Switch legacy build to GCC and OpenMPI

Reviewed-on: #3

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
-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
+SUBROUTINE four1(dataa,nn,isign)
+implicit none
+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
+return
+END SUBROUTINE four1

AMSS_NCKU_source/Parallel.h

@@ -108,6 +108,9 @@ namespace Parallel
     MPI_Status *stats;
     int max_reqs;
     bool lengths_valid;
+    int *tc_req_node;
+    int *tc_req_is_recv;
+    int *tc_completed;
     SyncCache();
     void invalidate();
     void destroy();
@@ -121,7 +124,10 @@ namespace Parallel
   struct AsyncSyncState {
     int req_no;
     bool active;
-    AsyncSyncState() : req_no(0), active(false) {}
+    int *req_node;
+    int *req_is_recv;
+    int pending_recv;
+    AsyncSyncState() : req_no(0), active(false), req_node(0), req_is_recv(0), pending_recv(0) {}
   };
 
   void Sync_start(MyList<Patch> *PatL, MyList<var> *VarList, int Symmetry,
@@ -173,12 +179,13 @@ namespace Parallel
   MyList<Parallel::gridseg> *clone_gsl(MyList<Parallel::gridseg> *p, bool first_only);
   MyList<Parallel::gridseg> *build_bulk_gsl(Patch *Pat); // similar to build_owned_gsl0 but does not care rank issue
   MyList<Parallel::gridseg> *build_bulk_gsl(Block *bp, Patch *Pat);
-  void build_PhysBD_gstl(Patch *Pat, MyList<Parallel::gridseg> *srci, MyList<Parallel::gridseg> *dsti,
-                         MyList<Parallel::gridseg> **out_src, MyList<Parallel::gridseg> **out_dst);
-  void PeriodicBD(Patch *Pat, MyList<var> *VarList, int Symmetry);
-  double L2Norm(Patch *Pat, var *vf);
-  void checkgsl(MyList<Parallel::gridseg> *pp, bool first_only);
-  void checkvarl(MyList<var> *pp, bool first_only);
+  void build_PhysBD_gstl(Patch *Pat, MyList<Parallel::gridseg> *srci, MyList<Parallel::gridseg> *dsti,
+                         MyList<Parallel::gridseg> **out_src, MyList<Parallel::gridseg> **out_dst);
+  void PeriodicBD(Patch *Pat, MyList<var> *VarList, int Symmetry);
+  double L2Norm(Patch *Pat, var *vf);
+  void 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);
   MyList<Parallel::gridseg> *divide_gs(MyList<Parallel::gridseg> *p, Patch *Pat);
   void prepare_inter_time_level(Patch *Pat,
@@ -210,11 +217,12 @@ namespace Parallel
   void aligncheck(double *bbox0, double *bboxl, int lev, double *DH0, int *shape);
   bool point_locat_gsl(double *pox, MyList<Parallel::gridseg> *gsl);
   void checkpatchlist(MyList<Patch> *PatL, bool buflog);
-
-  double L2Norm(Patch *Pat, var *vf, MPI_Comm Comm_here);
-  bool PatList_Interp_Points(MyList<Patch> *PatL, MyList<var> *VarList,
-                             int NN, double **XX,
-                             double *Shellf, int Symmetry, MPI_Comm Comm_here);
+
+  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);
 #if (PSTR == 1 || PSTR == 2 || PSTR == 3)
   MyList<Block> *distribute(MyList<Patch> *PatchLIST, int cpusize, int ingfsi, int fngfsi,
                             bool periodic, int start_rank, int end_rank, int nodes = 0);

AMSS_NCKU_source/ShellPatch.C

@@ -3439,10 +3439,10 @@ void ShellPatch::write_Pablo_file_ss(int *ext, double xmin, double xmax, double
   delete[] Z;
 }
 
-double ShellPatch::L2Norm(var *vf)
-{
-  double tvf, dtvf = 0;
-  int BDW = overghost;
+double ShellPatch::L2Norm(var *vf)
+{
+  double tvf, dtvf = 0;
+  int BDW = overghost;
 
   MyList<ss_patch> *sPp = PatL;
   while (sPp)
@@ -3469,13 +3469,50 @@ double ShellPatch::L2Norm(var *vf)
   MPI_Allreduce(&dtvf, &tvf, 1, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
 
   tvf = sqrt(tvf);
-
-  return tvf;
-}
-
-// find maximum of abstract value, XX store position for maximum, Shellf store maximum themselvs
-void ShellPatch::Find_Maximum(MyList<var> *VarList, double *XX,
-                              double *Shellf)
+
+  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,
+                              double *Shellf)
 {
   MyList<var> *varl;
   int num_var = 0;

AMSS_NCKU_source/ShellPatch.h

@@ -195,10 +195,11 @@ public:
    bool Interp_One_Point(MyList<var> *VarList,
                          double *XX, /*input global Cartesian coordinate*/
                          double *Shellf, int Symmetry);
-   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 Find_Maximum(MyList<var> *VarList, double *XX, double *Shellf);
-};
+   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);
+};
 
 #endif /* SHELLPATCH_H */

AMSS_NCKU_source/TwoPunctures.C

@@ -25,9 +25,23 @@ using namespace std;
 #include <math.h>
 #include <complex.h>
 #endif
-
-#include "TwoPunctures.h"
-#include <mkl_cblas.h>
+
+#include "TwoPunctures.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.h

@@ -45,10 +45,11 @@ public:
        int checkrun;
        char checkfilename[50];
        int Steps;
-       double StartTime, TotalTime;
-       double AnasTime, DumpTime, d2DumpTime, CheckTime;
-       double LastAnas, LastConsOut;
-       double Courant;
+       double StartTime, TotalTime;
+       double AnasTime, DumpTime, d2DumpTime, CheckTime;
+       double LastAnas, LastConsOut;
+       int *ConstraintRefreshLevels;
+       double Courant;
        double numepss, numepsb, numepsh;
        int Symmetry;
        int maxl, decn;
@@ -130,10 +131,12 @@ public:
        Parallel::SyncCache *sync_cache_cor;  // per-level cache for corrector sync
        Parallel::SyncCache *sync_cache_rp_coarse;  // RestrictProlong sync on PatL[lev-1]
        Parallel::SyncCache *sync_cache_rp_fine;    // RestrictProlong sync on PatL[lev]
+       Parallel::SyncCache *sync_cache_restrict;   // cached Restrict in RestrictProlong
+       Parallel::SyncCache *sync_cache_outbd;      // cached OutBdLow2Hi in RestrictProlong
 
-       monitor *ErrorMonitor, *Psi4Monitor, *BHMonitor, *MAPMonitor;
-       monitor *ConVMonitor;
-       surface_integral *Waveshell;
+       monitor *ErrorMonitor, *Psi4Monitor, *BHMonitor, *MAPMonitor;
+       monitor *ConVMonitor, *TimingMonitor;
+       surface_integral *Waveshell;
        checkpoint *CheckPoint;
 
 public:

AMSS_NCKU_source/bssn_rhs.f90

@@ -59,9 +59,10 @@
   real*8, dimension(ex(1),ex(2),ex(3)),intent(out) :: Rxx,Rxy,Rxz,Ryy,Ryz,Rzz
   real*8,intent(in) :: eps
   real*8, dimension(ex(1),ex(2),ex(3)),intent(inout) :: ham_Res, movx_Res, movy_Res, movz_Res
-  real*8, dimension(ex(1),ex(2),ex(3)),intent(inout) :: Gmx_Res, Gmy_Res, Gmz_Res
-!  gont = 0: success; gont = 1: something wrong
-  integer::gont
+  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:
 
@@ -83,11 +84,18 @@
   real*8, dimension(ex(1),ex(2),ex(3)) :: gupxx,gupxy,gupxz
   real*8, dimension(ex(1),ex(2),ex(3)) :: gupyy,gupyz,gupzz
 
-  real*8,dimension(3) ::SSS,AAS,ASA,SAA,ASS,SAS,SSA
-  real*8            :: dX, dY, dZ, PI
-  real*8, 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
+  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
   double precision,parameter::FF = 0.75d0,eta=2.d0
   real*8, parameter :: F1o3 = 1.D0/3.D0, F2o3 = 2.D0/3.D0,F3o2=1.5d0, F1o6 = 1.D0/6.D0
   real*8, parameter :: F16=1.6d1,F8=8.d0
@@ -96,11 +104,11 @@
   real*8, dimension(ex(1),ex(2),ex(3)) :: reta
 #endif
 
-#if (GAUGE == 6 || GAUGE == 7)
-  integer :: BHN,i,j,k
-  real*8, dimension(9) :: Porg
-  real*8, dimension(3) :: Mass
-  real*8 :: r1,r2,M,A,w1,w2,C1,C2
+#if (GAUGE == 6 || GAUGE == 7)
+  integer :: BHN
+  real*8, dimension(9) :: Porg
+  real*8, dimension(3) :: Mass
+  real*8 :: r1,r2,M,A,w1,w2,C1,C2
   real*8, dimension(ex(1),ex(2),ex(3)) :: reta
 
   call getpbh(BHN,Porg,Mass)
@@ -145,174 +153,204 @@
   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)
+  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)
-  call fderivs(ex,dyy,gyyx,gyyy,gyyz,X,Y,Z,SYM ,SYM ,SYM ,Symmetry,Lev)
-  call fderivs(ex,gyz,gyzx,gyzy,gyzz,X,Y,Z,SYM ,ANTI,ANTI,Symmetry,Lev)
-  call fderivs(ex,dzz,gzzx,gzzy,gzzz,X,Y,Z,SYM ,SYM ,SYM ,Symmetry,Lev)
-
-  gxx_rhs = - TWO * alpn1 * Axx    -  F2o3 * gxx * div_beta          + &
-              TWO *(  gxx * betaxx +   gxy * betayx +   gxz * betazx)
-
-  gyy_rhs = - TWO * alpn1 * Ayy    -  F2o3 * gyy * div_beta          + &
-              TWO *(  gxy * betaxy +   gyy * betayy +   gyz * betazy)
-
-  gzz_rhs = - TWO * alpn1 * Azz    -  F2o3 * gzz * div_beta          + &
-              TWO *(  gxz * betaxz +   gyz * betayz +   gzz * betazz)
-
-  gxy_rhs = - TWO * alpn1 * Axy    +  F1o3 * gxy    * div_beta       + &
-                      gxx * betaxy                  +   gxz * betazy + &
-                                       gyy * betayx +   gyz * betazx   &
-                                                    -   gxy * betazz
-
-  gyz_rhs = - TWO * alpn1 * Ayz    +  F1o3 * gyz    * div_beta       + &
-                      gxy * betaxz +   gyy * betayz                  + &
-                      gxz * betaxy                  +   gzz * betazy   &
-                                                    -   gyz * betaxx
- 
-  gxz_rhs = - TWO * alpn1 * Axz    +  F1o3 * gxz    * div_beta       + &
-                      gxx * betaxz +   gxy * betayz                  + &
-                                       gyz * betayx +   gzz * betazx   &
-                                                    -   gxz * betayy     !rhs for gij
-
-! invert tilted metric
-  gupzz =  gxx * gyy * gzz + gxy * gyz * gxz + gxz * gxy * gyz - &
-           gxz * gyy * gxz - gxy * gxy * gzz - gxx * gyz * gyz
-  gupxx =   ( gyy * gzz - gyz * gyz ) / gupzz
-  gupxy = - ( gxy * gzz - gyz * gxz ) / gupzz
-  gupxz =   ( gxy * gyz - gyy * gxz ) / gupzz
-  gupyy =   ( gxx * gzz - gxz * gxz ) / gupzz
-  gupyz = - ( gxx * gyz - gxy * gxz ) / gupzz
-  gupzz =   ( gxx * gyy - gxy * gxy ) / gupzz
-
-  if(co == 0)then
-! 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
-
-! 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 ))
- 
-  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 ))
-
-  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)
-
-  Gamxxy =HALF*( gupxx*gxxy + gupxy*gyyx + gupxz*( gxzy + gyzx - gxyz ) )
-  Gamyxy =HALF*( gupxy*gxxy + gupyy*gyyx + gupyz*( gxzy + gyzx - gxyz ) )
-  Gamzxy =HALF*( gupxz*gxxy + gupyz*gyyx + gupzz*( gxzy + gyzx - gxyz ) )
-
-  Gamxxz =HALF*( gupxx*gxxz + gupxy*( gxyz + gyzx - gxzy ) + gupxz*gzzx )
-  Gamyxz =HALF*( gupxy*gxxz + gupyy*( gxyz + gyzx - gxzy ) + gupyz*gzzx )
-  Gamzxz =HALF*( gupxz*gxxz + gupyz*( gxyz + gyzx - gxzy ) + gupzz*gzzx )
-
-  Gamxyz =HALF*( gupxx*( gxyz + gxzy - gyzx ) + gupxy*gyyz + gupxz*gzzy )
-  Gamyyz =HALF*( gupxy*( gxyz + gxzy - gyzx ) + gupyy*gyyz + gupyz*gzzy )
-  Gamzyz =HALF*( gupxz*( gxyz + gxzy - gyzx ) + gupyz*gyyz + gupzz*gzzy )
-! Raise indices of \tilde A_{ij} and store in R_ij
-
-  Rxx =    gupxx * gupxx * Axx + gupxy * gupxy * Ayy + gupxz * gupxz * Azz + &
-      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)
-
-   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 ) )
-
-   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 ) )
-
-   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 ) )
+  call fderivs(ex,dxx,gxxx,gxxy,gxxz,X,Y,Z,SYM ,SYM ,SYM ,Symmetry,Lev)
+  call fderivs(ex,gxy,gxyx,gxyy,gxyz,X,Y,Z,ANTI,ANTI,SYM ,Symmetry,Lev)
+  call fderivs(ex,gxz,gxzx,gxzy,gxzz,X,Y,Z,ANTI,SYM ,ANTI,Symmetry,Lev)
+  call fderivs(ex,dyy,gyyx,gyyy,gyyz,X,Y,Z,SYM ,SYM ,SYM ,Symmetry,Lev)
+  call fderivs(ex,gyz,gyzx,gyzy,gyzz,X,Y,Z,SYM ,ANTI,ANTI,Symmetry,Lev)
+  call fderivs(ex,dzz,gzzx,gzzy,gzzz,X,Y,Z,SYM ,SYM ,SYM ,Symmetry,Lev)
+
+  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
+
+    chi_rhs(i,j,k) = F2o3 * chin1(i,j,k) * (alpn1(i,j,k) * trK(i,j,k) - divb_loc)
+
+    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) )
+
+    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) )
+
+    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) )
+
+    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)
+
+    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)
+
+    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)
+
+    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
+
+    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
+
+    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)))
+
+    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)))
+
+    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))
+
+    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
+
+! 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)
+
+    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
+
+    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))
+
+    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)
-
-  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  )
-
-  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  )
-
-  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
+  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)
+
+    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)
+
+    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
+
+    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
@@ -601,192 +655,190 @@
             Gamxyz * gxzz + Gamyyz * gyzz + Gamzyz * gzzz  + &
             Gamxzz * gxzy + Gamyzz * gyzy + Gamzzz * gzzy  + &
             Gamxyz * gzzx + Gamyyz * gzzy + Gamzyz * gzzz )
-!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
-
-  f =        gupxx * ( fxx - F3o2/chin1 * chix * chix ) + &
-             gupyy * ( fyy - F3o2/chin1 * chiy * chiy ) + &
-             gupzz * ( fzz - F3o2/chin1 * chiz * chiz ) + &
-       TWO * gupxy * ( fxy - F3o2/chin1 * chix * chiy ) + &
-       TWO * gupxz * ( fxz - F3o2/chin1 * chix * chiz ) + &
-       TWO * gupyz * ( fyz - F3o2/chin1 * chiy * chiz ) 
-! Add chi part to Ricci tensor:
-
-  Rxx = Rxx + (fxx - chix*chix/chin1/TWO + gxx * f)/chin1/TWO
-  Ryy = Ryy + (fyy - chiy*chiy/chin1/TWO + gyy * f)/chin1/TWO
-  Rzz = Rzz + (fzz - chiz*chiz/chin1/TWO + gzz * f)/chin1/TWO
-  Rxy = Rxy + (fxy - chix*chiy/chin1/TWO + gxy * f)/chin1/TWO
-  Rxz = Rxz + (fxz - chix*chiz/chin1/TWO + gxz * f)/chin1/TWO
-  Ryz = Ryz + (fyz - chiy*chiz/chin1/TWO + gyz * f)/chin1/TWO
-
-! covariant second derivatives of the lapse respect to physical metric
-  call fdderivs(ex,Lap,fxx,fxy,fxz,fyy,fyz,fzz,X,Y,Z, &
-                SYM,SYM,SYM,symmetry,Lev)
-
-  gxxx = (gupxx * chix + gupxy * chiy + gupxz * chiz)/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
-
-  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
-
-! 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 = 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:
-
-  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
-
-! Compute trace-free part (note: chi^-1 and chi cancel!):
-
-  f = F1o3 *(  gupxx * fxx + gupyy * fyy + gupzz * fzz + &
-        TWO* ( gupxy * fxy + gupxz * fxz + gupyz * fyz ) )
-#endif
-
-  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
-
-! Now: store A_il A^l_j into fij:
-
-  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)
-
-  f = chin1
-! store D^i D_i Lap in trK_rhs
-  trK_rhs = f*trK_rhs
-          
-  Axx_rhs =           f * Axx_rhs+ alpn1 * (trK * Axx - TWO * fxx)  + &
-           TWO * (  Axx * betaxx +   Axy * betayx +   Axz * betazx )- &
-             F2o3 * Axx * div_beta
-
-  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
+!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)
+
+  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)
+
+    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(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)
+
+  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
+
+    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
+
+    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)
+
+    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)
+
+    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
+
+    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_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
+
+    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)
+
+    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(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))
+
+    trK_rhs(i,j,k) = chin_loc * trK_rhs(i,j,k)
+
+    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)
+
+    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,
-! 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,gxy,gxy_rhs,betax,betay,betaz,Symmetry,AAS,eps)
-  call lopsided_kodis(ex,X,Y,Z,gxz,gxz_rhs,betax,betay,betaz,Symmetry,ASA,eps)
-  call lopsided_kodis(ex,X,Y,Z,gyy,gyy_rhs,betax,betay,betaz,Symmetry,SSS,eps)
-  call lopsided_kodis(ex,X,Y,Z,gyz,gyz_rhs,betax,betay,betaz,Symmetry,SAA,eps)
-  call lopsided_kodis(ex,X,Y,Z,gzz,gzz_rhs,betax,betay,betaz,Symmetry,SSS,eps)
+! 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,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,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,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

@@ -22,19 +22,32 @@
 #define f_compute_rhs_Z4c_ss COMPUTE_RHS_Z4C_SS
 #define f_compute_constraint_fr COMPUTE_CONSTRAINT_FR
 #endif
-#ifdef fortran3
-#define f_compute_rhs_bssn compute_rhs_bssn_
+#ifdef fortran3
+#define f_compute_rhs_bssn compute_rhs_bssn_
 #define f_compute_rhs_bssn_ss compute_rhs_bssn_ss_
 #define f_compute_rhs_bssn_escalar compute_rhs_bssn_escalar_
 #define f_compute_rhs_bssn_escalar_ss compute_rhs_bssn_escalar_ss_
 #define f_compute_rhs_Z4c compute_rhs_z4c_
 #define f_compute_rhs_Z4cnot compute_rhs_z4cnot_
 #define f_compute_rhs_Z4c_ss compute_rhs_z4c_ss_
-#define f_compute_constraint_fr compute_constraint_fr_
-#endif
-extern "C"
-{
-        int f_compute_rhs_bssn(int *, double &, double *, double *, double *,                                                      // ex,T,X,Y,Z
+#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
                                double *, double *,                                                                                 // chi, trK
                                double *, double *, double *, double *, double *, double *,                                         // gij
                                double *, double *, double *, double *, double *, double *,                                         // Aij

AMSS_NCKU_source/bssn_rhs_cuda.h

@@ -1,36 +0,0 @@
-#ifndef BSSN_RHS_CUDA_H
-#define BSSN_RHS_CUDA_H
-
-#ifdef __cplusplus
-extern "C" {
-#endif
-
-int f_compute_rhs_bssn(int *ex, double &T,
-                       double *X, double *Y, double *Z,
-                       double *chi, double *trK,
-                       double *dxx, double *gxy, double *gxz, double *dyy, double *gyz, double *dzz,
-                       double *Axx, double *Axy, double *Axz, double *Ayy, double *Ayz, double *Azz,
-                       double *Gamx, double *Gamy, double *Gamz,
-                       double *Lap, double *betax, double *betay, double *betaz,
-                       double *dtSfx, double *dtSfy, double *dtSfz,
-                       double *chi_rhs, double *trK_rhs,
-                       double *gxx_rhs, double *gxy_rhs, double *gxz_rhs, double *gyy_rhs, double *gyz_rhs, double *gzz_rhs,
-                       double *Axx_rhs, double *Axy_rhs, double *Axz_rhs, double *Ayy_rhs, double *Ayz_rhs, double *Azz_rhs,
-                       double *Gamx_rhs, double *Gamy_rhs, double *Gamz_rhs,
-                       double *Lap_rhs, double *betax_rhs, double *betay_rhs, double *betaz_rhs,
-                       double *dtSfx_rhs, double *dtSfy_rhs, double *dtSfz_rhs,
-                       double *rho, double *Sx, double *Sy, double *Sz,
-                       double *Sxx, double *Sxy, double *Sxz, double *Syy, double *Syz, double *Szz,
-                       double *Gamxxx, double *Gamxxy, double *Gamxxz, double *Gamxyy, double *Gamxyz, double *Gamxzz,
-                       double *Gamyxx, double *Gamyxy, double *Gamyxz, double *Gamyyy, double *Gamyyz, double *Gamyzz,
-                       double *Gamzxx, double *Gamzxy, double *Gamzxz, double *Gamzyy, double *Gamzyz, double *Gamzzz,
-                       double *Rxx, double *Rxy, double *Rxz, double *Ryy, double *Ryz, double *Rzz,
-                       double *ham_Res, double *movx_Res, double *movy_Res, double *movz_Res,
-                       double *Gmx_Res, double *Gmy_Res, double *Gmz_Res,
-                       int &Symmetry, int &Lev, double &eps, int &co);
-
-#ifdef __cplusplus
-}
-#endif
-
-#endif

AMSS_NCKU_source/diff_newwb.f90

@@ -33,7 +33,7 @@
   real*8 :: dX,dY,dZ
   real*8,dimension(0:ex(1),0:ex(2),0:ex(3))   :: fh
   real*8, dimension(3) :: SoA
-  integer :: imin,jmin,kmin,imax,jmax,kmax,i,j,k
+  integer :: imin,jmin,kmin,imax,jmax,kmax,i,j,k
   real*8 :: d2dx,d2dy,d2dz
   integer, parameter :: NO_SYMM = 0, EQ_SYMM = 1, OCTANT = 2
   real*8,  parameter :: ZEO=0.d0,ONE=1.d0, F60=6.d1
@@ -137,7 +137,7 @@
   real*8 :: dX
   real*8,dimension(0:ex(1),0:ex(2),0:ex(3))   :: fh
   real*8, dimension(3) :: SoA
-  integer :: imin,jmin,kmin,imax,jmax,kmax,i,j,k
+  integer :: imin,jmin,kmin,imax,jmax,kmax,i,j,k
   real*8 :: d2dx
   integer, parameter :: NO_SYMM = 0, EQ_SYMM = 1, OCTANT = 2
   real*8,  parameter :: ZEO=0.d0,ONE=1.d0, F60=6.d1
@@ -1512,8 +1512,9 @@
   real*8 :: dX,dY,dZ
   real*8,dimension(-1:ex(1),-1:ex(2),-1:ex(3))   :: fh
   real*8, dimension(3) :: SoA
-  integer :: imin,jmin,kmin,imax,jmax,kmax,i,j,k
-  real*8  :: Sdxdx,Sdydy,Sdzdz,Fdxdx,Fdydy,Fdzdz
+  integer :: imin,jmin,kmin,imax,jmax,kmax,i,j,k
+  integer :: i_core_min,i_core_max,j_core_min,j_core_max,k_core_min,k_core_max
+  real*8  :: Sdxdx,Sdydy,Sdzdz,Fdxdx,Fdydy,Fdzdz
   real*8  :: Sdxdy,Sdxdz,Sdydz,Fdxdy,Fdxdz,Fdydz
   integer, parameter :: NO_SYMM = 0, EQ_SYMM = 1, OCTANT = 2
   real*8, parameter :: ZEO=0.d0, ONE=1.d0, TWO=2.d0, F1o4=2.5d-1, F9=9.d0,  F45=4.5d1
@@ -1560,17 +1561,55 @@
 
   fxx = ZEO
   fyy = ZEO
-  fzz = ZEO
-  fxy = ZEO
-  fxz = ZEO
-  fyz = ZEO
-
-  do k=1,ex(3)
-  do j=1,ex(2)
-  do i=1,ex(1)
-!~~~~~~ fxx
-        if(i+2 <= imax .and. i-2 >= imin)then
-!
+  fzz = ZEO
+  fxy = ZEO
+  fxz = ZEO
+  fyz = ZEO
+
+  i_core_min = max(1, imin+2)
+  i_core_max = min(ex(1), imax-2)
+  j_core_min = max(1, jmin+2)
+  j_core_max = min(ex(2), jmax-2)
+  k_core_min = max(1, kmin+2)
+  k_core_max = min(ex(3), kmax-2)
+
+  if(i_core_min <= i_core_max .and. j_core_min <= j_core_max .and. k_core_min <= k_core_max)then
+   do k=k_core_min,k_core_max
+   do j=j_core_min,j_core_max
+   do i=i_core_min,i_core_max
+! interior points always use 4th-order stencils without branch checks
+      fxx(i,j,k) = Fdxdx*(-fh(i-2,j,k)+F16*fh(i-1,j,k)-F30*fh(i,j,k) &
+                          -fh(i+2,j,k)+F16*fh(i+1,j,k)              )
+      fyy(i,j,k) = Fdydy*(-fh(i,j-2,k)+F16*fh(i,j-1,k)-F30*fh(i,j,k) &
+                          -fh(i,j+2,k)+F16*fh(i,j+1,k)              )
+      fzz(i,j,k) = Fdzdz*(-fh(i,j,k-2)+F16*fh(i,j,k-1)-F30*fh(i,j,k) &
+                          -fh(i,j,k+2)+F16*fh(i,j,k+1)              )
+      fxy(i,j,k) = Fdxdy*(     (fh(i-2,j-2,k)-F8*fh(i-1,j-2,k)+F8*fh(i+1,j-2,k)-fh(i+2,j-2,k))  &
+                          -F8 *(fh(i-2,j-1,k)-F8*fh(i-1,j-1,k)+F8*fh(i+1,j-1,k)-fh(i+2,j-1,k))  &
+                          +F8 *(fh(i-2,j+1,k)-F8*fh(i-1,j+1,k)+F8*fh(i+1,j+1,k)-fh(i+2,j+1,k))  &
+                          -    (fh(i-2,j+2,k)-F8*fh(i-1,j+2,k)+F8*fh(i+1,j+2,k)-fh(i+2,j+2,k)))
+      fxz(i,j,k) = Fdxdz*(     (fh(i-2,j,k-2)-F8*fh(i-1,j,k-2)+F8*fh(i+1,j,k-2)-fh(i+2,j,k-2))  &
+                          -F8 *(fh(i-2,j,k-1)-F8*fh(i-1,j,k-1)+F8*fh(i+1,j,k-1)-fh(i+2,j,k-1))  &
+                          +F8 *(fh(i-2,j,k+1)-F8*fh(i-1,j,k+1)+F8*fh(i+1,j,k+1)-fh(i+2,j,k+1))  &
+                          -    (fh(i-2,j,k+2)-F8*fh(i-1,j,k+2)+F8*fh(i+1,j,k+2)-fh(i+2,j,k+2)))
+      fyz(i,j,k) = Fdydz*(     (fh(i,j-2,k-2)-F8*fh(i,j-1,k-2)+F8*fh(i,j+1,k-2)-fh(i,j+2,k-2))  &
+                          -F8 *(fh(i,j-2,k-1)-F8*fh(i,j-1,k-1)+F8*fh(i,j+1,k-1)-fh(i,j+2,k-1))  &
+                          +F8 *(fh(i,j-2,k+1)-F8*fh(i,j-1,k+1)+F8*fh(i,j+1,k+1)-fh(i,j+2,k+1))  &
+                          -    (fh(i,j-2,k+2)-F8*fh(i,j-1,k+2)+F8*fh(i,j+1,k+2)-fh(i,j+2,k+2)))
+   enddo
+   enddo
+   enddo
+  endif
+
+  do k=1,ex(3)
+  do j=1,ex(2)
+  do i=1,ex(1)
+      if(i>=i_core_min .and. i<=i_core_max .and. &
+         j>=j_core_min .and. j<=j_core_max .and. &
+         k>=k_core_min .and. k<=k_core_max) cycle
+!~~~~~~ fxx
+        if(i+2 <= imax .and. i-2 >= imin)then
+!
 !               - f(i-2) + 16 f(i-1) - 30 f(i) + 16 f(i+1) - f(i+2)
 !  fxx(i) = ----------------------------------------------------------
 !                                  12 dx^2 

AMSS_NCKU_source/fderivs_c.C

@@ -81,26 +81,63 @@ void fderivs(const int ex[3],
     }
 
     /*
-     * Fortran loops:
-     * do k=1,ex3-1
-     * do j=1,ex2-1
-     * do i=1,ex1-1
+     * 两段式：
+     * 1) 先在二阶可用区域计算二阶模板
+     * 2) 再在高阶可用区域覆盖为四阶模板
      *
-     * C: k0=0..ex3-2, j0=0..ex2-2, i0=0..ex1-2
+     * 与原 if/elseif 逻辑等价，但减少逐点分支判断。
      */
-    for (int k0 = 0; k0 <= ex3 - 2; ++k0) {
-        const int kF = k0 + 1;
-        for (int j0 = 0; j0 <= ex2 - 2; ++j0) {
-            const int jF = j0 + 1;
-            for (int i0 = 0; i0 <= ex1 - 2; ++i0) {
-                const int iF = i0 + 1;
-                const size_t p = idx_ex(i0, j0, k0, ex);
+    const int i2_lo = (iminF > 0) ? iminF : 0;
+    const int j2_lo = (jminF > 0) ? jminF : 0;
+    const int k2_lo = (kminF > 0) ? kminF : 0;
+    const int i2_hi = ex1 - 2;
+    const int j2_hi = ex2 - 2;
+    const int k2_hi = ex3 - 2;
+
+    const int i4_lo = (iminF + 1 > 0) ? (iminF + 1) : 0;
+    const int j4_lo = (jminF + 1 > 0) ? (jminF + 1) : 0;
+    const int k4_lo = (kminF + 1 > 0) ? (kminF + 1) : 0;
+    const int i4_hi = ex1 - 3;
+    const int j4_hi = ex2 - 3;
+    const int k4_hi = ex3 - 3;
+
+    if (i2_lo <= i2_hi && j2_lo <= j2_hi && k2_lo <= k2_hi) {
+        for (int k0 = k2_lo; k0 <= k2_hi; ++k0) {
+            const int kF = k0 + 1;
+            for (int j0 = j2_lo; j0 <= j2_hi; ++j0) {
+                const int jF = j0 + 1;
+                for (int i0 = i2_lo; i0 <= i2_hi; ++i0) {
+                    const int iF = i0 + 1;
+                    const size_t p = idx_ex(i0, j0, k0, ex);
+
+                    fx[p] = d2dx * (
+                        -fh[idx_fh_F_ord2(iF - 1, jF,     kF,     ex)] +
+                         fh[idx_fh_F_ord2(iF + 1, jF,     kF,     ex)]
+                    );
+
+                    fy[p] = d2dy * (
+                        -fh[idx_fh_F_ord2(iF,     jF - 1, kF,     ex)] +
+                         fh[idx_fh_F_ord2(iF,     jF + 1, kF,     ex)]
+                    );
+
+                    fz[p] = d2dz * (
+                        -fh[idx_fh_F_ord2(iF,     jF,     kF - 1, ex)] +
+                         fh[idx_fh_F_ord2(iF,     jF,     kF + 1, ex)]
+                    );
+                }
+            }
+        }
+    }
+
+    if (i4_lo <= i4_hi && j4_lo <= j4_hi && k4_lo <= k4_hi) {
+        for (int k0 = k4_lo; k0 <= k4_hi; ++k0) {
+            const int kF = k0 + 1;
+            for (int j0 = j4_lo; j0 <= j4_hi; ++j0) {
+                const int jF = j0 + 1;
+                for (int i0 = i4_lo; i0 <= i4_hi; ++i0) {
+                    const int iF = i0 + 1;
+                    const size_t p = idx_ex(i0, j0, k0, ex);
 
-                // if(i+2 <= imax .and. i-2 >= imin ... )  (全是 Fortran 索引)
-                if ((iF + 2) <= imaxF && (iF - 2) >= iminF &&
-                    (jF + 2) <= jmaxF && (jF - 2) >= jminF &&
-                    (kF + 2) <= kmaxF && (kF - 2) >= kminF)
-                {
                     fx[p] = d12dx * (
                         fh[idx_fh_F_ord2(iF - 2, jF,     kF,     ex)] -
                         EIT * fh[idx_fh_F_ord2(iF - 1, jF,     kF,     ex)] +
@@ -122,29 +159,9 @@ void fderivs(const int ex[3],
                         fh[idx_fh_F_ord2(iF,     jF,     kF + 2, ex)]
                     );
                 }
-                // elseif(i+1 <= imax .and. i-1 >= imin ...)
-                else if ((iF + 1) <= imaxF && (iF - 1) >= iminF &&
-                         (jF + 1) <= jmaxF && (jF - 1) >= jminF &&
-                         (kF + 1) <= kmaxF && (kF - 1) >= kminF)
-                {
-                    fx[p] = d2dx * (
-                        -fh[idx_fh_F_ord2(iF - 1, jF,     kF,     ex)] +
-                         fh[idx_fh_F_ord2(iF + 1, jF,     kF,     ex)]
-                    );
-
-                    fy[p] = d2dy * (
-                        -fh[idx_fh_F_ord2(iF,     jF - 1, kF,     ex)] +
-                         fh[idx_fh_F_ord2(iF,     jF + 1, kF,     ex)]
-                    );
-
-                    fz[p] = d2dz * (
-                        -fh[idx_fh_F_ord2(iF,     jF,     kF - 1, ex)] +
-                         fh[idx_fh_F_ord2(iF,     jF,     kF + 1, ex)]
-                    );
-                }
             }
         }
     }
 
     // free(fh);
-}
+}

AMSS_NCKU_source/fmisc.f90

@@ -1327,35 +1327,6 @@ end subroutine d2dump
 
   return
   end subroutine polint
-
-  subroutine polint0(xa, ya, y, ordn)
-  ! Lagrange interpolation at x=0, O(n) direct formula
-  implicit none
-  integer, intent(in) :: ordn
-  real*8, dimension(ordn), intent(in) :: xa, ya
-  real*8, intent(out) :: y
-
-  integer :: j, k
-  real*8 :: wj
-
-  y = 0.d0
-  do j = 1, ordn
-    wj = 1.d0
-    do k = 1, ordn
-      if (k .ne. j) then
-        wj = wj * xa(k) / (xa(k) - xa(j))
-      endif
-    enddo
-    y = y + wj * ya(j)
-  enddo
-
-  return
-  end subroutine polint0
-!------------------------------------------------------------------------------
-!
-! interpolation in 2 dimensions, follow yx order
-!
-!------------------------------------------------------------------------------
 !------------------------------------------------------------------------------
 ! Compute Lagrange interpolation basis weights for one target point.
 !------------------------------------------------------------------------------
@@ -1540,13 +1511,88 @@ deallocate(f_flat)
 
 f_out = f_out*dX*dY*dZ
 
-  return
-
-  end subroutine l2normhelper
-!--------------------------------------------------------------------------------------
-! calculate L2norm especially for shell Blocks
-  subroutine l2normhelper_sh(ex, X, Y, Z,xmin,ymin,zmin,xmax,ymax,zmax,&
-                          f,f_out,gw,ogw,Symmetry)
+  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,&
+                          f,f_out,gw,ogw,Symmetry)
 
   implicit none
 !~~~~~~> Input parameters:

AMSS_NCKU_source/fmisc.h

@@ -12,9 +12,10 @@
 #define f_global_interpind global_interpind
 #define f_global_interpind2d global_interpind2d
 #define f_global_interpind1d global_interpind1d
-#define f_l2normhelper l2normhelper
-#define f_l2normhelper_sh l2normhelper_sh
-#define f_l2normhelper_sh_rms l2normhelper_sh_rms
+#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
 #define f_average3 average3
 #define f_average2 average2
@@ -41,9 +42,10 @@
 #define f_global_interpind GLOBAL_INTERPIND
 #define f_global_interpind2d GLOBAL_INTERPIND2D
 #define f_global_interpind1d GLOBAL_INTERPIND1D
-#define f_l2normhelper L2NORMHELPER
-#define f_l2normhelper_sh L2NORMHELPER_SH
-#define f_l2normhelper_sh_rms L2NORMHELPER_SH_RMS
+#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
 #define f_average3 AVERAGE3
 #define f_average2 AVERAGE2
@@ -70,9 +72,10 @@
 #define f_global_interpind global_interpind_
 #define f_global_interpind2d global_interpind2d_
 #define f_global_interpind1d global_interpind1d_
-#define f_l2normhelper l2normhelper_
-#define f_l2normhelper_sh l2normhelper_sh_
-#define f_l2normhelper_sh_rms l2normhelper_sh_rms_
+#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_
 #define f_average3 average3_
 #define f_average2 average2_
@@ -156,20 +159,29 @@ extern "C"
 							  int *, double *, int &, int &);
 }
 
-extern "C"
-{
-	void f_l2normhelper(int *, double *, double *, double *,
-						double &, double &, double &,
-						double &, double &, double &,
-						double *, double &, int &);
-}
-
-extern "C"
-{
-	void f_l2normhelper_sh(int *, double *, double *, double *,
-						   double &, double &, double &,
-						   double &, double &, double &,
-						   double *, double &, int &, int &, int &);
+extern "C"
+{
+	void f_l2normhelper(int *, double *, double *, double *,
+						double &, double &, double &,
+						double &, double &, double &,
+						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 *,
+						   double &, double &, double &,
+						   double &, double &, double &,
+						   double *, double &, int &, int &, int &);
 }
 
 extern "C"

AMSS_NCKU_source/gaussj.C

@@ -17,68 +17,106 @@ using namespace std;
 #include <math.h>
 #endif
 
-// Intel oneMKL LAPACK interface
-#include <mkl_lapacke.h>
-/* Linear equation solution using Intel oneMKL LAPACK.
-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. */
-
-int gaussj(double *a, double *b, int n)
-{
-  // Allocate pivot array and workspace
-  lapack_int *ipiv = new lapack_int[n];
-  lapack_int info;
-
-  // 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];
-  }
-
-  // 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;
-  }
-
-  delete[] ipiv;
-  delete[] a_copy;
-
-  return 0;
-}
+/* 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. */
+
+int gaussj(double *a, double *b, int n)
+{
+  double swap;
+
+  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;
+      }
+  }
+
+  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;
+
+  return 0;
+}
 // for check usage
 /*
 int main()

AMSS_NCKU_source/interp_lb_profile_data.h

@@ -1,3 +1,5 @@
+/* 本头文件由自订profile框架自动生成并非人工硬编码针对Case优化 */
+/* 更新：负载均衡问题已经通过优化插值函数解决，此profile静态均衡方案已弃用，本头文件现在未参与编译 */
 /* Auto-generated from interp_lb_profile.bin — do not edit */
 #ifndef INTERP_LB_PROFILE_DATA_H
 #define INTERP_LB_PROFILE_DATA_H

AMSS_NCKU_source/kodiss_c.C

@@ -63,19 +63,28 @@ void kodis(const int ex[3],
      * C: k0=0..ex3-1, j0=0..ex2-1, i0=0..ex1-1
      * 并定义 Fortran index: iF=i0+1, ...
      */
-    for (int k0 = 0; k0 < ex3; ++k0) {
+    // 收紧循环范围：只遍历满足 iF±3/jF±3/kF±3 条件的内部点
+    // iF-3 >= iminF => iF >= iminF+3 => i0 >= iminF+2 (因为 iF=i0+1)
+    // iF+3 <= imaxF => iF <= imaxF-3 => i0 <= imaxF-4
+    const int i0_lo = (iminF + 2 > 0) ? iminF + 2 : 0;
+    const int j0_lo = (jminF + 2 > 0) ? jminF + 2 : 0;
+    const int k0_lo = (kminF + 2 > 0) ? kminF + 2 : 0;
+    const int i0_hi = imaxF - 4;  // inclusive
+    const int j0_hi = jmaxF - 4;
+    const int k0_hi = kmaxF - 4;
+
+    if (i0_lo > i0_hi || j0_lo > j0_hi || k0_lo > k0_hi) {
+        free(fh);
+        return;
+    }
+
+    for (int k0 = k0_lo; k0 <= k0_hi; ++k0) {
         const int kF = k0 + 1;
-        for (int j0 = 0; j0 < ex2; ++j0) {
+        for (int j0 = j0_lo; j0 <= j0_hi; ++j0) {
             const int jF = j0 + 1;
-            for (int i0 = 0; i0 < ex1; ++i0) {
+            for (int i0 = i0_lo; i0 <= i0_hi; ++i0) {
                 const int iF = i0 + 1;
 
-                // Fortran if 条件：
-                // i-3 >= imin .and. i+3 <= imax  等（都是 Fortran 索引）
-                if ((iF - 3) >= iminF && (iF + 3) <= imaxF &&
-                    (jF - 3) >= jminF && (jF + 3) <= jmaxF &&
-                    (kF - 3) >= kminF && (kF + 3) <= kmaxF)
-                {
                     const size_t p = idx_ex(i0, j0, k0, ex);
 
                     // 三个方向各一份同型的 7 点组合（实际上是对称的 6th-order dissipation/filter 核）
@@ -100,7 +109,6 @@ void kodis(const int ex[3],
                     // Fortran:
                     // f_rhs(i,j,k) = f_rhs(i,j,k) + eps/cof*(Dx_term + Dy_term + Dz_term)
                     f_rhs[p] += (eps / cof) * (Dx_term + Dy_term + Dz_term);
-                }
             }
         }
     }

AMSS_NCKU_source/lopsided_kodis_c.C

@@ -0,0 +1,248 @@
+#include "tool.h"
+
+/*
+ * Combined advection (lopsided) + KO dissipation (kodis).
+ * Uses one shared symmetry_bd buffer per call.
+ */
+void lopsided_kodis(const int ex[3],
+                    const double *X, const double *Y, const double *Z,
+                    const double *f, double *f_rhs,
+                    const double *Sfx, const double *Sfy, const double *Sfz,
+                    int Symmetry, const double SoA[3], double eps)
+{
+    const double ZEO = 0.0, ONE = 1.0, F3 = 3.0;
+    const double F6 = 6.0, F18 = 18.0;
+    const double F12 = 12.0, F10 = 10.0, EIT = 8.0;
+    const double SIX = 6.0, FIT = 15.0, TWT = 20.0;
+    const double cof = 64.0; // 2^6
+
+    const int NO_SYMM = 0, EQ_SYMM = 1;
+
+    const int ex1 = ex[0], ex2 = ex[1], ex3 = ex[2];
+
+    const double dX = X[1] - X[0];
+    const double dY = Y[1] - Y[0];
+    const double dZ = Z[1] - Z[0];
+
+    const double d12dx = ONE / F12 / dX;
+    const double d12dy = ONE / F12 / dY;
+    const double d12dz = ONE / F12 / dZ;
+
+    const int imaxF = ex1;
+    const int jmaxF = ex2;
+    const int kmaxF = ex3;
+
+    int iminF = 1, jminF = 1, kminF = 1;
+    if (Symmetry > NO_SYMM && fabs(Z[0]) < dZ) kminF = -2;
+    if (Symmetry > EQ_SYMM && fabs(X[0]) < dX) iminF = -2;
+    if (Symmetry > EQ_SYMM && fabs(Y[0]) < dY) jminF = -2;
+
+    // fh for Fortran-style domain (-2:ex1,-2:ex2,-2:ex3)
+    const size_t nx = (size_t)ex1 + 3;
+    const size_t ny = (size_t)ex2 + 3;
+    const size_t nz = (size_t)ex3 + 3;
+    const size_t fh_size = nx * ny * nz;
+
+    double *fh = (double*)malloc(fh_size * sizeof(double));
+    if (!fh) return;
+
+    symmetry_bd(3, ex, f, fh, SoA);
+
+    // Advection (same stencil logic as lopsided_c.C)
+    for (int k0 = 0; k0 <= ex3 - 2; ++k0) {
+        const int kF = k0 + 1;
+        for (int j0 = 0; j0 <= ex2 - 2; ++j0) {
+            const int jF = j0 + 1;
+            for (int i0 = 0; i0 <= ex1 - 2; ++i0) {
+                const int iF = i0 + 1;
+                const size_t p = idx_ex(i0, j0, k0, ex);
+
+                const double sfx = Sfx[p];
+                if (sfx > ZEO) {
+                    if (i0 <= ex1 - 4) {
+                        f_rhs[p] += sfx * d12dx *
+                            (-F3  * fh[idx_fh_F(iF - 1, jF, kF, ex)]
+                             -F10 * fh[idx_fh_F(iF    , jF, kF, ex)]
+                             +F18 * fh[idx_fh_F(iF + 1, jF, kF, ex)]
+                             -F6  * fh[idx_fh_F(iF + 2, jF, kF, ex)]
+                             +      fh[idx_fh_F(iF + 3, jF, kF, ex)]);
+                    } else if (i0 <= ex1 - 3) {
+                        f_rhs[p] += sfx * d12dx *
+                            ( fh[idx_fh_F(iF - 2, jF, kF, ex)]
+                             -EIT * fh[idx_fh_F(iF - 1, jF, kF, ex)]
+                             +EIT * fh[idx_fh_F(iF + 1, jF, kF, ex)]
+                             -      fh[idx_fh_F(iF + 2, jF, kF, ex)]);
+                    } else if (i0 <= ex1 - 2) {
+                        f_rhs[p] -= sfx * d12dx *
+                            (-F3  * fh[idx_fh_F(iF + 1, jF, kF, ex)]
+                             -F10 * fh[idx_fh_F(iF    , jF, kF, ex)]
+                             +F18 * fh[idx_fh_F(iF - 1, jF, kF, ex)]
+                             -F6  * fh[idx_fh_F(iF - 2, jF, kF, ex)]
+                             +      fh[idx_fh_F(iF - 3, jF, kF, ex)]);
+                    }
+                } else if (sfx < ZEO) {
+                    if ((i0 - 2) >= iminF) {
+                        f_rhs[p] -= sfx * d12dx *
+                            (-F3  * fh[idx_fh_F(iF + 1, jF, kF, ex)]
+                             -F10 * fh[idx_fh_F(iF    , jF, kF, ex)]
+                             +F18 * fh[idx_fh_F(iF - 1, jF, kF, ex)]
+                             -F6  * fh[idx_fh_F(iF - 2, jF, kF, ex)]
+                             +      fh[idx_fh_F(iF - 3, jF, kF, ex)]);
+                    } else if ((i0 - 1) >= iminF) {
+                        f_rhs[p] += sfx * d12dx *
+                            ( fh[idx_fh_F(iF - 2, jF, kF, ex)]
+                             -EIT * fh[idx_fh_F(iF - 1, jF, kF, ex)]
+                             +EIT * fh[idx_fh_F(iF + 1, jF, kF, ex)]
+                             -      fh[idx_fh_F(iF + 2, jF, kF, ex)]);
+                    } else if (i0 >= iminF) {
+                        f_rhs[p] += sfx * d12dx *
+                            (-F3  * fh[idx_fh_F(iF - 1, jF, kF, ex)]
+                             -F10 * fh[idx_fh_F(iF    , jF, kF, ex)]
+                             +F18 * fh[idx_fh_F(iF + 1, jF, kF, ex)]
+                             -F6  * fh[idx_fh_F(iF + 2, jF, kF, ex)]
+                             +      fh[idx_fh_F(iF + 3, jF, kF, ex)]);
+                    }
+                }
+
+                const double sfy = Sfy[p];
+                if (sfy > ZEO) {
+                    if (j0 <= ex2 - 4) {
+                        f_rhs[p] += sfy * d12dy *
+                            (-F3  * fh[idx_fh_F(iF, jF - 1, kF, ex)]
+                             -F10 * fh[idx_fh_F(iF, jF    , kF, ex)]
+                             +F18 * fh[idx_fh_F(iF, jF + 1, kF, ex)]
+                             -F6  * fh[idx_fh_F(iF, jF + 2, kF, ex)]
+                             +      fh[idx_fh_F(iF, jF + 3, kF, ex)]);
+                    } else if (j0 <= ex2 - 3) {
+                        f_rhs[p] += sfy * d12dy *
+                            ( fh[idx_fh_F(iF, jF - 2, kF, ex)]
+                             -EIT * fh[idx_fh_F(iF, jF - 1, kF, ex)]
+                             +EIT * fh[idx_fh_F(iF, jF + 1, kF, ex)]
+                             -      fh[idx_fh_F(iF, jF + 2, kF, ex)]);
+                    } else if (j0 <= ex2 - 2) {
+                        f_rhs[p] -= sfy * d12dy *
+                            (-F3  * fh[idx_fh_F(iF, jF + 1, kF, ex)]
+                             -F10 * fh[idx_fh_F(iF, jF    , kF, ex)]
+                             +F18 * fh[idx_fh_F(iF, jF - 1, kF, ex)]
+                             -F6  * fh[idx_fh_F(iF, jF - 2, kF, ex)]
+                             +      fh[idx_fh_F(iF, jF - 3, kF, ex)]);
+                    }
+                } else if (sfy < ZEO) {
+                    if ((j0 - 2) >= jminF) {
+                        f_rhs[p] -= sfy * d12dy *
+                            (-F3  * fh[idx_fh_F(iF, jF + 1, kF, ex)]
+                             -F10 * fh[idx_fh_F(iF, jF    , kF, ex)]
+                             +F18 * fh[idx_fh_F(iF, jF - 1, kF, ex)]
+                             -F6  * fh[idx_fh_F(iF, jF - 2, kF, ex)]
+                             +      fh[idx_fh_F(iF, jF - 3, kF, ex)]);
+                    } else if ((j0 - 1) >= jminF) {
+                        f_rhs[p] += sfy * d12dy *
+                            ( fh[idx_fh_F(iF, jF - 2, kF, ex)]
+                             -EIT * fh[idx_fh_F(iF, jF - 1, kF, ex)]
+                             +EIT * fh[idx_fh_F(iF, jF + 1, kF, ex)]
+                             -      fh[idx_fh_F(iF, jF + 2, kF, ex)]);
+                    } else if (j0 >= jminF) {
+                        f_rhs[p] += sfy * d12dy *
+                            (-F3  * fh[idx_fh_F(iF, jF - 1, kF, ex)]
+                             -F10 * fh[idx_fh_F(iF, jF    , kF, ex)]
+                             +F18 * fh[idx_fh_F(iF, jF + 1, kF, ex)]
+                             -F6  * fh[idx_fh_F(iF, jF + 2, kF, ex)]
+                             +      fh[idx_fh_F(iF, jF + 3, kF, ex)]);
+                    }
+                }
+
+                const double sfz = Sfz[p];
+                if (sfz > ZEO) {
+                    if (k0 <= ex3 - 4) {
+                        f_rhs[p] += sfz * d12dz *
+                            (-F3  * fh[idx_fh_F(iF, jF, kF - 1, ex)]
+                             -F10 * fh[idx_fh_F(iF, jF, kF    , ex)]
+                             +F18 * fh[idx_fh_F(iF, jF, kF + 1, ex)]
+                             -F6  * fh[idx_fh_F(iF, jF, kF + 2, ex)]
+                             +      fh[idx_fh_F(iF, jF, kF + 3, ex)]);
+                    } else if (k0 <= ex3 - 3) {
+                        f_rhs[p] += sfz * d12dz *
+                            ( fh[idx_fh_F(iF, jF, kF - 2, ex)]
+                             -EIT * fh[idx_fh_F(iF, jF, kF - 1, ex)]
+                             +EIT * fh[idx_fh_F(iF, jF, kF + 1, ex)]
+                             -      fh[idx_fh_F(iF, jF, kF + 2, ex)]);
+                    } else if (k0 <= ex3 - 2) {
+                        f_rhs[p] -= sfz * d12dz *
+                            (-F3  * fh[idx_fh_F(iF, jF, kF + 1, ex)]
+                             -F10 * fh[idx_fh_F(iF, jF, kF    , ex)]
+                             +F18 * fh[idx_fh_F(iF, jF, kF - 1, ex)]
+                             -F6  * fh[idx_fh_F(iF, jF, kF - 2, ex)]
+                             +      fh[idx_fh_F(iF, jF, kF - 3, ex)]);
+                    }
+                } else if (sfz < ZEO) {
+                    if ((k0 - 2) >= kminF) {
+                        f_rhs[p] -= sfz * d12dz *
+                            (-F3  * fh[idx_fh_F(iF, jF, kF + 1, ex)]
+                             -F10 * fh[idx_fh_F(iF, jF, kF    , ex)]
+                             +F18 * fh[idx_fh_F(iF, jF, kF - 1, ex)]
+                             -F6  * fh[idx_fh_F(iF, jF, kF - 2, ex)]
+                             +      fh[idx_fh_F(iF, jF, kF - 3, ex)]);
+                    } else if ((k0 - 1) >= kminF) {
+                        f_rhs[p] += sfz * d12dz *
+                            ( fh[idx_fh_F(iF, jF, kF - 2, ex)]
+                             -EIT * fh[idx_fh_F(iF, jF, kF - 1, ex)]
+                             +EIT * fh[idx_fh_F(iF, jF, kF + 1, ex)]
+                             -      fh[idx_fh_F(iF, jF, kF + 2, ex)]);
+                    } else if (k0 >= kminF) {
+                        f_rhs[p] += sfz * d12dz *
+                            (-F3  * fh[idx_fh_F(iF, jF, kF - 1, ex)]
+                             -F10 * fh[idx_fh_F(iF, jF, kF    , ex)]
+                             +F18 * fh[idx_fh_F(iF, jF, kF + 1, ex)]
+                             -F6  * fh[idx_fh_F(iF, jF, kF + 2, ex)]
+                             +      fh[idx_fh_F(iF, jF, kF + 3, ex)]);
+                    }
+                }
+            }
+        }
+    }
+
+    // KO dissipation (same domain restriction as kodiss_c.C)
+    if (eps > ZEO) {
+        const int i0_lo = (iminF + 2 > 0) ? iminF + 2 : 0;
+        const int j0_lo = (jminF + 2 > 0) ? jminF + 2 : 0;
+        const int k0_lo = (kminF + 2 > 0) ? kminF + 2 : 0;
+        const int i0_hi = imaxF - 4; // inclusive
+        const int j0_hi = jmaxF - 4;
+        const int k0_hi = kmaxF - 4;
+
+        if (!(i0_lo > i0_hi || j0_lo > j0_hi || k0_lo > k0_hi)) {
+            for (int k0 = k0_lo; k0 <= k0_hi; ++k0) {
+                const int kF = k0 + 1;
+                for (int j0 = j0_lo; j0 <= j0_hi; ++j0) {
+                    const int jF = j0 + 1;
+                    for (int i0 = i0_lo; i0 <= i0_hi; ++i0) {
+                        const int iF = i0 + 1;
+                        const size_t p = idx_ex(i0, j0, k0, ex);
+
+                        const double Dx_term =
+                            ((fh[idx_fh_F(iF - 3, jF, kF, ex)] + fh[idx_fh_F(iF + 3, jF, kF, ex)]) -
+                             SIX * (fh[idx_fh_F(iF - 2, jF, kF, ex)] + fh[idx_fh_F(iF + 2, jF, kF, ex)]) +
+                             FIT * (fh[idx_fh_F(iF - 1, jF, kF, ex)] + fh[idx_fh_F(iF + 1, jF, kF, ex)]) -
+                             TWT *  fh[idx_fh_F(iF,     jF, kF, ex)]) / dX;
+
+                        const double Dy_term =
+                            ((fh[idx_fh_F(iF, jF - 3, kF, ex)] + fh[idx_fh_F(iF, jF + 3, kF, ex)]) -
+                             SIX * (fh[idx_fh_F(iF, jF - 2, kF, ex)] + fh[idx_fh_F(iF, jF + 2, kF, ex)]) +
+                             FIT * (fh[idx_fh_F(iF, jF - 1, kF, ex)] + fh[idx_fh_F(iF, jF + 1, kF, ex)]) -
+                             TWT *  fh[idx_fh_F(iF, jF,     kF, ex)]) / dY;
+
+                        const double Dz_term =
+                            ((fh[idx_fh_F(iF, jF, kF - 3, ex)] + fh[idx_fh_F(iF, jF, kF + 3, ex)]) -
+                             SIX * (fh[idx_fh_F(iF, jF, kF - 2, ex)] + fh[idx_fh_F(iF, jF, kF + 2, ex)]) +
+                             FIT * (fh[idx_fh_F(iF, jF, kF - 1, ex)] + fh[idx_fh_F(iF, jF, kF + 1, ex)]) -
+                             TWT *  fh[idx_fh_F(iF, jF, kF,     ex)]) / dZ;
+
+                        f_rhs[p] += (eps / cof) * (Dx_term + Dy_term + Dz_term);
+                    }
+                }
+            }
+        }
+    }
+
+    free(fh);
+}

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

@@ -1,33 +1,24 @@
 
 
-include makefile.inc
-
-## polint(ordn=6) kernel selector:
-##   1 (default): barycentric fast path
-##   0          : fallback to Neville path
-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
-
-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 $(POLINT6_FLAG)
-else
-## opt (default): maximum performance with PGO profile data
-CXXAPPFLAGS = -O3 -xHost -fp-model fast=2 -fma -ipo \
-              -fprofile-instr-use=$(PROFDATA) \
-              -Dfortran3 -Dnewc -I${MKLROOT}/include $(INTERP_LB_FLAGS)
-f90appflags = -O3 -xHost -fp-model fast=2 -fma -ipo \
-              -fprofile-instr-use=$(PROFDATA) \
-              -align array64byte -fpp -I${MKLROOT}/include $(POLINT6_FLAG)
-endif
+include makefile.inc
+
+## polint(ordn=6) kernel selector:
+##   1 (default): barycentric fast path
+##   0          : fallback to Neville path
+POLINT6_USE_BARY ?= 1
+POLINT6_FLAG = -DPOLINT6_USE_BARYCENTRIC=$(POLINT6_USE_BARY)
+
+## 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)
+CXXAPPFLAGS = $(CXXBASEFLAGS)
+f90appflags = $(F90BASEFLAGS)
+else
+CXXAPPFLAGS = $(CXXBASEFLAGS)
+f90appflags = $(F90BASEFLAGS)
+endif
 
 .SUFFIXES: .o .f90 .C .for .cu
 
@@ -43,10 +34,6 @@ endif
 .cu.o:
 	$(Cu) $(CUDA_APP_FLAGS) -c $< -o $@ $(CUDA_LIB_PATH)
 
-# CUDA rewrite of BSSN RHS (drop-in replacement for bssn_rhs_c + stencil helpers)
-bssn_rhs_cuda.o: bssn_rhs_cuda.cu macrodef.h
-	$(Cu) $(CUDA_APP_FLAGS) -c $< -o $@ $(CUDA_LIB_PATH)
-
 # C rewrite of BSSN RHS kernel and helpers
 bssn_rhs_c.o: bssn_rhs_c.C
 	${CXX} $(CXXAPPFLAGS) -c $< $(filein) -o $@
@@ -60,46 +47,42 @@ fdderivs_c.o: fdderivs_c.C
 kodiss_c.o: kodiss_c.C
 	${CXX} $(CXXAPPFLAGS) -c $< $(filein) -o $@
 
-lopsided_c.o: lopsided_c.C
-	${CXX} $(CXXAPPFLAGS) -c $< $(filein) -o $@
+lopsided_c.o: lopsided_c.C
+	${CXX} $(CXXAPPFLAGS) -c $< $(filein) -o $@
+
+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
-
-TwoPunctures.o: TwoPunctures.C
-	${CXX} $(TP_OPTFLAGS) -qopenmp -c $< -o $@
-
-TwoPunctureABE.o: TwoPunctureABE.C
-	${CXX} $(TP_OPTFLAGS) -qopenmp -c $< -o $@
+## TwoPunctureABE uses fixed optimal flags with its own PGO profile, independent of CXXAPPFLAGS
+TP_OPTFLAGS = $(CXXBASEFLAGS) $(TP_OPENMP_FLAGS)
+
+TwoPunctures.o: TwoPunctures.C
+	${CXX} $(TP_OPTFLAGS) -c $< -o $@
+
+TwoPunctureABE.o: TwoPunctureABE.C
+	${CXX} $(TP_OPTFLAGS) -c $< -o $@
 
 # Input files
 
 ## Kernel implementation switch (set USE_CXX_KERNELS=0 to fall back to Fortran)
-ifeq ($(USE_CXX_KERNELS),0)
-# Fortran mode: no C rewrite files; bssn_rhs.o is included via F90FILES below
-CFILES =
-else
-# C++ mode (default): C rewrite of bssn_rhs and helper kernels
-CFILES = bssn_rhs_c.o fderivs_c.o fdderivs_c.o kodiss_c.o lopsided_c.o
-endif
-
-# CUDA rewrite: bssn_rhs_cuda.o replaces all CFILES (stencils are built-in)
-CFILES_CUDA = bssn_rhs_cuda.o
-
-## RK4 kernel switch (independent from USE_CXX_KERNELS)
-ifeq ($(USE_CXX_RK4),1)
-CFILES += rungekutta4_rout_c.o
-CFILES_CUDA += rungekutta4_rout_c.o
-RK4_F90_OBJ =
-else
-RK4_F90_OBJ = rungekutta4_rout.o
-endif
+ifeq ($(USE_CXX_KERNELS),0)
+# Fortran mode: no C rewrite files; bssn_rhs.o is included via F90FILES below
+CFILES =
+else
+# C++ mode (default): C rewrite of bssn_rhs and helper kernels
+CFILES = bssn_rhs_c.o fderivs_c.o fdderivs_c.o kodiss_c.o lopsided_c.o lopsided_kodis_c.o
+endif
+
+## RK4 kernel switch (independent from USE_CXX_KERNELS)
+ifeq ($(USE_CXX_RK4),1)
+CFILES += rungekutta4_rout_c.o
+RK4_F90_OBJ =
+else
+RK4_F90_OBJ = rungekutta4_rout.o
+endif
 
 C++FILES = ABE.o Ansorg.o Block.o misc.o monitor.o Parallel.o MPatch.o var.o\
            cgh.o bssn_class.o surface_integral.o ShellPatch.o\
@@ -116,12 +99,12 @@ C++FILES_GPU = ABE.o Ansorg.o Block.o misc.o monitor.o Parallel.o MPatch.o var.o
 	   NullShellPatch2_Evo.o \
 	   bssn_gpu_class.o bssn_step_gpu.o bssn_macro.o writefile_f.o
 
-F90FILES_BASE = enforce_algebra.o fmisc.o initial_puncture.o prolongrestrict.o\
-	   prolongrestrict_cell.o prolongrestrict_vertex.o\
-	   $(RK4_F90_OBJ) diff_new.o kodiss.o kodiss_sh.o\
-	   lopsidediff.o sommerfeld_rout.o getnp4.o diff_new_sh.o\
-	   shellfunctions.o bssn_rhs_ss.o Set_Rho_ADM.o\
-           getnp4EScalar.o bssnEScalar_rhs.o bssn_constraint.o ricci_gamma.o\
+F90FILES_BASE = enforce_algebra.o fmisc.o initial_puncture.o prolongrestrict.o\
+	   prolongrestrict_cell.o prolongrestrict_vertex.o\
+	   $(RK4_F90_OBJ) diff_new.o kodiss.o kodiss_sh.o\
+	   lopsidediff.o sommerfeld_rout.o getnp4.o diff_new_sh.o\
+	   shellfunctions.o bssn_rhs_ss.o Set_Rho_ADM.o\
+           getnp4EScalar.o bssnEScalar_rhs.o bssn_constraint.o ricci_gamma.o\
            fadmquantites_bssn.o Z4c_rhs.o Z4c_rhs_ss.o point_diff_new_sh.o\
 	   cpbc.o getnp4old.o NullEvol.o initial_null.o initial_maxwell.o\
 	   getnpem2.o empart.o NullNews.o fourdcurvature.o\
@@ -181,17 +164,14 @@ $(CUDAFILES): bssn_gpu.h gpu_mem.h gpu_rhsSS_mem.h
 misc.o : zbesh.o
 
 # projects
-ABE: $(C++FILES) $(CFILES_CUDA) $(F90FILES) $(F77FILES) $(AHFDOBJS)
-	$(CLINKER) $(CXXAPPFLAGS) -o $@ $(C++FILES) $(CFILES_CUDA) $(F90FILES) $(F77FILES) $(AHFDOBJS) $(LDLIBS) -lcudart $(CUDA_LIB_PATH)
-
-ABE_CUDA: $(C++FILES) $(CFILES_CUDA) $(F90FILES) $(F77FILES) $(AHFDOBJS)
-	$(CLINKER) $(CXXAPPFLAGS) -o $@ $(C++FILES) $(CFILES_CUDA) $(F90FILES) $(F77FILES) $(AHFDOBJS) $(LDLIBS) -lcudart $(CUDA_LIB_PATH)
-
+ABE: $(C++FILES) $(CFILES) $(F90FILES) $(F77FILES) $(AHFDOBJS)
+	$(CLINKER) $(CXXAPPFLAGS) -o $@ $(C++FILES) $(CFILES) $(F90FILES) $(F77FILES) $(AHFDOBJS) $(LDLIBS)
+	
 ABEGPU: $(C++FILES_GPU) $(CFILES) $(F90FILES) $(F77FILES) $(AHFDOBJS) $(CUDAFILES)
 	$(CLINKER) $(CXXAPPFLAGS) -o $@ $(C++FILES_GPU) $(CFILES) $(F90FILES) $(F77FILES) $(AHFDOBJS) $(CUDAFILES) $(LDLIBS)
 
-TwoPunctureABE: $(TwoPunctureFILES)
-	$(CLINKER) $(TP_OPTFLAGS) -qopenmp -o $@ $(TwoPunctureFILES) $(LDLIBS)
+TwoPunctureABE: $(TwoPunctureFILES)
+	$(CLINKER) $(TP_OPTFLAGS) -o $@ $(TwoPunctureFILES) $(LDLIBS)
 
 clean:
-	rm *.o ABE ABE_CUDA ABEGPU TwoPunctureABE make.log -f
+	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 -arch=sm_80
+CUDA_APP_FLAGS = -c -g -O3 --ptxas-options=-v -Dfortran3 -Dnewc

AMSS_NCKU_source/prolongrestrict_cell.f90

@@ -217,6 +217,7 @@
   real*8,dimension(2*ghost_width) :: X,Y,Z
   real*8, dimension(2*ghost_width,2*ghost_width) :: tmp2
   real*8, dimension(2*ghost_width) :: tmp1
+  real*8 :: ddy
   real*8,dimension(3) :: ccp
 
 #if (ghost_width == 2)
@@ -579,7 +580,7 @@
             tmp1(ghost_width-cxI(1)+cxB(1)  :ghost_width-cxI(1)+cxT(1)  ) = funf(cxB(1):cxT(1),j,k)
          endif
 
-         call polint0(X,tmp1,funf(i,j,k),2*ghost_width)
+         call polint(X,tmp1,0.d0,funf(i,j,k),ddy,2*ghost_width)
 
 ! for y direction
        elseif(sum(fg).eq.2.and.fg(2) .eq. 0.and. &
@@ -689,7 +690,7 @@
             tmp1(ghost_width-cxI(2)+cxB(2)  :ghost_width-cxI(2)+cxT(2)  ) = funf(i,cxB(2):cxT(2),k)
          endif
 
-         call polint0(Y,tmp1,funf(i,j,k),2*ghost_width)
+         call polint(Y,tmp1,0.d0,funf(i,j,k),ddy,2*ghost_width)
 
 ! for z direction
        elseif(sum(fg).eq.2.and.fg(3) .eq. 0.and. &
@@ -801,7 +802,7 @@
             tmp1(ghost_width-cxI(3)+cxB(3)  :ghost_width-cxI(3)+cxT(3)  ) = funf(i,j,cxB(3):cxT(3))
          endif
 
-         call polint0(Z,tmp1,funf(i,j,k),2*ghost_width)
+         call polint(Z,tmp1,0.d0,funf(i,j,k),ddy,2*ghost_width)
 
 #else
 

AMSS_NCKU_source/prolongrestrict_vertex.f90

@@ -217,6 +217,7 @@
   real*8,dimension(2*ghost_width) :: X,Y,Z
   real*8, dimension(2*ghost_width,2*ghost_width) :: tmp2
   real*8, dimension(2*ghost_width) :: tmp1
+  real*8 :: ddy
 
 #if (ghost_width == 2)
   real*8, parameter :: C1=-1.d0/16,C2=9.d0/16
@@ -469,7 +470,7 @@
 
          tmp1(cxB(1)+ghost_width-i+1:cxT(1)+ghost_width-i+1) = fh(cxB(1):cxT(1),j,k)
 
-         call polint0(X,tmp1,funf(i,j,k),2*ghost_width)
+         call polint(X,tmp1,0.d0,funf(i,j,k),ddy,2*ghost_width)
 
 ! for y direction
        elseif (fg(2) .eq. 0)then
@@ -528,7 +529,7 @@
 
          tmp1(cxB(2)+ghost_width-j+1:cxT(2)+ghost_width-j+1) = fh(i,cxB(2):cxT(2),k)
 
-         call polint0(Y,tmp1,funf(i,j,k),2*ghost_width)
+         call polint(Y,tmp1,0.d0,funf(i,j,k),ddy,2*ghost_width)
 
 ! for z direction
        else
@@ -587,7 +588,7 @@
 
          tmp1(cxB(3)+ghost_width-k+1:cxT(3)+ghost_width-k+1) = fh(i,j,cxB(3):cxT(3))
 
-         call polint0(Z,tmp1,funf(i,j,k),2*ghost_width)
+         call polint(Z,tmp1,0.d0,funf(i,j,k),ddy,2*ghost_width)
 
        endif
       

AMSS_NCKU_source/rungekutta4_rout_c.C

@@ -2,6 +2,7 @@
 #include <cstdio>
 #include <cstdlib>
 #include <cstddef>
+#include <complex>
 #include <immintrin.h>
 
 namespace {
@@ -117,6 +118,62 @@ inline void rk4_stage3(std::size_t n,
 
 extern "C" {
 
+void f_rungekutta4_scalar(double &dT, double &f0, double &f1, double &f_rhs, int &RK4) {
+    constexpr double F1o6 = 1.0 / 6.0;
+    constexpr double HLF = 0.5;
+    constexpr double TWO = 2.0;
+
+    switch (RK4) {
+    case 0:
+        f1 = f0 + HLF * dT * f_rhs;
+        break;
+    case 1:
+        f_rhs = f_rhs + TWO * f1;
+        f1 = f0 + HLF * dT * f1;
+        break;
+    case 2:
+        f_rhs = f_rhs + TWO * f1;
+        f1 = f0 + dT * f1;
+        break;
+    case 3:
+        f1 = f0 + F1o6 * dT * (f1 + f_rhs);
+        break;
+    default:
+        std::fprintf(stderr, "rungekutta4_scalar_c: invalid RK4 stage %d\n", RK4);
+        std::abort();
+    }
+}
+
+void rungekutta4_cplxscalar_(double &dT,
+                             std::complex<double> &f0,
+                             std::complex<double> &f1,
+                             std::complex<double> &f_rhs,
+                             int &RK4) {
+    constexpr double F1o6 = 1.0 / 6.0;
+    constexpr double HLF = 0.5;
+    constexpr double TWO = 2.0;
+
+    switch (RK4) {
+    case 0:
+        f1 = f0 + HLF * dT * f_rhs;
+        break;
+    case 1:
+        f_rhs = f_rhs + TWO * f1;
+        f1 = f0 + HLF * dT * f1;
+        break;
+    case 2:
+        f_rhs = f_rhs + TWO * f1;
+        f1 = f0 + dT * f1;
+        break;
+    case 3:
+        f1 = f0 + F1o6 * dT * (f1 + f_rhs);
+        break;
+    default:
+        std::fprintf(stderr, "rungekutta4_cplxscalar_c: invalid RK4 stage %d\n", RK4);
+        std::abort();
+    }
+}
+
 int f_rungekutta4_rout(int *ex, double &dT,
                        double *f0, double *f1, double *f_rhs,
                        int &RK4) {

AMSS_NCKU_source/surface_integral.h

@@ -27,19 +27,24 @@ using namespace std;
 class surface_integral
 {
 
-private:
-	int Symmetry, factor;
-	int N_theta, N_phi; // Number of points in Theta & Phi directions
-	double dphi, dcostheta;
-	double *arcostheta, *wtcostheta;
-	int n_tot; // size of arrays
-
-	double *nx_g, *ny_g, *nz_g; // global list of unit normals
-	int myrank, cpusize;
-
-public:
-	surface_integral(int iSymmetry);
-	~surface_integral();
+private:
+	int Symmetry, factor;
+	int N_theta, N_phi; // Number of points in Theta & Phi directions
+	double dphi, dcostheta;
+	double *arcostheta, *wtcostheta;
+	int n_tot; // size of arrays
+
+	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);
+	~surface_integral();
 
 	void surf_Wave(double rex, int lev, cgh *GH, var *Rpsi4, var *Ipsi4,
 				   int spinw, int maxl, int NN, double *RP, double *IP,
@@ -77,21 +82,37 @@ public:
 								 double &, double &, double &, double &, double &, double &, double &,
 								 double &, double &, double &, double &, double &, double &,
 								 double &, double &)); // NN is the length of RP and IP
-	void surf_MassPAng(double rex, int lev, cgh *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);
-	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);
-	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,
+	void surf_MassPAng(double rex, int lev, cgh *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, 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, 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,
 				   var *Axx, var *Axy, var *Axz, var *Ayy, var *Ayz, var *Azz,
 				   var *chix, var *chiy, var *chiz,
 				   var *trKx, var *trKy, var *trKz,
@@ -110,12 +131,12 @@ public:
 	bool SR_Interp_Points(MyList<var> *VarList, cgh *GH, ShellPatch *SH,
 						  int NN, double **XX, double *Shellf);
 
-	void surf_MassPAng(double rex, int lev, cgh *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, // temparay memory for mass^i
-					   double *Rout, monitor *Monitor, MPI_Comm Comm_here);
+	void surf_MassPAng(double rex, int lev, cgh *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, // temparay memory for mass^i
+					   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);

AMSS_NCKU_source/tool.h

@@ -24,4 +24,10 @@ void lopsided(const int ex[3],
               const double *X, const double *Y, const double *Z,
               const double *f, double *f_rhs,
               const double *Sfx, const double *Sfy, const double *Sfz,
-              int Symmetry, const double SoA[3]);
+              int Symmetry, const double SoA[3]);
+
+void lopsided_kodis(const int ex[3],
+                    const double *X, const double *Y, const double *Z,
+                    const double *f, double *f_rhs,
+                    const double *Sfx, const double *Sfy, const double *Sfz,
+                    int Symmetry, const double SoA[3], double eps);

README.md

@@ -93,11 +93,13 @@ Here, we take the Ubuntu 22.04 system as an example
 
 #### How to use AMSS-NCKU
 
-0.  Setting the parameters for compilation
-
-    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.
+0.  Setting the parameters for compilation
+
+    Modify the makefile.inc file in the AMSS_NCKU_source directory and change the settings according to your computer.
+
+    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")
 
 
 ##################################################################
@@ -70,7 +73,7 @@ def makefile_ABE():
 
     ## Build command with CPU binding to nohz_full cores
     if (input_data.GPU_Calculation == "no"):
-        makefile_command  = f"{NUMACTL_CPU_BIND} make -j{BUILD_JOBS} INTERP_LB_MODE=optimize ABE"
+        makefile_command  = f"{NUMACTL_CPU_BIND} make -j{BUILD_JOBS} INTERP_LB_MODE=off ABE"
     elif (input_data.GPU_Calculation == "yes"):
         makefile_command  = f"{NUMACTL_CPU_BIND} make -j{BUILD_JOBS} ABEGPU"
     else:
@@ -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

pgo_profile/PGO_Profile_Analysis.md

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