tutorials/multirate/ex6.c

/*
  Note:
    -hratio is the ratio between mesh size of corse grids and fine grids
    -ts_rk_dtratio is the ratio between the large stepsize and the small stepsize
*/

static const char help[] = "1D periodic Finite Volume solver in slope-limiter form with semidiscrete time stepping.\n"
  "  advection   - Constant coefficient scalar advection\n"
  "                u_t       + (a*u)_x               = 0\n"
  "  for this toy problem, we choose different meshsizes for different sub-domains (slow-fast-slow), say\n"
  "                hxs  = hratio*hxf \n"
  "  where hxs and hxf are the grid spacings for coarse and fine grids respectively.\n"
  "  exact       - Exact Riemann solver which usually needs to perform a Newton iteration to connect\n"
  "                the states across shocks and rarefactions\n"
  "  simulation  - use reference solution which is generated by smaller time step size to be true solution,\n"
  "                also the reference solution should be generated by user and stored in a binary file.\n"
  "  characteristic - Limit the characteristic variables, this is usually preferred (default)\n"
  "Several initial conditions can be chosen with -initial N\n\n"
  "The problem size should be set with -da_grid_x M\n\n";

#include <petscts.h>
#include <petscdm.h>
#include <petscdmda.h>
#include <petscdraw.h>
#include "finitevolume1d.h"

PETSC_STATIC_INLINE PetscReal RangeMod(PetscReal a,PetscReal xmin,PetscReal xmax) { PetscReal range = xmax-xmin; return xmin +PetscFmodReal(range+PetscFmodReal(a,range),range); }

/* --------------------------------- Advection ----------------------------------- */
typedef struct {
  PetscReal a;                  /* advective velocity */
} AdvectCtx;

static PetscErrorCode PhysicsRiemann_Advect(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
{
  AdvectCtx *ctx = (AdvectCtx*)vctx;
  PetscReal speed;

  PetscFunctionBeginUser;
  speed     = ctx->a;
  flux[0]   = PetscMax(0,speed)*uL[0] + PetscMin(0,speed)*uR[0];
  *maxspeed = speed;
  PetscFunctionReturn(0);
}

static PetscErrorCode PhysicsCharacteristic_Advect(void *vctx,PetscInt m,const PetscScalar *u,PetscScalar *X,PetscScalar *Xi,PetscReal *speeds)
{
  AdvectCtx *ctx = (AdvectCtx*)vctx;

  PetscFunctionBeginUser;
  X[0]      = 1.;
  Xi[0]     = 1.;
  speeds[0] = ctx->a;
  PetscFunctionReturn(0);
}

static PetscErrorCode PhysicsSample_Advect(void *vctx,PetscInt initial,FVBCType bctype,PetscReal xmin,PetscReal xmax,PetscReal t,PetscReal x,PetscReal *u)
{
  AdvectCtx *ctx = (AdvectCtx*)vctx;
  PetscReal a    = ctx->a,x0;

  PetscFunctionBeginUser;
  switch (bctype) {
    case FVBC_OUTFLOW:   x0 = x-a*t; break;
    case FVBC_PERIODIC: x0 = RangeMod(x-a*t,xmin,xmax); break;
    default: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_UNKNOWN_TYPE,"unknown BCType");
  }
  switch (initial) {
    case 0: u[0] = (x0 < 0) ? 1 : -1; break;
    case 1: u[0] = (x0 < 0) ? -1 : 1; break;
    case 2: u[0] = (0 < x0 && x0 < 1) ? 1 : 0; break;
    case 3: u[0] = PetscSinReal(2*PETSC_PI*x0); break;
    case 4: u[0] = PetscAbs(x0); break;
    case 5: u[0] = (x0 < 0 || x0 > 0.5) ? 0 : PetscSqr(PetscSinReal(2*PETSC_PI*x0)); break;
    case 6: u[0] = (x0 < 0) ? 0 : ((x0 < 1) ? x0 : ((x0 < 2) ? 2-x0 : 0)); break;
    case 7: u[0] = PetscPowReal(PetscSinReal(PETSC_PI*x0),10.0);break;
    default: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_UNKNOWN_TYPE,"unknown initial condition");
  }
  PetscFunctionReturn(0);
}

static PetscErrorCode PhysicsCreate_Advect(FVCtx *ctx)
{
  PetscErrorCode ierr;
  AdvectCtx      *user;

  PetscFunctionBeginUser;
  ierr = PetscNew(&user);CHKERRQ(ierr);
  ctx->physics2.sample2         = PhysicsSample_Advect;
  ctx->physics2.riemann2        = PhysicsRiemann_Advect;
  ctx->physics2.characteristic2 = PhysicsCharacteristic_Advect;
  ctx->physics2.destroy         = PhysicsDestroy_SimpleFree;
  ctx->physics2.user            = user;
  ctx->physics2.dof             = 1;
  ierr = PetscStrallocpy("u",&ctx->physics2.fieldname[0]);CHKERRQ(ierr);
  user->a = 1;
  ierr = PetscOptionsBegin(ctx->comm,ctx->prefix,"Options for advection","");CHKERRQ(ierr);
  {
    ierr = PetscOptionsReal("-physics_advect_a","Speed","",user->a,&user->a,NULL);CHKERRQ(ierr);
  }
  ierr = PetscOptionsEnd();CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode FVSample_2WaySplit(FVCtx *ctx,DM da,PetscReal time,Vec U)
{
  PetscErrorCode  ierr;
  PetscScalar     *u,*uj,xj,xi;
  PetscInt        i,j,k,dof,xs,xm,Mx;
  const PetscInt  N = 200;
  PetscReal       hs,hf;

  PetscFunctionBeginUser;
  if (!ctx->physics2.sample2) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Physics has not provided a sampling function");
  ierr = DMDAGetInfo(da,0, &Mx,0,0, 0,0,0, &dof,0,0,0,0,0);CHKERRQ(ierr);
  ierr = DMDAGetCorners(da,&xs,0,0,&xm,0,0);CHKERRQ(ierr);
  ierr = DMDAVecGetArray(da,U,&u);CHKERRQ(ierr);
  ierr = PetscMalloc1(dof,&uj);CHKERRQ(ierr);
  hs   = (ctx->xmax-ctx->xmin)*3.0/8.0/ctx->sf;
  hf   = (ctx->xmax-ctx->xmin)/4.0/(ctx->fs-ctx->sf);
  for (i=xs; i<xs+xm; i++) {
    if (i < ctx->sf) {
      xi = ctx->xmin+0.5*hs+i*hs;
      /* Integrate over cell i using trapezoid rule with N points. */
      for (k=0; k<dof; k++) u[i*dof+k] = 0;
      for (j=0; j<N+1; j++) {
        xj = xi+hs*(j-N/2)/(PetscReal)N;
        ierr = (*ctx->physics2.sample2)(ctx->physics2.user,ctx->initial,ctx->bctype,ctx->xmin,ctx->xmax,time,xj,uj);CHKERRQ(ierr);
        for (k=0; k<dof; k++) u[i*dof+k] += ((j==0 || j==N) ? 0.5 : 1.0)*uj[k]/N;
      }
    } else if (i < ctx->fs) {
      xi = ctx->xmin+ctx->sf*hs+0.5*hf+(i-ctx->sf)*hf;
      /* Integrate over cell i using trapezoid rule with N points. */
      for (k=0; k<dof; k++) u[i*dof+k] = 0;
      for (j=0; j<N+1; j++) {
        xj = xi+hf*(j-N/2)/(PetscReal)N;
        ierr = (*ctx->physics2.sample2)(ctx->physics2.user,ctx->initial,ctx->bctype,ctx->xmin,ctx->xmax,time,xj,uj);CHKERRQ(ierr);
        for (k=0; k<dof; k++) u[i*dof+k] += ((j==0 || j==N) ? 0.5 : 1.0)*uj[k]/N;
      }
    } else {
      xi = ctx->xmin+ctx->sf*hs+(ctx->fs-ctx->sf)*hf+0.5*hs+(i-ctx->fs)*hs;
      /* Integrate over cell i using trapezoid rule with N points. */
      for (k=0; k<dof; k++) u[i*dof+k] = 0;
      for (j=0; j<N+1; j++) {
        xj = xi+hs*(j-N/2)/(PetscReal)N;
        ierr = (*ctx->physics2.sample2)(ctx->physics2.user,ctx->initial,ctx->bctype,ctx->xmin,ctx->xmax,time,xj,uj);CHKERRQ(ierr);
        for (k=0; k<dof; k++) u[i*dof+k] += ((j==0 || j==N) ? 0.5 : 1.0)*uj[k]/N;
      }
    }
  }
  ierr = DMDAVecRestoreArray(da,U,&u);CHKERRQ(ierr);
  ierr = PetscFree(uj);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode SolutionErrorNorms_2WaySplit(FVCtx *ctx,DM da,PetscReal t,Vec X,PetscReal *nrm1)
{
  PetscErrorCode    ierr;
  Vec               Y;
  PetscInt          i,Mx;
  const PetscScalar *ptr_X,*ptr_Y;
  PetscReal         hs,hf;

  PetscFunctionBeginUser;
  ierr = VecGetSize(X,&Mx);CHKERRQ(ierr);
  ierr = VecDuplicate(X,&Y);CHKERRQ(ierr);
  ierr = FVSample_2WaySplit(ctx,da,t,Y);CHKERRQ(ierr);
  hs   = (ctx->xmax-ctx->xmin)*3.0/8.0/ctx->sf;
  hf   = (ctx->xmax-ctx->xmin)/4.0/(ctx->fs-ctx->sf);
  ierr = VecGetArrayRead(X,&ptr_X);CHKERRQ(ierr);
  ierr = VecGetArrayRead(Y,&ptr_Y);CHKERRQ(ierr);
  for (i=0; i<Mx; i++) {
    if (i < ctx->sf || i > ctx->fs -1) *nrm1 +=  hs*PetscAbs(ptr_X[i]-ptr_Y[i]);
    else *nrm1 += hf*PetscAbs(ptr_X[i]-ptr_Y[i]);
  }
  ierr = VecRestoreArrayRead(X,&ptr_X);CHKERRQ(ierr);
  ierr = VecRestoreArrayRead(Y,&ptr_Y);CHKERRQ(ierr);
  ierr = VecDestroy(&Y);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode FVRHSFunction_2WaySplit(TS ts,PetscReal time,Vec X,Vec F,void *vctx)
{
  FVCtx          *ctx = (FVCtx*)vctx;
  PetscErrorCode ierr;
  PetscInt       i,j,k,Mx,dof,xs,xm,sf = ctx->sf,fs = ctx->fs;
  PetscReal      hxf,hxs,cfl_idt = 0;
  PetscScalar    *x,*f,*slope;
  Vec            Xloc;
  DM             da;

  PetscFunctionBeginUser;
  ierr = TSGetDM(ts,&da);CHKERRQ(ierr);
  ierr = DMGetLocalVector(da,&Xloc);CHKERRQ(ierr);                          /* Xloc contains ghost points                                     */
  ierr = DMDAGetInfo(da,0, &Mx,0,0, 0,0,0, &dof,0,0,0,0,0);CHKERRQ(ierr);   /* Mx is the number of center points                              */
  hxs  = (ctx->xmax-ctx->xmin)*3.0/8.0/ctx->sf;
  hxf  = (ctx->xmax-ctx->xmin)/4.0/(ctx->fs-ctx->sf);
  ierr = DMGlobalToLocalBegin(da,X,INSERT_VALUES,Xloc);CHKERRQ(ierr);       /* X is solution vector which does not contain ghost points       */
  ierr = DMGlobalToLocalEnd(da,X,INSERT_VALUES,Xloc);CHKERRQ(ierr);

  ierr = VecZeroEntries(F);CHKERRQ(ierr);                                   /* F is the right hand side function corresponds to center points */

  ierr = DMDAVecGetArray(da,Xloc,&x);CHKERRQ(ierr);
  ierr = DMDAVecGetArray(da,F,&f);CHKERRQ(ierr);
  ierr = DMDAGetArray(da,PETSC_TRUE,&slope);CHKERRQ(ierr);                  /* contains ghost points                                           */

  ierr = DMDAGetCorners(da,&xs,0,0,&xm,0,0);CHKERRQ(ierr);

  if (ctx->bctype == FVBC_OUTFLOW) {
    for (i=xs-2; i<0; i++) {
      for (j=0; j<dof; j++) x[i*dof+j] = x[j];
    }
    for (i=Mx; i<xs+xm+2; i++) {
      for (j=0; j<dof; j++) x[i*dof+j] = x[(xs+xm-1)*dof+j];
    }
  }
  for (i=xs-1; i<xs+xm+1; i++) {
    struct _LimitInfo info;
    PetscScalar       *cjmpL,*cjmpR;
    /* Determine the right eigenvectors R, where A = R \Lambda R^{-1} */
    ierr = (*ctx->physics2.characteristic2)(ctx->physics2.user,dof,&x[i*dof],ctx->R,ctx->Rinv,ctx->speeds);CHKERRQ(ierr);
    /* Evaluate jumps across interfaces (i-1, i) and (i, i+1), put in characteristic basis */
    ierr  = PetscArrayzero(ctx->cjmpLR,2*dof);CHKERRQ(ierr);
    cjmpL = &ctx->cjmpLR[0];
    cjmpR = &ctx->cjmpLR[dof];
    for (j=0; j<dof; j++) {
      PetscScalar jmpL,jmpR;
      jmpL = x[(i+0)*dof+j]-x[(i-1)*dof+j];
      jmpR = x[(i+1)*dof+j]-x[(i+0)*dof+j];
      for (k=0; k<dof; k++) {
        cjmpL[k] += ctx->Rinv[k+j*dof]*jmpL;
        cjmpR[k] += ctx->Rinv[k+j*dof]*jmpR;
      }
    }
    /* Apply limiter to the left and right characteristic jumps */
    info.m  = dof;
    info.hxs = hxs;
    info.hxf = hxf;
    (*ctx->limit2)(&info,cjmpL,cjmpR,ctx->sf,ctx->fs,i,ctx->cslope);
    for (j=0; j<dof; j++) {
      PetscScalar tmp = 0;
      for (k=0; k<dof; k++) tmp += ctx->R[j+k*dof]*ctx->cslope[k];
      slope[i*dof+j] = tmp;
    }
  }

  for (i=xs; i<xs+xm+1; i++) {
    PetscReal   maxspeed;
    PetscScalar *uL,*uR;
    uL = &ctx->uLR[0];
    uR = &ctx->uLR[dof];
    if (i < sf) { /* slow region */
      for (j=0; j<dof; j++) {
        uL[j] = x[(i-1)*dof+j]+slope[(i-1)*dof+j]*hxs/2;
        uR[j] = x[(i-0)*dof+j]-slope[(i-0)*dof+j]*hxs/2;
      }
      ierr    = (*ctx->physics2.riemann2)(ctx->physics2.user,dof,uL,uR,ctx->flux,&maxspeed);CHKERRQ(ierr);
      if (i > xs) {
        for (j=0; j<dof; j++) f[(i-1)*dof+j] -= ctx->flux[j]/hxs;
      }
      if (i < xs+xm) {
        for (j=0; j<dof; j++) f[i*dof+j] += ctx->flux[j]/hxs;
      }
    } else if (i == sf) { /* interface between the slow region and the fast region */
      for (j=0; j<dof; j++) {
        uL[j] = x[(i-1)*dof+j]+slope[(i-1)*dof+j]*hxs/2;
        uR[j] = x[(i-0)*dof+j]-slope[(i-0)*dof+j]*hxf/2;
      }
      ierr    = (*ctx->physics2.riemann2)(ctx->physics2.user,dof,uL,uR,ctx->flux,&maxspeed);CHKERRQ(ierr);
      if (i > xs) {
        for (j=0; j<dof; j++) f[(i-1)*dof+j] -= ctx->flux[j]/hxs;
      }
      if (i < xs+xm) {
        for (j=0; j<dof; j++) f[i*dof+j] += ctx->flux[j]/hxf;
      }
    } else if (i > sf && i < fs) { /* fast region */
      for (j=0; j<dof; j++) {
        uL[j] = x[(i-1)*dof+j]+slope[(i-1)*dof+j]*hxf/2;
        uR[j] = x[(i-0)*dof+j]-slope[(i-0)*dof+j]*hxf/2;
      }
      ierr    = (*ctx->physics2.riemann2)(ctx->physics2.user,dof,uL,uR,ctx->flux,&maxspeed);CHKERRQ(ierr);
      if (i > xs) {
        for (j=0; j<dof; j++) f[(i-1)*dof+j] -= ctx->flux[j]/hxf;
      }
      if (i < xs+xm) {
        for (j=0; j<dof; j++) f[i*dof+j] += ctx->flux[j]/hxf;
      }
    } else if (i == fs) { /* interface between the fast region and the slow region */
      for (j=0; j<dof; j++) {
        uL[j] = x[(i-1)*dof+j]+slope[(i-1)*dof+j]*hxf/2;
        uR[j] = x[(i-0)*dof+j]-slope[(i-0)*dof+j]*hxs/2;
      }
      ierr    = (*ctx->physics2.riemann2)(ctx->physics2.user,dof,uL,uR,ctx->flux,&maxspeed);CHKERRQ(ierr);
      if (i > xs) {
        for (j=0; j<dof; j++) f[(i-1)*dof+j] -= ctx->flux[j]/hxf;
      }
      if (i < xs+xm) {
        for (j=0; j<dof; j++) f[i*dof+j] += ctx->flux[j]/hxs;
      }
    } else { /* slow region */
      for (j=0; j<dof; j++) {
        uL[j] = x[(i-1)*dof+j]+slope[(i-1)*dof+j]*hxs/2;
        uR[j] = x[(i-0)*dof+j]-slope[(i-0)*dof+j]*hxs/2;
      }
      ierr    = (*ctx->physics2.riemann2)(ctx->physics2.user,dof,uL,uR,ctx->flux,&maxspeed);CHKERRQ(ierr);
      cfl_idt = PetscMax(cfl_idt,PetscAbsScalar(maxspeed/hxs)); /* Max allowable value of 1/Delta t */
      if (i > xs) {
        for (j=0; j<dof; j++) f[(i-1)*dof+j] -= ctx->flux[j]/hxs;
      }
      if (i < xs+xm) {
        for (j=0; j<dof; j++) f[i*dof+j] += ctx->flux[j]/hxs;
      }
    }
  }
  ierr = DMDAVecRestoreArray(da,Xloc,&x);CHKERRQ(ierr);
  ierr = DMDAVecRestoreArray(da,F,&f);CHKERRQ(ierr);
  ierr = DMDARestoreArray(da,PETSC_TRUE,&slope);CHKERRQ(ierr);
  ierr = DMRestoreLocalVector(da,&Xloc);CHKERRQ(ierr);
  ierr = MPI_Allreduce(&cfl_idt,&ctx->cfl_idt,1,MPIU_SCALAR,MPIU_MAX,PetscObjectComm((PetscObject)da));CHKERRQ(ierr);
  if (0) {
    /* We need to a way to inform the TS of a CFL constraint, this is a debugging fragment */
    PetscReal dt,tnow;
    ierr = TSGetTimeStep(ts,&dt);CHKERRQ(ierr);
    ierr = TSGetTime(ts,&tnow);CHKERRQ(ierr);
    if (dt > 0.5/ctx->cfl_idt) {
      if (1) {
        ierr = PetscPrintf(ctx->comm,"Stability constraint exceeded at t=%g, dt %g > %g\n",(double)tnow,(double)dt,(double)(0.5/ctx->cfl_idt));CHKERRQ(ierr);
      } else SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_UNKNOWN_TYPE,"Stability constraint exceeded, %g > %g",(double)dt,(double)(ctx->cfl/ctx->cfl_idt));
    }
  }
  PetscFunctionReturn(0);
}

/* --------------------------------- Finite Volume Solver for slow components ----------------------------------- */
PetscErrorCode FVRHSFunctionslow_2WaySplit(TS ts,PetscReal time,Vec X,Vec F,void *vctx)
{
  FVCtx          *ctx = (FVCtx*)vctx;
  PetscErrorCode ierr;
  PetscInt       i,j,k,Mx,dof,xs,xm,islow = 0,sf = ctx->sf,fs = ctx->fs,lsbwidth = ctx->lsbwidth,rsbwidth = ctx->rsbwidth;
  PetscReal      hxs,hxf,cfl_idt = 0;
  PetscScalar    *x,*f,*slope;
  Vec            Xloc;
  DM             da;

  PetscFunctionBeginUser;
  ierr = TSGetDM(ts,&da);CHKERRQ(ierr);
  ierr = DMGetLocalVector(da,&Xloc);CHKERRQ(ierr);
  ierr = DMDAGetInfo(da,0, &Mx,0,0, 0,0,0, &dof,0,0,0,0,0);CHKERRQ(ierr);
  hxs  = (ctx->xmax-ctx->xmin)*3.0/8.0/ctx->sf;
  hxf  = (ctx->xmax-ctx->xmin)/4.0/(ctx->fs-ctx->sf);
  ierr = DMGlobalToLocalBegin(da,X,INSERT_VALUES,Xloc);CHKERRQ(ierr);
  ierr = DMGlobalToLocalEnd(da,X,INSERT_VALUES,Xloc);CHKERRQ(ierr);
  ierr = VecZeroEntries(F);CHKERRQ(ierr);
  ierr = DMDAVecGetArray(da,Xloc,&x);CHKERRQ(ierr);
  ierr = VecGetArray(F,&f);CHKERRQ(ierr);
  ierr = DMDAGetArray(da,PETSC_TRUE,&slope);CHKERRQ(ierr);
  ierr = DMDAGetCorners(da,&xs,0,0,&xm,0,0);CHKERRQ(ierr);

  if (ctx->bctype == FVBC_OUTFLOW) {
    for (i=xs-2; i<0; i++) {
      for (j=0; j<dof; j++) x[i*dof+j] = x[j];
    }
    for (i=Mx; i<xs+xm+2; i++) {
      for (j=0; j<dof; j++) x[i*dof+j] = x[(xs+xm-1)*dof+j];
    }
  }
  for (i=xs-1; i<xs+xm+1; i++) {
    struct _LimitInfo info;
    PetscScalar       *cjmpL,*cjmpR;
    if (i < sf-lsbwidth+1 || i > fs+rsbwidth-2) { /* slow components and the first and last fast components */
      /* Determine the right eigenvectors R, where A = R \Lambda R^{-1} */
      ierr = (*ctx->physics2.characteristic2)(ctx->physics2.user,dof,&x[i*dof],ctx->R,ctx->Rinv,ctx->speeds);CHKERRQ(ierr);
      /* Evaluate jumps across interfaces (i-1, i) and (i, i+1), put in characteristic basis */
      ierr  = PetscArrayzero(ctx->cjmpLR,2*dof);CHKERRQ(ierr);
      cjmpL = &ctx->cjmpLR[0];
      cjmpR = &ctx->cjmpLR[dof];
      for (j=0; j<dof; j++) {
        PetscScalar jmpL,jmpR;
        jmpL = x[(i+0)*dof+j]-x[(i-1)*dof+j];
        jmpR = x[(i+1)*dof+j]-x[(i+0)*dof+j];
        for (k=0; k<dof; k++) {
          cjmpL[k] += ctx->Rinv[k+j*dof]*jmpL;
          cjmpR[k] += ctx->Rinv[k+j*dof]*jmpR;
        }
      }
      /* Apply limiter to the left and right characteristic jumps */
      info.m  = dof;
      info.hxs = hxs;
      info.hxf = hxf;
      (*ctx->limit2)(&info,cjmpL,cjmpR,ctx->sf,ctx->fs,i,ctx->cslope);
      for (j=0; j<dof; j++) {
        PetscScalar tmp = 0;
        for (k=0; k<dof; k++) tmp += ctx->R[j+k*dof]*ctx->cslope[k];
          slope[i*dof+j] = tmp;
      }
    }
  }

  for (i=xs; i<xs+xm+1; i++) {
    PetscReal   maxspeed;
    PetscScalar *uL,*uR;
    uL = &ctx->uLR[0];
    uR = &ctx->uLR[dof];
    if (i < sf-lsbwidth) { /* slow region */
      for (j=0; j<dof; j++) {
        uL[j] = x[(i-1)*dof+j]+slope[(i-1)*dof+j]*hxs/2;
        uR[j] = x[(i-0)*dof+j]-slope[(i-0)*dof+j]*hxs/2;
      }
      ierr    = (*ctx->physics2.riemann2)(ctx->physics2.user,dof,uL,uR,ctx->flux,&maxspeed);CHKERRQ(ierr);
      cfl_idt = PetscMax(cfl_idt,PetscAbsScalar(maxspeed/hxs)); /* Max allowable value of 1/Delta t */
      if (i > xs) {
        for (j=0; j<dof; j++) f[(islow-1)*dof+j] -= ctx->flux[j]/hxs;
      }
      if (i < xs+xm) {
        for (j=0; j<dof; j++) f[islow*dof+j] += ctx->flux[j]/hxs;
        islow++;
      }
    }
    if (i == sf-lsbwidth) { /* interface between the slow region and the fast region */
      for (j=0; j<dof; j++) {
        uL[j] = x[(i-1)*dof+j]+slope[(i-1)*dof+j]*hxs/2;
        uR[j] = x[(i-0)*dof+j]-slope[(i-0)*dof+j]*hxs/2;
      }
      ierr = (*ctx->physics2.riemann2)(ctx->physics2.user,dof,uL,uR,ctx->flux,&maxspeed);CHKERRQ(ierr);
      if (i > xs) {
        for (j=0; j<dof; j++) f[(islow-1)*dof+j] -= ctx->flux[j]/hxs;
      }
    }
    if (i == fs+rsbwidth) { /* slow region */
      for (j=0; j<dof; j++) {
        uL[j] = x[(i-1)*dof+j]+slope[(i-1)*dof+j]*hxs/2;
        uR[j] = x[(i-0)*dof+j]-slope[(i-0)*dof+j]*hxs/2;
      }
      ierr = (*ctx->physics2.riemann2)(ctx->physics2.user,dof,uL,uR,ctx->flux,&maxspeed);CHKERRQ(ierr);
      if (i < xs+xm) {
        for (j=0; j<dof; j++) f[islow*dof+j] += ctx->flux[j]/hxs;
        islow++;
      }
    }
    if (i > fs+rsbwidth) { /* slow region */
      for (j=0; j<dof; j++) {
        uL[j] = x[(i-1)*dof+j]+slope[(i-1)*dof+j]*hxs/2;
        uR[j] = x[(i-0)*dof+j]-slope[(i-0)*dof+j]*hxs/2;
      }
      ierr = (*ctx->physics2.riemann2)(ctx->physics2.user,dof,uL,uR,ctx->flux,&maxspeed);CHKERRQ(ierr);
      if (i > xs) {
        for (j=0; j<dof; j++) f[(islow-1)*dof+j] -= ctx->flux[j]/hxs;
      }
      if (i < xs+xm) {
        for (j=0; j<dof; j++) f[islow*dof+j] += ctx->flux[j]/hxs;
        islow++;
      }
    }
  }
  ierr = DMDAVecRestoreArray(da,Xloc,&x);CHKERRQ(ierr);
  ierr = VecRestoreArray(F,&f);CHKERRQ(ierr);
  ierr = DMDARestoreArray(da,PETSC_TRUE,&slope);CHKERRQ(ierr);
  ierr = DMRestoreLocalVector(da,&Xloc);CHKERRQ(ierr);
  ierr = MPI_Allreduce(&cfl_idt,&ctx->cfl_idt,1,MPIU_SCALAR,MPIU_MAX,PetscObjectComm((PetscObject)da));CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode FVRHSFunctionslowbuffer_2WaySplit(TS ts,PetscReal time,Vec X,Vec F,void *vctx)
{
  FVCtx          *ctx = (FVCtx*)vctx;
  PetscErrorCode ierr;
  PetscInt       i,j,k,Mx,dof,xs,xm,islow = 0,sf = ctx->sf,fs = ctx->fs,lsbwidth = ctx->lsbwidth,rsbwidth = ctx->rsbwidth;
  PetscReal      hxs,hxf;
  PetscScalar    *x,*f,*slope;
  Vec            Xloc;
  DM             da;

  PetscFunctionBeginUser;
  ierr = TSGetDM(ts,&da);CHKERRQ(ierr);
  ierr = DMGetLocalVector(da,&Xloc);CHKERRQ(ierr);
  ierr = DMDAGetInfo(da,0, &Mx,0,0, 0,0,0, &dof,0,0,0,0,0);CHKERRQ(ierr);
  hxs  = (ctx->xmax-ctx->xmin)*3.0/8.0/ctx->sf;
  hxf  = (ctx->xmax-ctx->xmin)/4.0/(ctx->fs-ctx->sf);
  ierr = DMGlobalToLocalBegin(da,X,INSERT_VALUES,Xloc);CHKERRQ(ierr);
  ierr = DMGlobalToLocalEnd(da,X,INSERT_VALUES,Xloc);CHKERRQ(ierr);
  ierr = VecZeroEntries(F);CHKERRQ(ierr);
  ierr = DMDAVecGetArray(da,Xloc,&x);CHKERRQ(ierr);
  ierr = VecGetArray(F,&f);CHKERRQ(ierr);
  ierr = DMDAGetArray(da,PETSC_TRUE,&slope);CHKERRQ(ierr);
  ierr = DMDAGetCorners(da,&xs,0,0,&xm,0,0);CHKERRQ(ierr);

  if (ctx->bctype == FVBC_OUTFLOW) {
    for (i=xs-2; i<0; i++) {
      for (j=0; j<dof; j++) x[i*dof+j] = x[j];
    }
    for (i=Mx; i<xs+xm+2; i++) {
      for (j=0; j<dof; j++) x[i*dof+j] = x[(xs+xm-1)*dof+j];
    }
  }
  for (i=xs-1; i<xs+xm+1; i++) {
    struct _LimitInfo info;
    PetscScalar       *cjmpL,*cjmpR;
    if ((i > sf-lsbwidth-2 && i < sf+1) || (i > fs-2 && i < fs+rsbwidth+1)) {
      /* Determine the right eigenvectors R, where A = R \Lambda R^{-1} */
      ierr = (*ctx->physics2.characteristic2)(ctx->physics2.user,dof,&x[i*dof],ctx->R,ctx->Rinv,ctx->speeds);CHKERRQ(ierr);
      /* Evaluate jumps across interfaces (i-1, i) and (i, i+1), put in characteristic basis */
      ierr  = PetscArrayzero(ctx->cjmpLR,2*dof);CHKERRQ(ierr);
      cjmpL = &ctx->cjmpLR[0];
      cjmpR = &ctx->cjmpLR[dof];
      for (j=0; j<dof; j++) {
        PetscScalar jmpL,jmpR;
        jmpL = x[(i+0)*dof+j]-x[(i-1)*dof+j];
        jmpR = x[(i+1)*dof+j]-x[(i+0)*dof+j];
        for (k=0; k<dof; k++) {
          cjmpL[k] += ctx->Rinv[k+j*dof]*jmpL;
          cjmpR[k] += ctx->Rinv[k+j*dof]*jmpR;
        }
      }
      /* Apply limiter to the left and right characteristic jumps */
      info.m  = dof;
      info.hxs = hxs;
      info.hxf = hxf;
      (*ctx->limit2)(&info,cjmpL,cjmpR,ctx->sf,ctx->fs,i,ctx->cslope);
      for (j=0; j<dof; j++) {
        PetscScalar tmp = 0;
        for (k=0; k<dof; k++) tmp += ctx->R[j+k*dof]*ctx->cslope[k];
          slope[i*dof+j] = tmp;
      }
    }
  }

  for (i=xs; i<xs+xm+1; i++) {
    PetscReal   maxspeed;
    PetscScalar *uL,*uR;
    uL = &ctx->uLR[0];
    uR = &ctx->uLR[dof];
    if (i == sf-lsbwidth) {
      for (j=0; j<dof; j++) {
        uL[j] = x[(i-1)*dof+j]+slope[(i-1)*dof+j]*hxs/2;
        uR[j] = x[(i-0)*dof+j]-slope[(i-0)*dof+j]*hxs/2;
      }
      ierr    = (*ctx->physics2.riemann2)(ctx->physics2.user,dof,uL,uR,ctx->flux,&maxspeed);CHKERRQ(ierr);
      if (i < xs+xm) {
        for (j=0; j<dof; j++) f[islow*dof+j] += ctx->flux[j]/hxs;
        islow++;
      }
    }
    if (i > sf-lsbwidth && i < sf) {
      for (j=0; j<dof; j++) {
        uL[j] = x[(i-1)*dof+j]+slope[(i-1)*dof+j]*hxs/2;
        uR[j] = x[(i-0)*dof+j]-slope[(i-0)*dof+j]*hxs/2;
      }
      ierr    = (*ctx->physics2.riemann2)(ctx->physics2.user,dof,uL,uR,ctx->flux,&maxspeed);CHKERRQ(ierr);
      if (i > xs) {
        for (j=0; j<dof; j++) f[(islow-1)*dof+j] -= ctx->flux[j]/hxs;
      }
      if (i < xs+xm) {
        for (j=0; j<dof; j++) f[islow*dof+j] += ctx->flux[j]/hxs;
        islow++;
      }
    }
    if (i == sf) { /* interface between the slow region and the fast region */
      for (j=0; j<dof; j++) {
        uL[j] = x[(i-1)*dof+j]+slope[(i-1)*dof+j]*hxs/2;
        uR[j] = x[(i-0)*dof+j]-slope[(i-0)*dof+j]*hxf/2;
      }
      ierr = (*ctx->physics2.riemann2)(ctx->physics2.user,dof,uL,uR,ctx->flux,&maxspeed);CHKERRQ(ierr);
      if (i > xs) {
        for (j=0; j<dof; j++) f[(islow-1)*dof+j] -= ctx->flux[j]/hxs;
      }
    }
    if (i == fs) { /* interface between the fast region and the slow region */
      for (j=0; j<dof; j++) {
        uL[j] = x[(i-1)*dof+j]+slope[(i-1)*dof+j]*hxf/2;
        uR[j] = x[(i-0)*dof+j]-slope[(i-0)*dof+j]*hxs/2;
      }
      ierr = (*ctx->physics2.riemann2)(ctx->physics2.user,dof,uL,uR,ctx->flux,&maxspeed);CHKERRQ(ierr);
      if (i < xs+xm) {
        for (j=0; j<dof; j++) f[islow*dof+j] += ctx->flux[j]/hxs;
        islow++;
      }
    }
    if (i > fs && i < fs+rsbwidth) {
      for (j=0; j<dof; j++) {
        uL[j] = x[(i-1)*dof+j]+slope[(i-1)*dof+j]*hxs/2;
        uR[j] = x[(i-0)*dof+j]-slope[(i-0)*dof+j]*hxs/2;
      }
      ierr = (*ctx->physics2.riemann2)(ctx->physics2.user,dof,uL,uR,ctx->flux,&maxspeed);CHKERRQ(ierr);
      if (i > xs) {
        for (j=0; j<dof; j++) f[(islow-1)*dof+j] -= ctx->flux[j]/hxs;
      }
      if (i < xs+xm) {
        for (j=0; j<dof; j++) f[islow*dof+j] += ctx->flux[j]/hxs;
        islow++;
      }
    }
    if (i == fs+rsbwidth) {
      for (j=0; j<dof; j++) {
        uL[j] = x[(i-1)*dof+j]+slope[(i-1)*dof+j]*hxs/2;
        uR[j] = x[(i-0)*dof+j]-slope[(i-0)*dof+j]*hxs/2;
      }
      ierr = (*ctx->physics2.riemann2)(ctx->physics2.user,dof,uL,uR,ctx->flux,&maxspeed);CHKERRQ(ierr);
      if (i > xs) {
        for (j=0; j<dof; j++) f[(islow-1)*dof+j] -= ctx->flux[j]/hxs;
      }
    }
  }
  ierr = DMDAVecRestoreArray(da,Xloc,&x);CHKERRQ(ierr);
  ierr = VecRestoreArray(F,&f);CHKERRQ(ierr);
  ierr = DMDARestoreArray(da,PETSC_TRUE,&slope);CHKERRQ(ierr);
  ierr = DMRestoreLocalVector(da,&Xloc);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/* --------------------------------- Finite Volume Solver for fast  parts ----------------------------------- */
PetscErrorCode FVRHSFunctionfast_2WaySplit(TS ts,PetscReal time,Vec X,Vec F,void *vctx)
{
  FVCtx          *ctx = (FVCtx*)vctx;
  PetscErrorCode ierr;
  PetscInt       i,j,k,Mx,dof,xs,xm,ifast = 0,sf = ctx->sf,fs = ctx->fs;
  PetscReal      hxs,hxf;
  PetscScalar    *x,*f,*slope;
  Vec            Xloc;
  DM             da;

  PetscFunctionBeginUser;
  ierr = TSGetDM(ts,&da);CHKERRQ(ierr);
  ierr = DMGetLocalVector(da,&Xloc);CHKERRQ(ierr);
  ierr = DMDAGetInfo(da,0, &Mx,0,0, 0,0,0, &dof,0,0,0,0,0);CHKERRQ(ierr);
  hxs  = (ctx->xmax-ctx->xmin)*3.0/8.0/ctx->sf;
  hxf  = (ctx->xmax-ctx->xmin)/4.0/(ctx->fs-ctx->sf);
  ierr = DMGlobalToLocalBegin(da,X,INSERT_VALUES,Xloc);CHKERRQ(ierr);
  ierr = DMGlobalToLocalEnd(da,X,INSERT_VALUES,Xloc);CHKERRQ(ierr);
  ierr = VecZeroEntries(F);CHKERRQ(ierr);
  ierr = DMDAVecGetArray(da,Xloc,&x);CHKERRQ(ierr);
  ierr = VecGetArray(F,&f);CHKERRQ(ierr);
  ierr = DMDAGetArray(da,PETSC_TRUE,&slope);CHKERRQ(ierr);
  ierr = DMDAGetCorners(da,&xs,0,0,&xm,0,0);CHKERRQ(ierr);

  if (ctx->bctype == FVBC_OUTFLOW) {
    for (i=xs-2; i<0; i++) {
      for (j=0; j<dof; j++) x[i*dof+j] = x[j];
    }
    for (i=Mx; i<xs+xm+2; i++) {
      for (j=0; j<dof; j++) x[i*dof+j] = x[(xs+xm-1)*dof+j];
    }
  }
  for (i=xs-1; i<xs+xm+1; i++) {
    struct _LimitInfo info;
    PetscScalar       *cjmpL,*cjmpR;
    if (i > sf-2 && i < fs+1) {
      ierr = (*ctx->physics2.characteristic2)(ctx->physics2.user,dof,&x[i*dof],ctx->R,ctx->Rinv,ctx->speeds);CHKERRQ(ierr);
      ierr  = PetscArrayzero(ctx->cjmpLR,2*dof);CHKERRQ(ierr);
      cjmpL = &ctx->cjmpLR[0];
      cjmpR = &ctx->cjmpLR[dof];
      for (j=0; j<dof; j++) {
        PetscScalar jmpL,jmpR;
        jmpL = x[(i+0)*dof+j]-x[(i-1)*dof+j];
        jmpR = x[(i+1)*dof+j]-x[(i+0)*dof+j];
        for (k=0; k<dof; k++) {
          cjmpL[k] += ctx->Rinv[k+j*dof]*jmpL;
          cjmpR[k] += ctx->Rinv[k+j*dof]*jmpR;
        }
      }
      /* Apply limiter to the left and right characteristic jumps */
      info.m  = dof;
      info.hxs = hxs;
      info.hxf = hxf;
      (*ctx->limit2)(&info,cjmpL,cjmpR,ctx->sf,ctx->fs,i,ctx->cslope);
      for (j=0; j<dof; j++) {
      PetscScalar tmp = 0;
      for (k=0; k<dof; k++) tmp += ctx->R[j+k*dof]*ctx->cslope[k];
        slope[i*dof+j] = tmp;
      }
    }
  }

  for (i=xs; i<xs+xm+1; i++) {
    PetscReal   maxspeed;
    PetscScalar *uL,*uR;
    uL = &ctx->uLR[0];
    uR = &ctx->uLR[dof];
    if (i == sf) { /* interface between the slow region and the fast region */
      for (j=0; j<dof; j++) {
        uL[j] = x[(i-1)*dof+j]+slope[(i-1)*dof+j]*hxs/2;
        uR[j] = x[(i-0)*dof+j]-slope[(i-0)*dof+j]*hxf/2;
      }
      ierr    = (*ctx->physics2.riemann2)(ctx->physics2.user,dof,uL,uR,ctx->flux,&maxspeed);CHKERRQ(ierr);
      if (i < xs+xm) {
        for (j=0; j<dof; j++) f[ifast*dof+j] += ctx->flux[j]/hxf;
        ifast++;
      }
    }
    if (i > sf && i < fs) { /* fast region */
      for (j=0; j<dof; j++) {
        uL[j] = x[(i-1)*dof+j]+slope[(i-1)*dof+j]*hxf/2;
        uR[j] = x[(i-0)*dof+j]-slope[(i-0)*dof+j]*hxf/2;
      }
      ierr    = (*ctx->physics2.riemann2)(ctx->physics2.user,dof,uL,uR,ctx->flux,&maxspeed);CHKERRQ(ierr);
      if (i > xs) {
        for (j=0; j<dof; j++) f[(ifast-1)*dof+j] -= ctx->flux[j]/hxf;
      }
      if (i < xs+xm) {
        for (j=0; j<dof; j++) f[ifast*dof+j] += ctx->flux[j]/hxf;
        ifast++;
      }
    }
    if (i == fs) { /* interface between the fast region and the slow region */
      for (j=0; j<dof; j++) {
        uL[j] = x[(i-1)*dof+j]+slope[(i-1)*dof+j]*hxf/2;
        uR[j] = x[(i-0)*dof+j]-slope[(i-0)*dof+j]*hxs/2;
      }
      ierr    = (*ctx->physics2.riemann2)(ctx->physics2.user,dof,uL,uR,ctx->flux,&maxspeed);CHKERRQ(ierr);
      if (i > xs) {
        for (j=0; j<dof; j++) f[(ifast-1)*dof+j] -= ctx->flux[j]/hxf;
      }
    }
  }
  ierr = DMDAVecRestoreArray(da,Xloc,&x);CHKERRQ(ierr);
  ierr = VecRestoreArray(F,&f);CHKERRQ(ierr);
  ierr = DMDARestoreArray(da,PETSC_TRUE,&slope);CHKERRQ(ierr);
  ierr = DMRestoreLocalVector(da,&Xloc);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

int main(int argc,char *argv[])
{
  char              lname[256] = "mc",physname[256] = "advect",final_fname[256] = "solution.m";
  PetscFunctionList limiters   = 0,physics = 0;
  MPI_Comm          comm;
  TS                ts;
  DM                da;
  Vec               X,X0,R;
  FVCtx             ctx;
  PetscInt          i,k,dof,xs,xm,Mx,draw = 0,count_slow,count_fast,islow = 0,ifast =0,islowbuffer = 0,*index_slow,*index_fast,*index_slowbuffer;
  PetscBool         view_final = PETSC_FALSE;
  PetscReal         ptime;
  PetscErrorCode    ierr;

  ierr = PetscInitialize(&argc,&argv,0,help);if (ierr) return ierr;
  comm = PETSC_COMM_WORLD;
  ierr = PetscMemzero(&ctx,sizeof(ctx));CHKERRQ(ierr);

  /* Register limiters to be available on the command line */
  ierr = PetscFunctionListAdd(&limiters,"upwind"              ,Limit2_Upwind);CHKERRQ(ierr);
  ierr = PetscFunctionListAdd(&limiters,"lax-wendroff"        ,Limit2_LaxWendroff);CHKERRQ(ierr);
  ierr = PetscFunctionListAdd(&limiters,"beam-warming"        ,Limit2_BeamWarming);CHKERRQ(ierr);
  ierr = PetscFunctionListAdd(&limiters,"fromm"               ,Limit2_Fromm);CHKERRQ(ierr);
  ierr = PetscFunctionListAdd(&limiters,"minmod"              ,Limit2_Minmod);CHKERRQ(ierr);
  ierr = PetscFunctionListAdd(&limiters,"superbee"            ,Limit2_Superbee);CHKERRQ(ierr);
  ierr = PetscFunctionListAdd(&limiters,"mc"                  ,Limit2_MC);CHKERRQ(ierr);
  ierr = PetscFunctionListAdd(&limiters,"koren3"              ,Limit2_Koren3);CHKERRQ(ierr);

  /* Register physical models to be available on the command line */
  ierr = PetscFunctionListAdd(&physics,"advect"          ,PhysicsCreate_Advect);CHKERRQ(ierr);

  ctx.comm = comm;
  ctx.cfl  = 0.9;
  ctx.bctype = FVBC_PERIODIC;
  ctx.xmin = -1.0;
  ctx.xmax = 1.0;
  ierr = PetscOptionsBegin(comm,NULL,"Finite Volume solver options","");CHKERRQ(ierr);
  ierr = PetscOptionsReal("-xmin","X min","",ctx.xmin,&ctx.xmin,NULL);CHKERRQ(ierr);
  ierr = PetscOptionsReal("-xmax","X max","",ctx.xmax,&ctx.xmax,NULL);CHKERRQ(ierr);
  ierr = PetscOptionsFList("-limit","Name of flux imiter to use","",limiters,lname,lname,sizeof(lname),NULL);CHKERRQ(ierr);
  ierr = PetscOptionsInt("-draw","Draw solution vector, bitwise OR of (1=initial,2=final,4=final error)","",draw,&draw,NULL);CHKERRQ(ierr);
  ierr = PetscOptionsString("-view_final","Write final solution in ASCII MATLAB format to given file name","",final_fname,final_fname,sizeof(final_fname),&view_final);CHKERRQ(ierr);
  ierr = PetscOptionsInt("-initial","Initial condition (depends on the physics)","",ctx.initial,&ctx.initial,NULL);CHKERRQ(ierr);
  ierr = PetscOptionsBool("-exact","Compare errors with exact solution","",ctx.exact,&ctx.exact,NULL);CHKERRQ(ierr);
  ierr = PetscOptionsBool("-simulation","Compare errors with reference solution","",ctx.simulation,&ctx.simulation,NULL);CHKERRQ(ierr);
  ierr = PetscOptionsReal("-cfl","CFL number to time step at","",ctx.cfl,&ctx.cfl,NULL);CHKERRQ(ierr);
  ierr = PetscOptionsEnum("-bc_type","Boundary condition","",FVBCTypes,(PetscEnum)ctx.bctype,(PetscEnum*)&ctx.bctype,NULL);CHKERRQ(ierr);
  ierr = PetscOptionsInt("-hratio","Spacing ratio","",ctx.hratio,&ctx.hratio,NULL);CHKERRQ(ierr);
  ierr = PetscOptionsEnd();CHKERRQ(ierr);

  /* Choose the limiter from the list of registered limiters */
  ierr = PetscFunctionListFind(limiters,lname,&ctx.limit2);CHKERRQ(ierr);
  if (!ctx.limit2) SETERRQ1(PETSC_COMM_SELF,1,"Limiter '%s' not found",lname);

  /* Choose the physics from the list of registered models */
  {
    PetscErrorCode (*r)(FVCtx*);
    ierr = PetscFunctionListFind(physics,physname,&r);CHKERRQ(ierr);
    if (!r) SETERRQ1(PETSC_COMM_SELF,1,"Physics '%s' not found",physname);
    /* Create the physics, will set the number of fields and their names */
    ierr = (*r)(&ctx);CHKERRQ(ierr);
  }

  /* Create a DMDA to manage the parallel grid */
  ierr = DMDACreate1d(comm,DM_BOUNDARY_PERIODIC,50,ctx.physics2.dof,2,NULL,&da);CHKERRQ(ierr);
  ierr = DMSetFromOptions(da);CHKERRQ(ierr);
  ierr = DMSetUp(da);CHKERRQ(ierr);
  /* Inform the DMDA of the field names provided by the physics. */
  /* The names will be shown in the title bars when run with -ts_monitor_draw_solution */
  for (i=0; i<ctx.physics2.dof; i++) {
    ierr = DMDASetFieldName(da,i,ctx.physics2.fieldname[i]);CHKERRQ(ierr);
  }
  ierr = DMDAGetInfo(da,0, &Mx,0,0, 0,0,0, &dof,0,0,0,0,0);CHKERRQ(ierr);
  ierr = DMDAGetCorners(da,&xs,0,0,&xm,0,0);CHKERRQ(ierr);

  /* Set coordinates of cell centers */
  ierr = DMDASetUniformCoordinates(da,ctx.xmin+0.5*(ctx.xmax-ctx.xmin)/Mx,ctx.xmax+0.5*(ctx.xmax-ctx.xmin)/Mx,0,0,0,0);CHKERRQ(ierr);

  /* Allocate work space for the Finite Volume solver (so it doesn't have to be reallocated on each function evaluation) */
  ierr = PetscMalloc4(dof*dof,&ctx.R,dof*dof,&ctx.Rinv,2*dof,&ctx.cjmpLR,1*dof,&ctx.cslope);CHKERRQ(ierr);
  ierr = PetscMalloc3(2*dof,&ctx.uLR,dof,&ctx.flux,dof,&ctx.speeds);CHKERRQ(ierr);

  /* Create a vector to store the solution and to save the initial state */
  ierr = DMCreateGlobalVector(da,&X);CHKERRQ(ierr);
  ierr = VecDuplicate(X,&X0);CHKERRQ(ierr);
  ierr = VecDuplicate(X,&R);CHKERRQ(ierr);

  /* create index for slow parts and fast parts,
     count_slow + count_fast = Mx, counts_slow*hs = 0.5, counts_fast*hf = 0.5 */
  count_slow = Mx/(1.0+ctx.hratio/3.0);
  if (count_slow%2) SETERRQ(PETSC_COMM_WORLD,PETSC_ERR_USER,"Please adjust grid size Mx (-da_grid_x) and hratio (-hratio) so that Mx/(1+hartio/3) is even");
  count_fast = Mx-count_slow;
  ctx.sf = count_slow/2;
  ctx.fs = ctx.sf+count_fast;
  ierr = PetscMalloc1(xm*dof,&index_slow);CHKERRQ(ierr);
  ierr = PetscMalloc1(xm*dof,&index_fast);CHKERRQ(ierr);
  ierr = PetscMalloc1(6*dof,&index_slowbuffer);CHKERRQ(ierr);
  if (((AdvectCtx*)ctx.physics2.user)->a > 0) {
    ctx.lsbwidth = 2;
    ctx.rsbwidth = 4;
  } else {
    ctx.lsbwidth = 4;
    ctx.rsbwidth = 2;
  }
  for (i=xs; i<xs+xm; i++) {
    if (i < ctx.sf-ctx.lsbwidth || i > ctx.fs+ctx.rsbwidth-1)
      for (k=0; k<dof; k++) index_slow[islow++] = i*dof+k;
    else if ((i >= ctx.sf-ctx.lsbwidth && i < ctx.sf) || (i > ctx.fs-1 && i <= ctx.fs+ctx.rsbwidth-1))
      for (k=0; k<dof; k++) index_slowbuffer[islowbuffer++] = i*dof+k;
    else
      for (k=0; k<dof; k++) index_fast[ifast++] = i*dof+k;
  }
  ierr = ISCreateGeneral(PETSC_COMM_WORLD,islow,index_slow,PETSC_COPY_VALUES,&ctx.iss);CHKERRQ(ierr);
  ierr = ISCreateGeneral(PETSC_COMM_WORLD,ifast,index_fast,PETSC_COPY_VALUES,&ctx.isf);CHKERRQ(ierr);
  ierr = ISCreateGeneral(PETSC_COMM_WORLD,islowbuffer,index_slowbuffer,PETSC_COPY_VALUES,&ctx.issb);CHKERRQ(ierr);

  /* Create a time-stepping object */
  ierr = TSCreate(comm,&ts);CHKERRQ(ierr);
  ierr = TSSetDM(ts,da);CHKERRQ(ierr);
  ierr = TSSetRHSFunction(ts,R,FVRHSFunction_2WaySplit,&ctx);CHKERRQ(ierr);
  ierr = TSRHSSplitSetIS(ts,"slow",ctx.iss);CHKERRQ(ierr);
  ierr = TSRHSSplitSetIS(ts,"slowbuffer",ctx.issb);CHKERRQ(ierr);
  ierr = TSRHSSplitSetIS(ts,"fast",ctx.isf);CHKERRQ(ierr);
  ierr = TSRHSSplitSetRHSFunction(ts,"slow",NULL,FVRHSFunctionslow_2WaySplit,&ctx);CHKERRQ(ierr);
  ierr = TSRHSSplitSetRHSFunction(ts,"fast",NULL,FVRHSFunctionfast_2WaySplit,&ctx);CHKERRQ(ierr);
  ierr = TSRHSSplitSetRHSFunction(ts,"slowbuffer",NULL,FVRHSFunctionslowbuffer_2WaySplit,&ctx);CHKERRQ(ierr);

  ierr = TSSetType(ts,TSSSP);CHKERRQ(ierr);
  /*ierr = TSSetType(ts,TSMPRK);CHKERRQ(ierr);*/
  ierr = TSSetMaxTime(ts,10);CHKERRQ(ierr);
  ierr = TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);CHKERRQ(ierr);

  /* Compute initial conditions and starting time step */
  ierr = FVSample_2WaySplit(&ctx,da,0,X0);CHKERRQ(ierr);
  ierr = FVRHSFunction_2WaySplit(ts,0,X0,X,(void*)&ctx);CHKERRQ(ierr); /* Initial function evaluation, only used to determine max speed */
  ierr = VecCopy(X0,X);CHKERRQ(ierr);                        /* The function value was not used so we set X=X0 again */
  ierr = TSSetTimeStep(ts,ctx.cfl/ctx.cfl_idt);CHKERRQ(ierr);
  ierr = TSSetFromOptions(ts);CHKERRQ(ierr); /* Take runtime options */
  ierr = SolutionStatsView(da,X,PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr);
  {
    PetscInt          steps;
    PetscScalar       mass_initial,mass_final,mass_difference,mass_differenceg;
    const PetscScalar *ptr_X,*ptr_X0;
    const PetscReal   hs = (ctx.xmax-ctx.xmin)*3.0/4.0/count_slow;
    const PetscReal   hf = (ctx.xmax-ctx.xmin)/4.0/count_fast;

    ierr = TSSolve(ts,X);CHKERRQ(ierr);
    ierr = TSGetSolveTime(ts,&ptime);CHKERRQ(ierr);
    ierr = TSGetStepNumber(ts,&steps);CHKERRQ(ierr);
    /* calculate the total mass at initial time and final time */
    mass_initial = 0.0;
    mass_final   = 0.0;
    ierr = DMDAVecGetArrayRead(da,X0,(void*)&ptr_X0);CHKERRQ(ierr);
    ierr = DMDAVecGetArrayRead(da,X,(void*)&ptr_X);CHKERRQ(ierr);
    for (i=xs;i<xs+xm;i++) {
      if (i < ctx.sf || i > ctx.fs-1) {
        for (k=0; k<dof; k++) {
          mass_initial = mass_initial + hs*ptr_X0[i*dof+k];
          mass_final = mass_final + hs*ptr_X[i*dof+k];
        }
      } else {
        for (k=0; k<dof; k++) {
          mass_initial = mass_initial + hf*ptr_X0[i*dof+k];
          mass_final = mass_final + hf*ptr_X[i*dof+k];
        }
      }
    }
    ierr = DMDAVecRestoreArrayRead(da,X0,(void*)&ptr_X0);CHKERRQ(ierr);
    ierr = DMDAVecRestoreArrayRead(da,X,(void*)&ptr_X);CHKERRQ(ierr);
    mass_difference = mass_final - mass_initial;
    ierr = MPI_Allreduce(&mass_difference,&mass_differenceg,1,MPIU_SCALAR,MPIU_SUM,comm);CHKERRQ(ierr);
    ierr = PetscPrintf(comm,"Mass difference %g\n",(double)mass_differenceg);CHKERRQ(ierr);
    ierr = PetscPrintf(comm,"Final time %g, steps %D\n",(double)ptime,steps);CHKERRQ(ierr);
    ierr = PetscPrintf(comm,"Maximum allowable stepsize according to CFL %g\n",(double)(1.0/ctx.cfl_idt));CHKERRQ(ierr);
    if (ctx.exact) {
      PetscReal nrm1=0;
      ierr = SolutionErrorNorms_2WaySplit(&ctx,da,ptime,X,&nrm1);CHKERRQ(ierr);
      ierr = PetscPrintf(comm,"Error ||x-x_e||_1 %g\n",(double)nrm1);CHKERRQ(ierr);
    }
    if (ctx.simulation) {
      PetscReal    nrm1=0;
      PetscViewer  fd;
      char         filename[PETSC_MAX_PATH_LEN] = "binaryoutput";
      Vec          XR;
      PetscBool    flg;
      const PetscScalar  *ptr_XR;
      ierr = PetscOptionsGetString(NULL,NULL,"-f",filename,sizeof(filename),&flg);CHKERRQ(ierr);
      if (!flg) SETERRQ(PETSC_COMM_WORLD,PETSC_ERR_USER,"Must indicate binary file with the -f option");
      ierr = PetscViewerBinaryOpen(PETSC_COMM_WORLD,filename,FILE_MODE_READ,&fd);CHKERRQ(ierr);
      ierr = VecDuplicate(X0,&XR);CHKERRQ(ierr);
      ierr = VecLoad(XR,fd);CHKERRQ(ierr);
      ierr = PetscViewerDestroy(&fd);CHKERRQ(ierr);
      ierr = VecGetArrayRead(X,&ptr_X);CHKERRQ(ierr);
      ierr = VecGetArrayRead(XR,&ptr_XR);CHKERRQ(ierr);
      for (i=xs;i<xs+xm;i++) {
        if (i < ctx.sf || i > ctx.fs-1)
          for (k=0; k<dof; k++) nrm1 = nrm1 + hs*PetscAbs(ptr_X[i*dof+k]-ptr_XR[i*dof+k]);
        else
          for (k=0; k<dof; k++) nrm1 = nrm1 + hf*PetscAbs(ptr_X[i*dof+k]-ptr_XR[i*dof+k]);
      }
      ierr = VecRestoreArrayRead(X,&ptr_X);CHKERRQ(ierr);
      ierr = VecRestoreArrayRead(XR,&ptr_XR);CHKERRQ(ierr);
      ierr = PetscPrintf(comm,"Error ||x-x_e||_1 %g\n",(double)nrm1);CHKERRQ(ierr);
      ierr = VecDestroy(&XR);CHKERRQ(ierr);
    }
  }

  ierr = SolutionStatsView(da,X,PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr);
  if (draw & 0x1) {ierr = VecView(X0,PETSC_VIEWER_DRAW_WORLD);CHKERRQ(ierr);}
  if (draw & 0x2) {ierr = VecView(X,PETSC_VIEWER_DRAW_WORLD);CHKERRQ(ierr);}
  if (draw & 0x4) {
    Vec Y;
    ierr = VecDuplicate(X,&Y);CHKERRQ(ierr);
    ierr = FVSample_2WaySplit(&ctx,da,ptime,Y);CHKERRQ(ierr);
    ierr = VecAYPX(Y,-1,X);CHKERRQ(ierr);
    ierr = VecView(Y,PETSC_VIEWER_DRAW_WORLD);CHKERRQ(ierr);
    ierr = VecDestroy(&Y);CHKERRQ(ierr);
  }

  if (view_final) {
    PetscViewer viewer;
    ierr = PetscViewerASCIIOpen(PETSC_COMM_WORLD,final_fname,&viewer);CHKERRQ(ierr);
    ierr = PetscViewerPushFormat(viewer,PETSC_VIEWER_ASCII_MATLAB);CHKERRQ(ierr);
    ierr = VecView(X,viewer);CHKERRQ(ierr);
    ierr = PetscViewerPopFormat(viewer);CHKERRQ(ierr);
    ierr = PetscViewerDestroy(&viewer);CHKERRQ(ierr);
  }

  /* Clean up */
  ierr = (*ctx.physics2.destroy)(ctx.physics2.user);CHKERRQ(ierr);
  for (i=0; i<ctx.physics2.dof; i++) {ierr = PetscFree(ctx.physics2.fieldname[i]);CHKERRQ(ierr);}
  ierr = PetscFree4(ctx.R,ctx.Rinv,ctx.cjmpLR,ctx.cslope);CHKERRQ(ierr);
  ierr = PetscFree3(ctx.uLR,ctx.flux,ctx.speeds);CHKERRQ(ierr);
  ierr = VecDestroy(&X);CHKERRQ(ierr);
  ierr = VecDestroy(&X0);CHKERRQ(ierr);
  ierr = VecDestroy(&R);CHKERRQ(ierr);
  ierr = DMDestroy(&da);CHKERRQ(ierr);
  ierr = TSDestroy(&ts);CHKERRQ(ierr);
  ierr = ISDestroy(&ctx.iss);CHKERRQ(ierr);
  ierr = ISDestroy(&ctx.isf);CHKERRQ(ierr);
  ierr = ISDestroy(&ctx.issb);CHKERRQ(ierr);
  ierr = PetscFree(index_slow);CHKERRQ(ierr);
  ierr = PetscFree(index_fast);CHKERRQ(ierr);
  ierr = PetscFree(index_slowbuffer);CHKERRQ(ierr);
  ierr = PetscFunctionListDestroy(&limiters);CHKERRQ(ierr);
  ierr = PetscFunctionListDestroy(&physics);CHKERRQ(ierr);
  ierr = PetscFinalize();
  return ierr;
}

/*TEST

    build:
      requires: !complex
      depends: finitevolume1d.c

    test:
      suffix: 1
      args: -da_grid_x 60 -initial 7 -xmin -1 -xmax 1 -hratio 2 -limit mc -ts_dt 0.025 -ts_max_steps 24 -ts_type mprk -ts_mprk_type 2a22

    test:
      suffix: 2
      args: -da_grid_x 60 -initial 7 -xmin -1 -xmax 1 -hratio 2 -limit mc -ts_dt 0.025 -ts_max_steps 24 -ts_type mprk -ts_mprk_type 2a22 -ts_use_splitrhsfunction 0
      output_file: output/ex6_1.out

TEST*/