gamxxz[i] * chix[i] - Gamyxz[i] * chiy[i] - Gamzxz[i] * chiz[i];
+            const double cyy = fyy[i] - Gamxyy[i] * chix[i] - Gamyyy[i] * chiy[i] - Gamzyy[i] * chiz[i];
+            const double cyz = fyz[i] - Gamxyz[i] * chix[i] - Gamyyz[i] * chiy[i] - Gamzyz[i] * chiz[i];
+            const double czz = fzz[i] - Gamxzz[i] * chix[i] - Gamyzz[i] * chiy[i] - Gamzzz[i] * chiz[i];
+            const double ricci_chi =
+                gupxx[i] * (cxx - scaled_inv * chix[i] * chix[i])
+                + gupyy[i] * (cyy - scaled_inv * chiy[i] * chiy[i])
+                + gupzz[i] * (czz - scaled_inv * chiz[i] * chiz[i])
+                + TWO * gupxy[i] * (cxy - scaled_inv * chix[i] * chiy[i])
+                + TWO * gupxz[i] * (cxz - scaled_inv * chix[i] * chiz[i])
+                + TWO * gupyz[i] * (cyz - scaled_inv * chiy[i] * chiz[i]);
+            f[i] = ricci_chi;
+            Rxx[i] = Rxx[i] + ( cxx - half_inv_chin1 * chix[i] * chix[i] + (dxx[i] + ONE) * ricci_chi ) * half_inv_chin1;
+            Ryy[i] = Ryy[i] + ( cyy - half_inv_chin1 * chiy[i] * chiy[i] + (dyy[i] + ONE) * ricci_chi ) * half_inv_chin1;
+            Rzz[i] = Rzz[i] + ( czz - half_inv_chin1 * chiz[i] * chiz[i] + (dzz[i] + ONE) * ricci_chi ) * half_inv_chin1;
 
-            Rxy[i] = Rxy[i] + ( fxy[i] - (chix[i] * chiy[i]) / (chin1[i] * TWO) + gxy[i] * f[i] ) / (chin1[i] * TWO);
-            Rxz[i] = Rxz[i] + ( fxz[i] - (chix[i] * chiz[i]) / (chin1[i] * TWO) + gxz[i] * f[i] ) / (chin1[i] * TWO);
-            Ryz[i] = Ryz[i] + ( fyz[i] - (chiy[i] * chiz[i]) / (chin1[i] * TWO) + gyz[i] * f[i] ) / (chin1[i] * TWO);
+            Rxy[i] = Rxy[i] + ( cxy - half_inv_chin1 * chix[i] * chiy[i] + gxy[i] * ricci_chi ) * half_inv_chin1;
+            Rxz[i] = Rxz[i] + ( cxz - half_inv_chin1 * chix[i] * chiz[i] + gxz[i] * ricci_chi ) * half_inv_chin1;
+            Ryz[i] = Ryz[i] + ( cyz - half_inv_chin1 * chiy[i] * chiz[i] + gyz[i] * ricci_chi ) * half_inv_chin1;
         }
 
         // 24ms //
         fdderivs(ex,Lap,fxx,fxy,fxz,fyy,fyz,fzz,X,Y,Z,SYM,SYM,SYM,Symmetry,Lev);
-        fderivs(ex,chi,dtSfx_rhs,dtSfy_rhs,dtSfz_rhs,X,Y,Z,SYM,SYM,SYM,Symmetry,Lev);
 
         // 6ms //
         for (int i=0;i<all;i+=1) {
-            /* gxxx,gxxy,gxxz (这里是“升指标后的chi导数/chi”那类量，你沿用原变量名即可) */
-            gxxx[i] = (gupxx[i] * chix[i] + gupxy[i] * chiy[i] + gupxz[i] * chiz[i]) / chin1[i];
-            gxxy[i] = (gupxy[i] * chix[i] + gupyy[i] * chiy[i] + gupyz[i] * chiz[i]) / chin1[i];
-            gxxz[i] = (gupxz[i] * chix[i] + gupyz[i] * chiy[i] + gupzz[i] * chiz[i]) / chin1[i];
+            const double inv_chin1 = ONE / chin1[i];
+            const double gchi_x = (gupxx[i] * chix[i] + gupxy[i] * chiy[i] + gupxz[i] * chiz[i]) * inv_chin1;
+            const double gchi_y = (gupxy[i] * chix[i] + gupyy[i] * chiy[i] + gupyz[i] * chiz[i]) * inv_chin1;
+            const double gchi_z = (gupxz[i] * chix[i] + gupyz[i] * chiy[i] + gupzz[i] * chiz[i]) * inv_chin1;
 
             /* Christoffel 修正项 */
-            Gamxxx[i] = Gamxxx[i] - ( ((chix[i] + chix[i]) / chin1[i]) - (dxx[i] + ONE) * gxxx[i] ) * HALF;
-            Gamyxx[i] = Gamyxx[i] - ( 0.0           - (dxx[i] + ONE) * gxxy[i] ) * HALF;  /* 原式只有 -gxx*gxxy */
-            Gamzxx[i] = Gamzxx[i] - ( 0.0           - (dxx[i] + ONE) * gxxz[i] ) * HALF;
+            Gamxxx[i] = Gamxxx[i] - ( ((chix[i] + chix[i]) * inv_chin1) - (dxx[i] + ONE) * gchi_x ) * HALF;
+            Gamyxx[i] = Gamyxx[i] - ( 0.0           - (dxx[i] + ONE) * gchi_y ) * HALF;  /* 原式只有 -gxx*gxxy */
+            Gamzxx[i] = Gamzxx[i] - ( 0.0           - (dxx[i] + ONE) * gchi_z ) * HALF;
 
-            Gamxyy[i] = Gamxyy[i] - ( 0.0           - (dyy[i] + ONE) * gxxx[i] ) * HALF;
-            Gamyyy[i] = Gamyyy[i] - ( ((chiy[i] + chiy[i]) / chin1[i]) - (dyy[i] + ONE) * gxxy[i] ) * HALF;
-            Gamzyy[i] = Gamzyy[i] - ( 0.0           - (dyy[i] + ONE) * gxxz[i] ) * HALF;
+            Gamxyy[i] = Gamxyy[i] - ( 0.0           - (dyy[i] + ONE) * gchi_x ) * HALF;
+            Gamyyy[i] = Gamyyy[i] - ( ((chiy[i] + chiy[i]) * inv_chin1) - (dyy[i] + ONE) * gchi_y ) * HALF;
+            Gamzyy[i] = Gamzyy[i] - ( 0.0           - (dyy[i] + ONE) * gchi_z ) * HALF;
 
-            Gamxzz[i] = Gamxzz[i] - ( 0.0           - (dzz[i] + ONE) * gxxx[i] ) * HALF;
-            Gamyzz[i] = Gamyzz[i] - ( 0.0           - (dzz[i] + ONE) * gxxy[i] ) * HALF;
-            Gamzzz[i] = Gamzzz[i] - ( ((chiz[i] + chiz[i]) / chin1[i]) - (dzz[i] + ONE) * gxxz[i] ) * HALF;
+            Gamxzz[i] = Gamxzz[i] - ( 0.0           - (dzz[i] + ONE) * gchi_x ) * HALF;
+            Gamyzz[i] = Gamyzz[i] - ( 0.0           - (dzz[i] + ONE) * gchi_y ) * HALF;
+            Gamzzz[i] = Gamzzz[i] - ( ((chiz[i] + chiz[i]) * inv_chin1) - (dzz[i] + ONE) * gchi_z ) * HALF;
 
-            Gamxxy[i] = Gamxxy[i] - ( ( chiy[i] / chin1[i])      - gxy[i] * gxxx[i] ) * HALF;
-            Gamyxy[i] = Gamyxy[i] - ( ( chix[i] / chin1[i])      - gxy[i] * gxxy[i] ) * HALF;
-            Gamzxy[i] = Gamzxy[i] - ( 0.0           - gxy[i] * gxxz[i] ) * HALF;
+            Gamxxy[i] = Gamxxy[i] - ( ( chiy[i] * inv_chin1)      - gxy[i] * gchi_x ) * HALF;
+            Gamyxy[i] = Gamyxy[i] - ( ( chix[i] * inv_chin1)      - gxy[i] * gchi_y ) * HALF;
+            Gamzxy[i] = Gamzxy[i] - ( 0.0           - gxy[i] * gchi_z ) * HALF;
 
-            Gamxxz[i] = Gamxxz[i] - ( ( chiz[i] / chin1[i])      - gxz[i] * gxxx[i] ) * HALF;
-            Gamyxz[i] = Gamyxz[i] - ( 0.0           - gxz[i] * gxxy[i] ) * HALF;
-            Gamzxz[i] = Gamzxz[i] - ( ( chix[i] / chin1[i])      - gxz[i] * gxxz[i] ) * HALF;
+            Gamxxz[i] = Gamxxz[i] - ( ( chiz[i] * inv_chin1)      - gxz[i] * gchi_x ) * HALF;
+            Gamyxz[i] = Gamyxz[i] - ( 0.0           - gxz[i] * gchi_y ) * HALF;
+            Gamzxz[i] = Gamzxz[i] - ( ( chix[i] * inv_chin1)      - gxz[i] * gchi_z ) * HALF;
 
-            Gamxyz[i] = Gamxyz[i] - ( 0.0           - gyz[i] * gxxx[i] ) * HALF;
-            Gamyyz[i] = Gamyyz[i] - ( ( chiz[i] / chin1[i])      - gyz[i] * gxxy[i] ) * HALF;
-            Gamzyz[i] = Gamzyz[i] - ( ( chiy[i] / chin1[i])      - gyz[i] * gxxz[i] ) * HALF;
+            Gamxyz[i] = Gamxyz[i] - ( 0.0           - gyz[i] * gchi_x ) * HALF;
+            Gamyyz[i] = Gamyyz[i] - ( ( chiz[i] * inv_chin1)      - gyz[i] * gchi_y ) * HALF;
+            Gamzyz[i] = Gamzyz[i] - ( ( chiy[i] * inv_chin1)      - gyz[i] * gchi_z ) * HALF;
 
             /* fxx..fyz 修正：减去 Γ * ∂Lap */
             fxx[i] = fxx[i] - Gamxxx[i] * Lapx[i] - Gamyxx[i] * Lapy[i] - Gamzxx[i] * Lapz[i];
@@ -762,6 +844,8 @@ int f_compute_rhs_bssn(int *ex, double &T,
             trK_rhs[i] =  gupxx[i] * fxx[i] + gupyy[i] * fyy[i] + gupzz[i] * fzz[i]
                     +  TWO * ( gupxy[i] * fxy[i] + gupxz[i] * fxz[i] + gupyz[i] * fyz[i] );
         }
+        RHS_KERNEL_TIMER_ADD(KB_CHI_LAPSE, timer_chi_lapse);
+        RHS_KERNEL_TIMER_DECL(timer_aij_trk_gauge);
         // 2.5ms //
         for (int i=0;i<all;i+=1) {
             const double divb = betaxx[i] + betayy[i] + betazz[i];
@@ -1062,6 +1146,8 @@ int f_compute_rhs_bssn(int *ex, double &T,
             dtSfz_rhs[i] = Gamz_rhs[i] - reta[i] * dtSfz[i];
             #endif
         }
+        RHS_KERNEL_TIMER_ADD(KB_AIJ_TRK_GAUGE, timer_aij_trk_gauge);
+        RHS_KERNEL_TIMER_DECL(timer_ko_constraint);
         // advection + KO dissipation with shared symmetry buffer
         lopsided_kodis(ex,X,Y,Z,dxx,gxx_rhs,betax,betay,betaz,Symmetry,SSS,eps);
         lopsided_kodis(ex,X,Y,Z,Gamz,Gamz_rhs,betax,betay,betaz,Symmetry,SSA,eps);
@@ -1139,60 +1225,61 @@ int f_compute_rhs_bssn(int *ex, double &T,
             fderivs(ex,Ayy,gyyx,gyyy,gyyz,X,Y,Z,SYM ,SYM ,SYM ,Symmetry,0);
             fderivs(ex,Ayz,gyzx,gyzy,gyzz,X,Y,Z,SYM ,ANTI,ANTI,Symmetry,0);
             fderivs(ex,Azz,gzzx,gzzy,gzzz,X,Y,Z,SYM ,SYM ,SYM ,Symmetry,0);
-        }
-        // 7ms //
-        for (int i=0;i<all;i+=1) {
-            gxxx[i] = gxxx[i] - (  Gamxxx[i] * Axx[i] + Gamyxx[i] * Axy[i] + Gamzxx[i] * Axz[i]
-                                  + Gamxxx[i] * Axx[i] + Gamyxx[i] * Axy[i] + Gamzxx[i] * Axz[i]) - chix[i]*Axx[i]/chin1[i];
-            gxyx[i] = gxyx[i] - (  Gamxxy[i] * Axx[i] + Gamyxy[i] * Axy[i] + Gamzxy[i] * Axz[i]
-                                  + Gamxxx[i] * Axy[i] + Gamyxx[i] * Ayy[i] + Gamzxx[i] * Ayz[i]) - chix[i]*Axy[i]/chin1[i];
-            gxzx[i] = gxzx[i] - (  Gamxxz[i] * Axx[i] + Gamyxz[i] * Axy[i] + Gamzxz[i] * Axz[i]
-                                  + Gamxxx[i] * Axz[i] + Gamyxx[i] * Ayz[i] + Gamzxx[i] * Azz[i]) - chix[i]*Axz[i]/chin1[i];
-            gyyx[i] = gyyx[i] - (  Gamxxy[i] * Axy[i] + Gamyxy[i] * Ayy[i] + Gamzxy[i] * Ayz[i]
-                                  + Gamxxy[i] * Axy[i] + Gamyxy[i] * Ayy[i] + Gamzxy[i] * Ayz[i]) - chix[i]*Ayy[i]/chin1[i];
-            gyzx[i] = gyzx[i] - (  Gamxxz[i] * Axy[i] + Gamyxz[i] * Ayy[i] + Gamzxz[i] * Ayz[i]
-                                  + Gamxxy[i] * Axz[i] + Gamyxy[i] * Ayz[i] + Gamzxy[i] * Azz[i]) - chix[i]*Ayz[i]/chin1[i];
-            gzzx[i] = gzzx[i] - (  Gamxxz[i] * Axz[i] + Gamyxz[i] * Ayz[i] + Gamzxz[i] * Azz[i]
-                                  + Gamxxz[i] * Axz[i] + Gamyxz[i] * Ayz[i] + Gamzxz[i] * Azz[i]) - chix[i]*Azz[i]/chin1[i];
-            gxxy[i] = gxxy[i] - (  Gamxxy[i] * Axx[i] + Gamyxy[i] * Axy[i] + Gamzxy[i] * Axz[i]
-                                  + Gamxxy[i] * Axx[i] + Gamyxy[i] * Axy[i] + Gamzxy[i] * Axz[i]) - chiy[i]*Axx[i]/chin1[i];
-            gxyy[i] = gxyy[i] - (  Gamxyy[i] * Axx[i] + Gamyyy[i] * Axy[i] + Gamzyy[i] * Axz[i]
-                                  + Gamxxy[i] * Axy[i] + Gamyxy[i] * Ayy[i] + Gamzxy[i] * Ayz[i]) - chiy[i]*Axy[i]/chin1[i];
-            gxzy[i] = gxzy[i] - (  Gamxyz[i] * Axx[i] + Gamyyz[i] * Axy[i] + Gamzyz[i] * Axz[i]
-                                  + Gamxxy[i] * Axz[i] + Gamyxy[i] * Ayz[i] + Gamzxy[i] * Azz[i]) - chiy[i]*Axz[i]/chin1[i];
-            gyyy[i] = gyyy[i] - (  Gamxyy[i] * Axy[i] + Gamyyy[i] * Ayy[i] + Gamzyy[i] * Ayz[i]
-                                  + Gamxyy[i] * Axy[i] + Gamyyy[i] * Ayy[i] + Gamzyy[i] * Ayz[i]) - chiy[i]*Ayy[i]/chin1[i];
-            gyzy[i] = gyzy[i] - (  Gamxyz[i] * Axy[i] + Gamyyz[i] * Ayy[i] + Gamzyz[i] * Ayz[i]
-                                  + Gamxyy[i] * Axz[i] + Gamyyy[i] * Ayz[i] + Gamzyy[i] * Azz[i]) - chiy[i]*Ayz[i]/chin1[i];
-            gzzy[i] = gzzy[i] - (  Gamxyz[i] * Axz[i] + Gamyyz[i] * Ayz[i] + Gamzyz[i] * Azz[i]
-                                  + Gamxyz[i] * Axz[i] + Gamyyz[i] * Ayz[i] + Gamzyz[i] * Azz[i]) - chiy[i]*Azz[i]/chin1[i];
-            gxxz[i] = gxxz[i] - (  Gamxxz[i] * Axx[i] + Gamyxz[i] * Axy[i] + Gamzxz[i] * Axz[i]
-                                  + Gamxxz[i] * Axx[i] + Gamyxz[i] * Axy[i] + Gamzxz[i] * Axz[i]) - chiz[i]*Axx[i]/chin1[i];
-            gxyz[i] = gxyz[i] - (  Gamxyz[i] * Axx[i] + Gamyyz[i] * Axy[i] + Gamzyz[i] * Axz[i]
-                                  + Gamxxz[i] * Axy[i] + Gamyxz[i] * Ayy[i] + Gamzxz[i] * Ayz[i]) - chiz[i]*Axy[i]/chin1[i];
-            gxzz[i] = gxzz[i] - (  Gamxzz[i] * Axx[i] + Gamyzz[i] * Axy[i] + Gamzzz[i] * Axz[i]
-                                  + Gamxxz[i] * Axz[i] + Gamyxz[i] * Ayz[i] + Gamzxz[i] * Azz[i]) - chiz[i]*Axz[i]/chin1[i];
-            gyyz[i] = gyyz[i] - (  Gamxyz[i] * Axy[i] + Gamyyz[i] * Ayy[i] + Gamzyz[i] * Ayz[i]
-                                  + Gamxyz[i] * Axy[i] + Gamyyz[i] * Ayy[i] + Gamzyz[i] * Ayz[i]) - chiz[i]*Ayy[i]/chin1[i];
-            gyzz[i] = gyzz[i] - (  Gamxzz[i] * Axy[i] + Gamyzz[i] * Ayy[i] + Gamzzz[i] * Ayz[i]
-                                  + Gamxyz[i] * Axz[i] + Gamyyz[i] * Ayz[i] + Gamzyz[i] * Azz[i]) - chiz[i]*Ayz[i]/chin1[i];
-            gzzz[i] = gzzz[i] - (  Gamxzz[i] * Axz[i] + Gamyzz[i] * Ayz[i] + Gamzzz[i] * Azz[i]
-                                  + Gamxzz[i] * Axz[i] + Gamyzz[i] * Ayz[i] + Gamzzz[i] * Azz[i]) - chiz[i]*Azz[i]/chin1[i];
+            // 7ms //
+            for (int i=0;i<all;i+=1) {
+                gxxx[i] = gxxx[i] - (  Gamxxx[i] * Axx[i] + Gamyxx[i] * Axy[i] + Gamzxx[i] * Axz[i]
+                                      + Gamxxx[i] * Axx[i] + Gamyxx[i] * Axy[i] + Gamzxx[i] * Axz[i]) - chix[i]*Axx[i]/chin1[i];
+                gxyx[i] = gxyx[i] - (  Gamxxy[i] * Axx[i] + Gamyxy[i] * Axy[i] + Gamzxy[i] * Axz[i]
+                                      + Gamxxx[i] * Axy[i] + Gamyxx[i] * Ayy[i] + Gamzxx[i] * Ayz[i]) - chix[i]*Axy[i]/chin1[i];
+                gxzx[i] = gxzx[i] - (  Gamxxz[i] * Axx[i] + Gamyxz[i] * Axy[i] + Gamzxz[i] * Axz[i]
+                                      + Gamxxx[i] * Axz[i] + Gamyxx[i] * Ayz[i] + Gamzxx[i] * Azz[i]) - chix[i]*Axz[i]/chin1[i];
+                gyyx[i] = gyyx[i] - (  Gamxxy[i] * Axy[i] + Gamyxy[i] * Ayy[i] + Gamzxy[i] * Ayz[i]
+                                      + Gamxxy[i] * Axy[i] + Gamyxy[i] * Ayy[i] + Gamzxy[i] * Ayz[i]) - chix[i]*Ayy[i]/chin1[i];
+                gyzx[i] = gyzx[i] - (  Gamxxz[i] * Axy[i] + Gamyxz[i] * Ayy[i] + Gamzxz[i] * Ayz[i]
+                                      + Gamxxy[i] * Axz[i] + Gamyxy[i] * Ayz[i] + Gamzxy[i] * Azz[i]) - chix[i]*Ayz[i]/chin1[i];
+                gzzx[i] = gzzx[i] - (  Gamxxz[i] * Axz[i] + Gamyxz[i] * Ayz[i] + Gamzxz[i] * Azz[i]
+                                      + Gamxxz[i] * Axz[i] + Gamyxz[i] * Ayz[i] + Gamzxz[i] * Azz[i]) - chix[i]*Azz[i]/chin1[i];
+                gxxy[i] = gxxy[i] - (  Gamxxy[i] * Axx[i] + Gamyxy[i] * Axy[i] + Gamzxy[i] * Axz[i]
+                                      + Gamxxy[i] * Axx[i] + Gamyxy[i] * Axy[i] + Gamzxy[i] * Axz[i]) - chiy[i]*Axx[i]/chin1[i];
+                gxyy[i] = gxyy[i] - (  Gamxyy[i] * Axx[i] + Gamyyy[i] * Axy[i] + Gamzyy[i] * Axz[i]
+                                      + Gamxxy[i] * Axy[i] + Gamyxy[i] * Ayy[i] + Gamzxy[i] * Ayz[i]) - chiy[i]*Axy[i]/chin1[i];
+                gxzy[i] = gxzy[i] - (  Gamxyz[i] * Axx[i] + Gamyyz[i] * Axy[i] + Gamzyz[i] * Axz[i]
+                                      + Gamxxy[i] * Axz[i] + Gamyxy[i] * Ayz[i] + Gamzxy[i] * Azz[i]) - chiy[i]*Axz[i]/chin1[i];
+                gyyy[i] = gyyy[i] - (  Gamxyy[i] * Axy[i] + Gamyyy[i] * Ayy[i] + Gamzyy[i] * Ayz[i]
+                                      + Gamxyy[i] * Axy[i] + Gamyyy[i] * Ayy[i] + Gamzyy[i] * Ayz[i]) - chiy[i]*Ayy[i]/chin1[i];
+                gyzy[i] = gyzy[i] - (  Gamxyz[i] * Axy[i] + Gamyyz[i] * Ayy[i] + Gamzyz[i] * Ayz[i]
+                                      + Gamxyy[i] * Axz[i] + Gamyyy[i] * Ayz[i] + Gamzyy[i] * Azz[i]) - chiy[i]*Ayz[i]/chin1[i];
+                gzzy[i] = gzzy[i] - (  Gamxyz[i] * Axz[i] + Gamyyz[i] * Ayz[i] + Gamzyz[i] * Azz[i]
+                                      + Gamxyz[i] * Axz[i] + Gamyyz[i] * Ayz[i] + Gamzyz[i] * Azz[i]) - chiy[i]*Azz[i]/chin1[i];
+                gxxz[i] = gxxz[i] - (  Gamxxz[i] * Axx[i] + Gamyxz[i] * Axy[i] + Gamzxz[i] * Axz[i]
+                                      + Gamxxz[i] * Axx[i] + Gamyxz[i] * Axy[i] + Gamzxz[i] * Axz[i]) - chiz[i]*Axx[i]/chin1[i];
+                gxyz[i] = gxyz[i] - (  Gamxyz[i] * Axx[i] + Gamyyz[i] * Axy[i] + Gamzyz[i] * Axz[i]
+                                      + Gamxxz[i] * Axy[i] + Gamyxz[i] * Ayy[i] + Gamzxz[i] * Ayz[i]) - chiz[i]*Axy[i]/chin1[i];
+                gxzz[i] = gxzz[i] - (  Gamxzz[i] * Axx[i] + Gamyzz[i] * Axy[i] + Gamzzz[i] * Axz[i]
+                                      + Gamxxz[i] * Axz[i] + Gamyxz[i] * Ayz[i] + Gamzxz[i] * Azz[i]) - chiz[i]*Axz[i]/chin1[i];
+                gyyz[i] = gyyz[i] - (  Gamxyz[i] * Axy[i] + Gamyyz[i] * Ayy[i] + Gamzyz[i] * Ayz[i]
+                                      + Gamxyz[i] * Axy[i] + Gamyyz[i] * Ayy[i] + Gamzyz[i] * Ayz[i]) - chiz[i]*Ayy[i]/chin1[i];
+                gyzz[i] = gyzz[i] - (  Gamxzz[i] * Axy[i] + Gamyzz[i] * Ayy[i] + Gamzzz[i] * Ayz[i]
+                                      + Gamxyz[i] * Axz[i] + Gamyyz[i] * Ayz[i] + Gamzyz[i] * Azz[i]) - chiz[i]*Ayz[i]/chin1[i];
+                gzzz[i] = gzzz[i] - (  Gamxzz[i] * Axz[i] + Gamyzz[i] * Ayz[i] + Gamzzz[i] * Azz[i]
+                                      + Gamxzz[i] * Axz[i] + Gamyzz[i] * Ayz[i] + Gamzzz[i] * Azz[i]) - chiz[i]*Azz[i]/chin1[i];
 
-            movx_Res[i] = gupxx[i]*gxxx[i] + gupyy[i]*gxyy[i] + gupzz[i]*gxzz[i]
-                        + gupxy[i]*gxyx[i] + gupxz[i]*gxzx[i] + gupyz[i]*gxzy[i]
-                        + gupxy[i]*gxxy[i] + gupxz[i]*gxxz[i] + gupyz[i]*gxyz[i];
-            movy_Res[i] = gupxx[i]*gxyx[i] + gupyy[i]*gyyy[i] + gupzz[i]*gyzz[i]
-                        + gupxy[i]*gyyx[i] + gupxz[i]*gyzx[i] + gupyz[i]*gyzy[i]
-                        + gupxy[i]*gxyy[i] + gupxz[i]*gxyz[i] + gupyz[i]*gyyz[i];
-            movz_Res[i] = gupxx[i]*gxzx[i] + gupyy[i]*gyzy[i] + gupzz[i]*gzzz[i]
-                        + gupxy[i]*gyzx[i] + gupxz[i]*gzzx[i] + gupyz[i]*gzzy[i]
-                        + gupxy[i]*gxzy[i] + gupxz[i]*gxzz[i] + gupyz[i]*gyzz[i];
+                movx_Res[i] = gupxx[i]*gxxx[i] + gupyy[i]*gxyy[i] + gupzz[i]*gxzz[i]
+                            + gupxy[i]*gxyx[i] + gupxz[i]*gxzx[i] + gupyz[i]*gxzy[i]
+                            + gupxy[i]*gxxy[i] + gupxz[i]*gxxz[i] + gupyz[i]*gxyz[i];
+                movy_Res[i] = gupxx[i]*gxyx[i] + gupyy[i]*gyyy[i] + gupzz[i]*gyzz[i]
+                            + gupxy[i]*gyyx[i] + gupxz[i]*gyzx[i] + gupyz[i]*gyzy[i]
+                            + gupxy[i]*gxyy[i] + gupxz[i]*gxyz[i] + gupyz[i]*gyyz[i];
+                movz_Res[i] = gupxx[i]*gxzx[i] + gupyy[i]*gyzy[i] + gupzz[i]*gzzz[i]
+                            + gupxy[i]*gyzx[i] + gupxz[i]*gzzx[i] + gupyz[i]*gzzy[i]
+                            + gupxy[i]*gxzy[i] + gupxz[i]*gxzz[i] + gupyz[i]*gyzz[i];
 
-            movx_Res[i] = movx_Res[i] - F2o3*Kx[i] - F8*PI*Sx[i];
-            movy_Res[i] = movy_Res[i] - F2o3*Ky[i] - F8*PI*Sy[i];
-            movz_Res[i] = movz_Res[i] - F2o3*Kz[i] - F8*PI*Sz[i];
+                movx_Res[i] = movx_Res[i] - F2o3*Kx[i] - F8*PI*Sx[i];
+                movy_Res[i] = movy_Res[i] - F2o3*Ky[i] - F8*PI*Sy[i];
+                movz_Res[i] = movz_Res[i] - F2o3*Kz[i] - F8*PI*Sz[i];
+            }
         }
+        RHS_KERNEL_TIMER_ADD(KB_KO_CONSTRAINT, timer_ko_constraint);
 
 
             

AMSS_NCKU_source/fmisc.f90

@@ -1514,6 +1514,81 @@ f_out = f_out*dX*dY*dZ
   return
 
   end subroutine l2normhelper
+!--------------------------------------------------------------------------------------
+  subroutine l2normhelper7(ex, X, Y, Z,xmin,ymin,zmin,xmax,ymax,zmax,&
+                           f1,f2,f3,f4,f5,f6,f7,f_out,gw)
+
+  implicit none
+!~~~~~~> Input parameters:
+  integer,intent(in ):: ex(1:3)
+  real*8, intent(in ):: X(1:ex(1)),Y(1:ex(2)),Z(1:ex(3)),xmin,ymin,zmin,xmax,ymax,zmax
+  integer,intent(in)::gw
+  real*8, dimension(ex(1),ex(2),ex(3)),intent(in) :: f1,f2,f3,f4,f5,f6,f7
+  real*8, intent(out) :: f_out(7)
+!~~~~~~> Other variables:
+
+  real*8            :: dX, dY, dZ
+  integer::imin,jmin,kmin
+  integer::imax,jmax,kmax
+  integer::i,j,k
+  real*8 :: s1,s2,s3,s4,s5,s6,s7
+
+  dX = X(2) - X(1)
+  dY = Y(2) - Y(1)
+  dZ = Z(2) - Z(1)
+
+! for ghost zone
+   imin = gw+1
+   jmin = gw+1
+   kmin = gw+1
+
+   imax = ex(1) - gw
+   jmax = ex(2) - gw
+   kmax = ex(3) - gw
+
+!for patch boundary (i.e., not ghost boundary)
+
+if(dabs(X(ex(1))-xmax) < dX) imax = ex(1)
+if(dabs(Y(ex(2))-ymax) < dY) jmax = ex(2)
+if(dabs(Z(ex(3))-zmax) < dZ) kmax = ex(3)
+if(dabs(X(1)-xmin) < dX) imin = 1
+if(dabs(Y(1)-ymin) < dY) jmin = 1
+if(dabs(Z(1)-zmin) < dZ) kmin = 1
+
+  s1 = 0.d0
+  s2 = 0.d0
+  s3 = 0.d0
+  s4 = 0.d0
+  s5 = 0.d0
+  s6 = 0.d0
+  s7 = 0.d0
+
+  do k=kmin,kmax
+    do j=jmin,jmax
+!DIR$ SIMD REDUCTION(+:s1,s2,s3,s4,s5,s6,s7)
+      do i=imin,imax
+        s1 = s1 + f1(i,j,k)*f1(i,j,k)
+        s2 = s2 + f2(i,j,k)*f2(i,j,k)
+        s3 = s3 + f3(i,j,k)*f3(i,j,k)
+        s4 = s4 + f4(i,j,k)*f4(i,j,k)
+        s5 = s5 + f5(i,j,k)*f5(i,j,k)
+        s6 = s6 + f6(i,j,k)*f6(i,j,k)
+        s7 = s7 + f7(i,j,k)*f7(i,j,k)
+      enddo
+    enddo
+  enddo
+
+  f_out(1) = s1*dX*dY*dZ
+  f_out(2) = s2*dX*dY*dZ
+  f_out(3) = s3*dX*dY*dZ
+  f_out(4) = s4*dX*dY*dZ
+  f_out(5) = s5*dX*dY*dZ
+  f_out(6) = s6*dX*dY*dZ
+  f_out(7) = s7*dX*dY*dZ
+
+  return
+
+  end subroutine l2normhelper7
 !--------------------------------------------------------------------------------------
 ! calculate L2norm especially for shell Blocks
   subroutine l2normhelper_sh(ex, X, Y, Z,xmin,ymin,zmin,xmax,ymax,zmax,&

AMSS_NCKU_source/fmisc.h

@@ -13,6 +13,7 @@
 #define f_global_interpind2d global_interpind2d
 #define f_global_interpind1d global_interpind1d
 #define f_l2normhelper l2normhelper
+#define f_l2normhelper7 l2normhelper7
 #define f_l2normhelper_sh l2normhelper_sh
 #define f_l2normhelper_sh_rms l2normhelper_sh_rms
 #define f_average average
@@ -42,6 +43,7 @@
 #define f_global_interpind2d GLOBAL_INTERPIND2D
 #define f_global_interpind1d GLOBAL_INTERPIND1D
 #define f_l2normhelper L2NORMHELPER
+#define f_l2normhelper7 L2NORMHELPER7
 #define f_l2normhelper_sh L2NORMHELPER_SH
 #define f_l2normhelper_sh_rms L2NORMHELPER_SH_RMS
 #define f_average AVERAGE
@@ -71,6 +73,7 @@
 #define f_global_interpind2d global_interpind2d_
 #define f_global_interpind1d global_interpind1d_
 #define f_l2normhelper l2normhelper_
+#define f_l2normhelper7 l2normhelper7_
 #define f_l2normhelper_sh l2normhelper_sh_
 #define f_l2normhelper_sh_rms l2normhelper_sh_rms_
 #define f_average average_
@@ -164,6 +167,15 @@ extern "C"
 						double *, double &, int &);
 }
 
+extern "C"
+{
+	void f_l2normhelper7(int *, double *, double *, double *,
+						 double &, double &, double &,
+						 double &, double &, double &,
+						 double *, double *, double *, double *,
+						 double *, double *, double *, double *, int &);
+}
+
 extern "C"
 {
 	void f_l2normhelper_sh(int *, double *, double *, double *,

AMSS_NCKU_source/gaussj.C

@@ -17,65 +17,103 @@ using namespace std;
 #include <math.h>
 #endif
 
-// Intel oneMKL LAPACK interface
-#include <mkl_lapacke.h>
-/* Linear equation solution using Intel oneMKL LAPACK.
+/* Linear equation solution by Gauss-Jordan elimination.
 a[0..n-1][0..n-1] is the input matrix. b[0..n-1] is input
 containing the right-hand side vectors. On output a is
 replaced by its matrix inverse, and b is replaced by the
-corresponding set of solution vectors.
-
-Mathematical equivalence:
-  Solves: A * x = b  =>  x = A^(-1) * b
-  Original Gauss-Jordan and LAPACK dgesv/dgetri produce identical results
-  within numerical precision. */
+corresponding set of solution vectors. */
 
 int gaussj(double *a, double *b, int n)
 {
-  // Allocate pivot array and workspace
-  lapack_int *ipiv = new lapack_int[n];
-  lapack_int info;
+  double swap;
 
-  // Make a copy of matrix a for solving (dgesv modifies it to LU form)
-  double *a_copy = new double[n * n];
-  for (int i = 0; i < n * n; i++) {
-    a_copy[i] = a[i];
+  int *indxc, *indxr, *ipiv;
+  indxc = new int[n];
+  indxr = new int[n];
+  ipiv = new int[n];
+
+  int i, icol, irow, j, k, l, ll;
+  double big, dum, pivinv;
+
+  for (j = 0; j < n; j++)
+    ipiv[j] = 0;
+  for (i = 0; i < n; i++)
+  {
+    big = 0.0;
+    for (j = 0; j < n; j++)
+      if (ipiv[j] != 1)
+        for (k = 0; k < n; k++)
+        {
+          if (ipiv[k] == 0)
+          {
+            if (fabs(a[j * n + k]) >= big)
+            {
+              big = fabs(a[j * n + k]);
+              irow = j;
+              icol = k;
+            }
+          }
+          else if (ipiv[k] > 1)
+          {
+            cout << "gaussj: Singular Matrix-1" << endl;
+            return 1;
+          }
+        }
+
+    ipiv[icol] = ipiv[icol] + 1;
+    if (irow != icol)
+    {
+      for (l = 0; l < n; l++)
+      {
+        swap = a[irow * n + l];
+        a[irow * n + l] = a[icol * n + l];
+        a[icol * n + l] = swap;
+      }
+
+      swap = b[irow];
+      b[irow] = b[icol];
+      b[icol] = swap;
+    }
+
+    indxr[i] = irow;
+    indxc[i] = icol;
+
+    if (a[icol * n + icol] == 0.0)
+    {
+      cout << "gaussj: Singular Matrix-2" << endl;
+      return 1;
+    }
+
+    pivinv = 1.0 / a[icol * n + icol];
+    a[icol * n + icol] = 1.0;
+    for (l = 0; l < n; l++)
+      a[icol * n + l] *= pivinv;
+    b[icol] *= pivinv;
+    for (ll = 0; ll < n; ll++)
+      if (ll != icol)
+      {
+        dum = a[ll * n + icol];
+        a[ll * n + icol] = 0.0;
+        for (l = 0; l < n; l++)
+          a[ll * n + l] -= a[icol * n + l] * dum;
+        b[ll] -= b[icol] * dum;
+      }
   }
 
-  // Step 1: Solve linear system A*x = b using LU decomposition
-  // LAPACKE_dgesv uses column-major by default, but we use row-major
-  info = LAPACKE_dgesv(LAPACK_ROW_MAJOR, n, 1, a_copy, n, ipiv, b, 1);
-
-  if (info != 0) {
-    cout << "gaussj: Singular Matrix (dgesv info=" << info << ")" << endl;
-    delete[] ipiv;
-    delete[] a_copy;
-    return 1;
-  }
-
-  // Step 2: Compute matrix inverse A^(-1) using LU factorization
-  // First do LU factorization of original matrix a
-  info = LAPACKE_dgetrf(LAPACK_ROW_MAJOR, n, n, a, n, ipiv);
-
-  if (info != 0) {
-    cout << "gaussj: Singular Matrix (dgetrf info=" << info << ")" << endl;
-    delete[] ipiv;
-    delete[] a_copy;
-    return 1;
-  }
-
-  // Then compute inverse from LU factorization
-  info = LAPACKE_dgetri(LAPACK_ROW_MAJOR, n, a, n, ipiv);
-
-  if (info != 0) {
-    cout << "gaussj: Singular Matrix (dgetri info=" << info << ")" << endl;
-    delete[] ipiv;
-    delete[] a_copy;
-    return 1;
+  for (l = n - 1; l >= 0; l--)
+  {
+    if (indxr[l] != indxc[l])
+      for (k = 0; k < n; k++)
+      {
+        swap = a[k * n + indxr[l]];
+        a[k * n + indxr[l]] = a[k * n + indxc[l]];
+        a[k * n + indxc[l]] = swap;
+      }
   }
 
+  delete[] indxc;
+  delete[] indxr;
   delete[] ipiv;
-  delete[] a_copy;
 
   return 0;
 }

AMSS_NCKU_source/macrodef.h

@@ -29,6 +29,16 @@
 
 #define REGLEV 0
 
+#define BSSN_FINE_TIMING 0
+
+#define BSSN_FINE_TIMING_EVERY 1
+
+#define BSSN_FINE_TIMING_TOPN 8
+
+#define BSSN_KERNEL_FINE_TIMING 0
+
+#define BSSN_ENABLE_STDIN_ABORT_POLL 0
+
 //#define USE_GPU
 
 //#define CHECKDETAIL
@@ -88,6 +98,21 @@
 //     0: for every level;
 //     1: for all
 //
+// define BSSN_FINE_TIMING
+//     enable fine-grained per-timestep timing monitor
+//
+// define BSSN_FINE_TIMING_EVERY
+//     report timing every N coarse timesteps
+//
+// define BSSN_FINE_TIMING_TOPN
+//     number of hottest timing buckets shown in stdout
+//
+// define BSSN_KERNEL_FINE_TIMING
+//     enable split timing inside compute_rhs_bssn
+//
+// define BSSN_ENABLE_STDIN_ABORT_POLL
+//     poll stdin and broadcast abort flag every coarse step
+//
 // define USE_GPU
 //     use gpu or not
 //
@@ -142,4 +167,3 @@
 #define TINY 1e-10
 
 #endif   /* MICRODEF_H */
-

AMSS_NCKU_source/makefile

@@ -8,27 +8,16 @@ include makefile.inc
 POLINT6_USE_BARY ?= 1
 POLINT6_FLAG = -DPOLINT6_USE_BARYCENTRIC=$(POLINT6_USE_BARY)
 
-## ABE build flags selected by PGO_MODE (set in makefile.inc, default: opt)
-##   make                        -> opt  (PGO-guided, maximum performance)
-##   make PGO_MODE=instrument    -> instrument (Phase 1: collect fresh profile data)
-PROFDATA = /home/$(shell whoami)/AMSS-NCKU/pgo_profile/default.profdata
+## Legacy GNU/OpenMPI flags
+CXXBASEFLAGS = -O3 -march=native -Wno-deprecated -Dfortran3 -Dnewc $(INTERP_LB_FLAGS)
+F90BASEFLAGS = -O3 -march=native -cpp -fallow-argument-mismatch $(POLINT6_FLAG)
 
 ifeq ($(PGO_MODE),instrument)
-## Phase 1: instrumentation — omit -ipo/-fp-model fast=2 for faster build and numerical stability
-CXXAPPFLAGS = -O3 -xHost -fma -fprofile-instr-generate -ipo \
-              -Dfortran3 -Dnewc -I${MKLROOT}/include $(INTERP_LB_FLAGS)
-f90appflags = -O3 -xHost -fma -fprofile-instr-generate -ipo \
-              -align array64byte -fpp -I${MKLROOT}/include $(POLINT6_FLAG)
+CXXAPPFLAGS = $(CXXBASEFLAGS)
+f90appflags = $(F90BASEFLAGS)
 else
-## opt (default): maximum performance with PGO profile data -fprofile-instr-use=$(PROFDATA) \
-## PGO has been turned off, now tested and found to be negative optimization
-## INTERP_LB_FLAGS has been turned off too, now tested and found to be negative optimization
-
-
-CXXAPPFLAGS = -O3 -xHost -fp-model fast=2 -fma -ipo \
-              -Dfortran3 -Dnewc -I${MKLROOT}/include $(INTERP_LB_FLAGS)
-f90appflags = -O3 -xHost -fp-model fast=2 -fma -ipo \
-              -align array64byte -fpp -I${MKLROOT}/include $(POLINT6_FLAG)
+CXXAPPFLAGS = $(CXXBASEFLAGS)
+f90appflags = $(F90BASEFLAGS)
 endif
 
 .SUFFIXES: .o .f90 .C .for .cu
@@ -64,20 +53,17 @@ lopsided_c.o: lopsided_c.C
 lopsided_kodis_c.o: lopsided_kodis_c.C
 	${CXX} $(CXXAPPFLAGS) -c $< $(filein) -o $@
 
-interp_lb_profile.o: interp_lb_profile.C interp_lb_profile.h
-	${CXX} $(CXXAPPFLAGS) -c $< $(filein) -o $@
+#interp_lb_profile.o: interp_lb_profile.C interp_lb_profile.h
+#	${CXX} $(CXXAPPFLAGS) -c $< $(filein) -o $@
 
 ## TwoPunctureABE uses fixed optimal flags with its own PGO profile, independent of CXXAPPFLAGS
-TP_PROFDATA = /home/$(shell whoami)/AMSS-NCKU/pgo_profile/TwoPunctureABE.profdata
-TP_OPTFLAGS = -O3 -xHost -fp-model fast=2 -fma -ipo \
-              -fprofile-instr-use=$(TP_PROFDATA) \
-              -Dfortran3 -Dnewc -I${MKLROOT}/include
+TP_OPTFLAGS = $(CXXBASEFLAGS) $(TP_OPENMP_FLAGS)
 
 TwoPunctures.o: TwoPunctures.C
-	${CXX} $(TP_OPTFLAGS) -qopenmp -c $< -o $@
+	${CXX} $(TP_OPTFLAGS) -c $< -o $@
 
 TwoPunctureABE.o: TwoPunctureABE.C
-	${CXX} $(TP_OPTFLAGS) -qopenmp -c $< -o $@
+	${CXX} $(TP_OPTFLAGS) -c $< -o $@
 
 # Input files
 
@@ -185,7 +171,7 @@ ABEGPU: $(C++FILES_GPU) $(CFILES) $(F90FILES) $(F77FILES) $(AHFDOBJS) $(CUDAFILE
 	$(CLINKER) $(CXXAPPFLAGS) -o $@ $(C++FILES_GPU) $(CFILES) $(F90FILES) $(F77FILES) $(AHFDOBJS) $(CUDAFILES) $(LDLIBS)
 
 TwoPunctureABE: $(TwoPunctureFILES)
-	$(CLINKER) $(TP_OPTFLAGS) -qopenmp -o $@ $(TwoPunctureFILES) $(LDLIBS)
+	$(CLINKER) $(TP_OPTFLAGS) -o $@ $(TwoPunctureFILES) $(LDLIBS)
 
 clean:
 	rm *.o ABE ABEGPU TwoPunctureABE make.log -f

AMSS_NCKU_source/makefile.inc

@@ -1,33 +1,27 @@
-## GCC version (commented out)
-## filein  = -I/usr/include -I/usr/lib/x86_64-linux-gnu/mpich/include -I/usr/lib/x86_64-linux-gnu/openmpi/lib/ -I/usr/lib/gcc/x86_64-linux-gnu/11/ -I/usr/include/c++/11/
-## filein  = -I/usr/include/ -I/usr/include/openmpi-x86_64/ -I/usr/lib/x86_64-linux-gnu/openmpi/include/ -I/usr/lib/x86_64-linux-gnu/openmpi/lib/ -I/usr/lib/gcc/x86_64-linux-gnu/11/ -I/usr/include/c++/11/
-## LDLIBS  = -L/usr/lib/x86_64-linux-gnu -L/usr/lib64 -L/usr/lib/gcc/x86_64-linux-gnu/11 -lgfortran -lmpi -lgfortran
+## Legacy GNU/OpenMPI toolchain configuration
 
-## Intel oneAPI version with oneMKL (Optimized for performance)
-filein  = -I/usr/include/ -I${MKLROOT}/include
+## OpenMPI wrappers are installed but may not be on PATH.
+OMPI_BIN ?= /usr/lib64/openmpi/bin
 
-## Using sequential MKL (OpenMP disabled for better single-threaded performance)
-## Added -lifcore for Intel Fortran runtime and -limf for Intel math library
-LDLIBS  = -L${MKLROOT}/lib -lmkl_intel_lp64 -lmkl_sequential -lmkl_core -lifcore -limf -lpthread -lm -ldl -liomp5
+## Wrapper compilers
+f90          = $(OMPI_BIN)/mpifort
+f77          = $(OMPI_BIN)/mpifort
+CXX          = $(OMPI_BIN)/mpicxx
+CC           = $(OMPI_BIN)/mpicc
+CLINKER      = $(OMPI_BIN)/mpicxx
 
-## Memory allocator switch
-##   1 (default) : link Intel oneTBB allocator (libtbbmalloc)
-##   0           : use system default allocator (ptmalloc)
-USE_TBBMALLOC ?= 1
-TBBMALLOC_SO ?= /home/intel/oneapi/2025.3/lib/libtbbmalloc.so
-ifneq ($(wildcard $(TBBMALLOC_SO)),)
-TBBMALLOC_LIBS = -Wl,--no-as-needed $(TBBMALLOC_SO) -Wl,--as-needed
-else
-TBBMALLOC_LIBS = -Wl,--no-as-needed -ltbbmalloc -Wl,--as-needed
-endif
-ifeq ($(USE_TBBMALLOC),1)
-LDLIBS := $(TBBMALLOC_LIBS) $(LDLIBS)
-endif
+## Extra include flags are not needed when using the OpenMPI wrappers.
+filein       =
 
-## PGO build mode switch (ABE only; TwoPunctureABE always uses opt flags)
-##   opt        : (default) maximum performance with PGO profile-guided optimization
-##   instrument : PGO Phase 1 instrumentation to collect fresh profile data
-PGO_MODE ?= opt
+## BLAS/LAPACK backend:
+## OpenBLAS on this system provides BLAS, CBLAS and LAPACK symbols.
+BLAS_LAPACK_LIB ?= /lib64/libopenblaso.so.0
+LDLIBS  = $(BLAS_LAPACK_LIB) -lgfortran -lpthread -lm -ldl
+
+## PGO build mode switch
+##   off        : default legacy GNU build without PGO
+##   instrument : accepted for compatibility, currently same as off
+PGO_MODE ?= off
 
 ## Interp_Points load balance profiling mode
 ##   off        : (default) no load balance instrumentation
@@ -49,17 +43,13 @@ endif
 USE_CXX_KERNELS ?= 1
 
 ## RK4 kernel implementation switch
-##   1 (default) : use C/C++ rewrite of rungekutta4_rout (for optimization experiments)
+##   1 (default) : use C/C++ rewrite of rungekutta4_rout
 ##   0           : use original Fortran rungekutta4_rout.o
 USE_CXX_RK4 ?= 1
 
-f90          = ifx
-f77          = ifx
-CXX          = icpx
-CC           = icx
-CLINKER      = mpiicpx
+## OpenMP is only used for TwoPunctures on the legacy toolchain.
+TP_OPENMP_FLAGS ?= -fopenmp
 
 Cu = nvcc
 CUDA_LIB_PATH = -L/usr/lib/cuda/lib64 -I/usr/include -I/usr/lib/cuda/include
-#CUDA_APP_FLAGS = -c -g -O3 --ptxas-options=-v -arch compute_13 -code compute_13,sm_13 -Dfortran3 -Dnewc
 CUDA_APP_FLAGS = -c -g -O3 --ptxas-options=-v -Dfortran3 -Dnewc

AMSS_NCKU_source/surface_integral.h

@@ -36,6 +36,11 @@ private:
 
 	double *nx_g, *ny_g, *nz_g; // global list of unit normals
 	int myrank, cpusize;
+	int wave_cache_spinw, wave_cache_maxl, wave_cache_modes;
+	double *wave_theta_pos, *wave_theta_neg;
+	double *wave_phi_cos, *wave_phi_sin;
+	void clear_wave_cache();
+	void build_wave_cache(int spinw, int maxl);
 
 public:
 	surface_integral(int iSymmetry);
@@ -82,13 +87,29 @@ public:
 					   var *Axx, var *Axy, var *Axz, var *Ayy, var *Ayz, var *Azz,
 					   var *Gmx, var *Gmy, var *Gmz,
 					   var *Sfx_rhs, var *Sfy_rhs, var *Sfz_rhs,
-					   double *Rout, monitor *Monitor);
+					   double *Rout, monitor *Monitor, bool refresh_mass_fields = true);
 	void surf_MassPAng(double rex, int lev, ShellPatch *GH, var *chi, var *trK,
 					   var *gxx, var *gxy, var *gxz, var *gyy, var *gyz, var *gzz,
 					   var *Axx, var *Axy, var *Axz, var *Ayy, var *Ayz, var *Azz,
 					   var *Gmx, var *Gmy, var *Gmz,
 					   var *Sfx_rhs, var *Sfy_rhs, var *Sfz_rhs,
-					   double *Rout, monitor *Monitor);
+					   double *Rout, monitor *Monitor, bool refresh_mass_fields = true);
+	void surf_WaveMassPAng(double rex, int lev, cgh *GH,
+					   var *Rpsi4, var *Ipsi4, int spinw, int maxl, int NN, double *RP, double *IP,
+					   var *chi, var *trK,
+					   var *gxx, var *gxy, var *gxz, var *gyy, var *gyz, var *gzz,
+					   var *Axx, var *Axy, var *Axz, var *Ayy, var *Ayz, var *Azz,
+					   var *Gmx, var *Gmy, var *Gmz,
+					   var *Sfx_rhs, var *Sfy_rhs, var *Sfz_rhs,
+					   double *Rout, monitor *Monitor, bool refresh_mass_fields = true);
+	void surf_WaveMassPAng(double rex, int lev, ShellPatch *GH,
+					   var *Rpsi4, var *Ipsi4, int spinw, int maxl, int NN, double *RP, double *IP,
+					   var *chi, var *trK,
+					   var *gxx, var *gxy, var *gxz, var *gyy, var *gyz, var *gzz,
+					   var *Axx, var *Axy, var *Axz, var *Ayy, var *Ayz, var *Azz,
+					   var *Gmx, var *Gmy, var *Gmz,
+					   var *Sfx_rhs, var *Sfy_rhs, var *Sfz_rhs,
+					   double *Rout, monitor *Monitor, bool refresh_mass_fields = true);
 	void surf_Wave(double rex, cgh *GH, ShellPatch *SH,
 				   var *chi, var *trK,
 				   var *gxx, var *gxy, var *gxz, var *gyy, var *gyz, var *gzz,
@@ -115,7 +136,7 @@ public:
 					   var *Axx, var *Axy, var *Axz, var *Ayy, var *Ayz, var *Azz,
 					   var *Gmx, var *Gmy, var *Gmz,
 					   var *Sfx_rhs, var *Sfy_rhs, var *Sfz_rhs, // temparay memory for mass^i
-					   double *Rout, monitor *Monitor, MPI_Comm Comm_here);
+					   double *Rout, monitor *Monitor, MPI_Comm Comm_here, bool refresh_mass_fields = true);
 	void surf_Wave(double rex, int lev, cgh *GH, var *Rpsi4, var *Ipsi4,
 				   int spinw, int maxl, int NN, double *RP, double *IP,
 				   monitor *Monitor, MPI_Comm Comm_here);

README.md

@@ -97,7 +97,9 @@ Here, we take the Ubuntu 22.04 system as an example
 
     Modify the makefile.inc file in the AMSS_NCKU_source directory and change the settings according to your computer.
 
-    The settings for the Ubuntu 22.04 system do not need to be modified.
+    The default configuration in this branch uses GNU compilers through the OpenMPI wrappers under `/usr/lib64/openmpi/bin`.
+
+    If your OpenMPI installation is in another location, update `OMPI_BIN` in `AMSS_NCKU_source/makefile.inc` or export `AMSS_OPENMPI_BIN` before running the Python launcher.
 
 1.  Enter the AMSS-NCKU Python code folder and modify the input.
 

generate_macrodef.py

@@ -144,6 +144,62 @@ def generate_macrodef_h():
     print( "#define REGLEV 0",      file=file1 )
     print(                          file=file1 )
 
+    # Define fine-grained timing/debug macros.
+    # All of them default to OFF so production builds do not pay profiling overhead.
+
+    fine_timing = getattr(input_data, "Fine_Timing",
+                  getattr(input_data, "Finegrained_Timing", "no"))
+    kernel_fine_timing = getattr(input_data, "Kernel_Fine_Timing",
+                          getattr(input_data, "BSSN_Kernel_Fine_Timing", "no"))
+    stdin_abort_poll = getattr(input_data, "Enable_Stdin_Abort_Poll",
+                       getattr(input_data, "Stdin_Abort_Poll", "no"))
+    timing_report_every = max(1, int(getattr(
+        input_data, "Timing_Every_Steps",
+        getattr(input_data, "Timing_Report_Every", 1))))
+    timing_top_hotspots = max(1, int(getattr(
+        input_data, "Timing_Top_Hotspots", 8)))
+
+    if ( fine_timing == "yes" ):
+        print( "#define BSSN_FINE_TIMING 1", file=file1 )
+        print(                               file=file1 )
+    elif ( fine_timing == "no" ):
+        print( "#define BSSN_FINE_TIMING 0", file=file1 )
+        print(                               file=file1 )
+    else:
+        print( "Fine_Timing setting error!!!" )
+        print()
+        print( "# Fine_Timing setting error!!!", file=file1 )
+        print(                                   file=file1 )
+
+    print( f"#define BSSN_FINE_TIMING_EVERY {timing_report_every}", file=file1 )
+    print(                                                          file=file1 )
+    print( f"#define BSSN_FINE_TIMING_TOPN {timing_top_hotspots}",  file=file1 )
+    print(                                                          file=file1 )
+
+    if ( kernel_fine_timing == "yes" ):
+        print( "#define BSSN_KERNEL_FINE_TIMING 1", file=file1 )
+        print(                                      file=file1 )
+    elif ( kernel_fine_timing == "no" ):
+        print( "#define BSSN_KERNEL_FINE_TIMING 0", file=file1 )
+        print(                                      file=file1 )
+    else:
+        print( "Kernel_Fine_Timing setting error!!!" )
+        print()
+        print( "# Kernel_Fine_Timing setting error!!!", file=file1 )
+        print(                                          file=file1 )
+
+    if ( stdin_abort_poll == "yes" ):
+        print( "#define BSSN_ENABLE_STDIN_ABORT_POLL 1", file=file1 )
+        print(                                           file=file1 )
+    elif ( stdin_abort_poll == "no" ):
+        print( "#define BSSN_ENABLE_STDIN_ABORT_POLL 0", file=file1 )
+        print(                                           file=file1 )
+    else:
+        print( "Enable_Stdin_Abort_Poll setting error!!!" )
+        print()
+        print( "# Enable_Stdin_Abort_Poll setting error!!!", file=file1 )
+        print(                                               file=file1 )
+
     # Define macro USE_GPU
     # use GPU or not
     
@@ -224,6 +280,21 @@ def generate_macrodef_h():
     print( "//     0: for every level;",                                                 file=file1 ) 
     print( "//     1: for all",                                                          file=file1 )
     print( "//",                                                                         file=file1 )
+    print( "// define BSSN_FINE_TIMING",                                                 file=file1 )
+    print( "//     enable fine-grained per-timestep timing monitor",                     file=file1 )
+    print( "//",                                                                         file=file1 )
+    print( "// define BSSN_FINE_TIMING_EVERY",                                           file=file1 )
+    print( "//     report timing every N coarse timesteps",                              file=file1 )
+    print( "//",                                                                         file=file1 )
+    print( "// define BSSN_FINE_TIMING_TOPN",                                            file=file1 )
+    print( "//     number of hottest timing buckets shown in stdout",                    file=file1 )
+    print( "//",                                                                         file=file1 )
+    print( "// define BSSN_KERNEL_FINE_TIMING",                                          file=file1 )
+    print( "//     enable split timing inside compute_rhs_bssn",                         file=file1 )
+    print( "//",                                                                         file=file1 )
+    print( "// define BSSN_ENABLE_STDIN_ABORT_POLL",                                     file=file1 )
+    print( "//     poll stdin and broadcast abort flag every coarse step",               file=file1 )
+    print( "//",                                                                         file=file1 )
     print( "// define USE_GPU",                                                          file=file1 )
     print( "//     use gpu or not",                                                      file=file1 )
     print( "//",                                                                         file=file1 )

makefile_and_run.py

@@ -9,6 +9,7 @@
 
 
 import AMSS_NCKU_Input as input_data
+import os
 import subprocess
 import time
 
@@ -52,6 +53,8 @@ NUMACTL_CPU_BIND = get_last_n_cores_per_socket(n=32)
 
 ## Build parallelism: match the number of bound cores
 BUILD_JOBS = 64
+OPENMPI_BIN = os.environ.get("AMSS_OPENMPI_BIN", "/usr/lib64/openmpi/bin")
+MPI_RUNNER = os.path.join(OPENMPI_BIN, "mpirun")
 
 
 ##################################################################
@@ -147,11 +150,11 @@ def run_ABE():
     ## 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         = NUMACTL_CPU_BIND + " " + MPI_RUNNER + " -np " + str(input_data.MPI_processes) + " ./ABE"
         #mpi_command         = " mpirun -np " + str(input_data.MPI_processes) + " ./ABE"
         mpi_command_outfile = "ABE_out.log"
     elif (input_data.GPU_Calculation == "yes"):
-        mpi_command         = NUMACTL_CPU_BIND + " mpirun -np " + str(input_data.MPI_processes) + " ./ABEGPU"
+        mpi_command         = NUMACTL_CPU_BIND + " " + MPI_RUNNER + " -np " + str(input_data.MPI_processes) + " ./ABEGPU"
         mpi_command_outfile = "ABEGPU_out.log"
  
     ## Execute the MPI command and stream output