tutorials/multirate/ex8.c

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-medium-fast-medium-slow), \n"
                           "  the meshsize ratio between two adjacient sub-domains is controlled with -hratio,\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"

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(PETSC_SUCCESS);
}

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(PETSC_SUCCESS);
}

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(PETSC_SUCCESS);
}

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

  PetscFunctionBeginUser;
  PetscCall(PetscNew(&user));
  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;
  PetscCall(PetscStrallocpy("u", &ctx->physics2.fieldname[0]));
  user->a = 1;
  PetscOptionsBegin(ctx->comm, ctx->prefix, "Options for advection", "");
  {
    PetscCall(PetscOptionsReal("-physics_advect_a", "Speed", "", user->a, &user->a, NULL));
  }
  PetscOptionsEnd();
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode FVSample_3WaySplit(FVCtx *ctx, DM da, PetscReal time, Vec U)
{
  PetscScalar   *u, *uj, xj, xi;
  PetscInt       i, j, k, dof, xs, xm, Mx;
  const PetscInt N = 200;
  PetscReal      hs, hm, hf;

  PetscFunctionBeginUser;
  PetscCheck(ctx->physics2.sample2, PETSC_COMM_SELF, PETSC_ERR_SUP, "Physics has not provided a sampling function");
  PetscCall(DMDAGetInfo(da, 0, &Mx, 0, 0, 0, 0, 0, &dof, 0, 0, 0, 0, 0));
  PetscCall(DMDAGetCorners(da, &xs, 0, 0, &xm, 0, 0));
  PetscCall(DMDAVecGetArray(da, U, &u));
  PetscCall(PetscMalloc1(dof, &uj));

  hs = (ctx->xmax - ctx->xmin) / 8.0 / ctx->sm;
  hm = (ctx->xmax - ctx->xmin) / 4.0 / (ctx->mf - ctx->sm);
  hf = (ctx->xmax - ctx->xmin) / 4.0 / (ctx->fm - ctx->mf);
  for (i = xs; i < xs + xm; i++) {
    if (i < ctx->sm) {
      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;
        PetscCall((*ctx->physics2.sample2)(ctx->physics2.user, ctx->initial, ctx->bctype, ctx->xmin, ctx->xmax, time, xj, uj));
        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->mf) {
      xi = ctx->xmin + ctx->sm * hs + 0.5 * hm + (i - ctx->sm) * hm;
      /* 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 + hm * (j - N / 2) / (PetscReal)N;
        PetscCall((*ctx->physics2.sample2)(ctx->physics2.user, ctx->initial, ctx->bctype, ctx->xmin, ctx->xmax, time, xj, uj));
        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->fm) {
      xi = ctx->xmin + ctx->sm * hs + (ctx->mf - ctx->sm) * hm + 0.5 * hf + (i - ctx->mf) * 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;
        PetscCall((*ctx->physics2.sample2)(ctx->physics2.user, ctx->initial, ctx->bctype, ctx->xmin, ctx->xmax, time, xj, uj));
        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->ms) {
      xi = ctx->xmin + ctx->sm * hs + (ctx->mf - ctx->sm) * hm + (ctx->fm - ctx->mf) * hf + 0.5 * hm + (i - ctx->fm) * hm;
      /* 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 + hm * (j - N / 2) / (PetscReal)N;
        PetscCall((*ctx->physics2.sample2)(ctx->physics2.user, ctx->initial, ctx->bctype, ctx->xmin, ctx->xmax, time, xj, uj));
        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->sm * hs + (ctx->mf - ctx->sm) * hm + (ctx->fm - ctx->mf) * hf + (ctx->ms - ctx->fm) * hm + 0.5 * hs + (i - ctx->ms) * 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;
        PetscCall((*ctx->physics2.sample2)(ctx->physics2.user, ctx->initial, ctx->bctype, ctx->xmin, ctx->xmax, time, xj, uj));
        for (k = 0; k < dof; k++) u[i * dof + k] += ((j == 0 || j == N) ? 0.5 : 1.0) * uj[k] / N;
      }
    }
  }
  PetscCall(DMDAVecRestoreArray(da, U, &u));
  PetscCall(PetscFree(uj));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode SolutionErrorNorms_3WaySplit(FVCtx *ctx, DM da, PetscReal t, Vec X, PetscReal *nrm1)
{
  Vec                Y;
  PetscInt           i, Mx;
  const PetscScalar *ptr_X, *ptr_Y;
  PetscReal          hs, hm, hf;

  PetscFunctionBeginUser;
  PetscCall(VecGetSize(X, &Mx));
  hs = (ctx->xmax - ctx->xmin) / 8.0 / ctx->sm;
  hm = (ctx->xmax - ctx->xmin) / 4.0 / (ctx->mf - ctx->sm);
  hf = (ctx->xmax - ctx->xmin) / 4.0 / (ctx->fm - ctx->mf);
  PetscCall(VecDuplicate(X, &Y));
  PetscCall(FVSample_3WaySplit(ctx, da, t, Y));
  PetscCall(VecGetArrayRead(X, &ptr_X));
  PetscCall(VecGetArrayRead(Y, &ptr_Y));
  for (i = 0; i < Mx; i++) {
    if (i < ctx->sm || i > ctx->ms - 1) *nrm1 += hs * PetscAbs(ptr_X[i] - ptr_Y[i]);
    else if (i < ctx->mf || i > ctx->fm - 1) *nrm1 += hm * PetscAbs(ptr_X[i] - ptr_Y[i]);
    else *nrm1 += hf * PetscAbs(ptr_X[i] - ptr_Y[i]);
  }
  PetscCall(VecRestoreArrayRead(X, &ptr_X));
  PetscCall(VecRestoreArrayRead(Y, &ptr_Y));
  PetscCall(VecDestroy(&Y));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode FVRHSFunction_3WaySplit(TS ts, PetscReal time, Vec X, Vec F, void *vctx)
{
  FVCtx       *ctx = (FVCtx *)vctx;
  PetscInt     i, j, k, Mx, dof, xs, xm, sm = ctx->sm, mf = ctx->mf, fm = ctx->fm, ms = ctx->ms;
  PetscReal    hxf, hxm, hxs, cfl_idt = 0;
  PetscScalar *x, *f, *slope;
  Vec          Xloc;
  DM           da;

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

  PetscCall(VecZeroEntries(F)); /* F is the right-hand side function corresponds to center points */

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

  PetscCall(DMDAGetCorners(da, &xs, 0, 0, &xm, 0, 0));

  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} */
    PetscCall((*ctx->physics2.characteristic2)(ctx->physics2.user, dof, &x[i * dof], ctx->R, ctx->Rinv, ctx->speeds));
    /* Evaluate jumps across interfaces (i-1, i) and (i, i+1), put in characteristic basis */
    PetscCall(PetscArrayzero(ctx->cjmpLR, 2 * dof));
    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.hxm = hxm;
    info.hxf = hxf;
    (*ctx->limit3)(&info, cjmpL, cjmpR, ctx->sm, ctx->mf, ctx->fm, ctx->ms, 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 < sm || i > ms) { /* 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;
      }
      PetscCall((*ctx->physics2.riemann2)(ctx->physics2.user, dof, uL, uR, ctx->flux, &maxspeed));
      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 == sm) { /* interface between slow and medium component */
      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] * hxm / 2;
      }
      PetscCall((*ctx->physics2.riemann2)(ctx->physics2.user, dof, uL, uR, ctx->flux, &maxspeed));
      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] / hxm;
      }
    } else if (i == ms) { /* interface between medium and slow regions */
      for (j = 0; j < dof; j++) {
        uL[j] = x[(i - 1) * dof + j] + slope[(i - 1) * dof + j] * hxm / 2;
        uR[j] = x[(i - 0) * dof + j] - slope[(i - 0) * dof + j] * hxs / 2;
      }
      PetscCall((*ctx->physics2.riemann2)(ctx->physics2.user, dof, uL, uR, ctx->flux, &maxspeed));
      if (i > xs) {
        for (j = 0; j < dof; j++) f[(i - 1) * dof + j] -= ctx->flux[j] / hxm;
      }
      if (i < xs + xm) {
        for (j = 0; j < dof; j++) f[i * dof + j] += ctx->flux[j] / hxs;
      }
    } else if (i < mf || i > fm) { /* medium region */
      for (j = 0; j < dof; j++) {
        uL[j] = x[(i - 1) * dof + j] + slope[(i - 1) * dof + j] * hxm / 2;
        uR[j] = x[(i - 0) * dof + j] - slope[(i - 0) * dof + j] * hxm / 2;
      }
      PetscCall((*ctx->physics2.riemann2)(ctx->physics2.user, dof, uL, uR, ctx->flux, &maxspeed));
      if (i > xs) {
        for (j = 0; j < dof; j++) f[(i - 1) * dof + j] -= ctx->flux[j] / hxm;
      }
      if (i < xs + xm) {
        for (j = 0; j < dof; j++) f[i * dof + j] += ctx->flux[j] / hxm;
      }
    } else if (i == mf) { /* interface between medium and fast regions */
      for (j = 0; j < dof; j++) {
        uL[j] = x[(i - 1) * dof + j] + slope[(i - 1) * dof + j] * hxm / 2;
        uR[j] = x[(i - 0) * dof + j] - slope[(i - 0) * dof + j] * hxf / 2;
      }
      PetscCall((*ctx->physics2.riemann2)(ctx->physics2.user, dof, uL, uR, ctx->flux, &maxspeed));
      if (i > xs) {
        for (j = 0; j < dof; j++) f[(i - 1) * dof + j] -= ctx->flux[j] / hxm;
      }
      if (i < xs + xm) {
        for (j = 0; j < dof; j++) f[i * dof + j] += ctx->flux[j] / hxf;
      }
    } else if (i == fm) { /* interface between fast and medium regions */
      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] * hxm / 2;
      }
      PetscCall((*ctx->physics2.riemann2)(ctx->physics2.user, dof, uL, uR, ctx->flux, &maxspeed));
      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] / hxm;
      }
    } else { /* 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;
      }
      PetscCall((*ctx->physics2.riemann2)(ctx->physics2.user, dof, uL, uR, ctx->flux, &maxspeed));
      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] / hxf;
      }
      if (i < xs + xm) {
        for (j = 0; j < dof; j++) f[i * dof + j] += ctx->flux[j] / hxf;
      }
    }
  }
  PetscCall(DMDAVecRestoreArray(da, Xloc, &x));
  PetscCall(DMDAVecRestoreArray(da, F, &f));
  PetscCall(DMDARestoreArray(da, PETSC_TRUE, &slope));
  PetscCall(DMRestoreLocalVector(da, &Xloc));
  PetscCall(MPIU_Allreduce(&cfl_idt, &ctx->cfl_idt, 1, MPIU_SCALAR, MPIU_MAX, PetscObjectComm((PetscObject)da)));
  if (0) {
    /* We need to a way to inform the TS of a CFL constraint, this is a debugging fragment */
    PetscReal dt, tnow;
    PetscCall(TSGetTimeStep(ts, &dt));
    PetscCall(TSGetTime(ts, &tnow));
    if (dt > 0.5 / ctx->cfl_idt) PetscCall(PetscPrintf(ctx->comm, "Stability constraint exceeded at t=%g, dt %g > %g\n", (double)tnow, (double)dt, (double)(0.5 / ctx->cfl_idt)));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/* --------------------------------- Finite Volume Solver for slow components ----------------------------------- */
PetscErrorCode FVRHSFunctionslow_3WaySplit(TS ts, PetscReal time, Vec X, Vec F, void *vctx)
{
  FVCtx       *ctx = (FVCtx *)vctx;
  PetscInt     i, j, k, Mx, dof, xs, xm, islow = 0, sm = ctx->sm, ms = ctx->ms, lsbwidth = ctx->lsbwidth, rsbwidth = ctx->rsbwidth;
  PetscReal    hxs, hxm, hxf, cfl_idt = 0;
  PetscScalar *x, *f, *slope;
  Vec          Xloc;
  DM           da;

  PetscFunctionBeginUser;
  PetscCall(TSGetDM(ts, &da));
  PetscCall(DMGetLocalVector(da, &Xloc));
  PetscCall(DMDAGetInfo(da, 0, &Mx, 0, 0, 0, 0, 0, &dof, 0, 0, 0, 0, 0));
  hxs = (ctx->xmax - ctx->xmin) / 8.0 / ctx->sm;
  hxm = (ctx->xmax - ctx->xmin) / 4.0 / (ctx->mf - ctx->sm);
  hxf = (ctx->xmax - ctx->xmin) / 4.0 / (ctx->fm - ctx->mf);
  PetscCall(DMGlobalToLocalBegin(da, X, INSERT_VALUES, Xloc));
  PetscCall(DMGlobalToLocalEnd(da, X, INSERT_VALUES, Xloc));
  PetscCall(VecZeroEntries(F));
  PetscCall(DMDAVecGetArray(da, Xloc, &x));
  PetscCall(VecGetArray(F, &f));
  PetscCall(DMDAGetArray(da, PETSC_TRUE, &slope));
  PetscCall(DMDAGetCorners(da, &xs, 0, 0, &xm, 0, 0));

  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 < sm - lsbwidth + 1 || i > ms + rsbwidth - 2) { /* slow components and the first and last fast components */
      /* Determine the right eigenvectors R, where A = R \Lambda R^{-1} */
      PetscCall((*ctx->physics2.characteristic2)(ctx->physics2.user, dof, &x[i * dof], ctx->R, ctx->Rinv, ctx->speeds));
      /* Evaluate jumps across interfaces (i-1, i) and (i, i+1), put in characteristic basis */
      PetscCall(PetscArrayzero(ctx->cjmpLR, 2 * dof));
      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.hxm = hxm;
      info.hxf = hxf;
      (*ctx->limit3)(&info, cjmpL, cjmpR, ctx->sm, ctx->mf, ctx->fm, ctx->ms, 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 < sm - 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;
      }
      PetscCall((*ctx->physics2.riemann2)(ctx->physics2.user, dof, uL, uR, ctx->flux, &maxspeed));
      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 == sm - lsbwidth) { /* interface between slow and medium regions */
      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;
      }
      PetscCall((*ctx->physics2.riemann2)(ctx->physics2.user, dof, uL, uR, ctx->flux, &maxspeed));
      if (i > xs) {
        for (j = 0; j < dof; j++) f[(islow - 1) * dof + j] -= ctx->flux[j] / hxs;
      }
    }
    if (i == ms + rsbwidth) { /* interface between medium and slow regions */
      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;
      }
      PetscCall((*ctx->physics2.riemann2)(ctx->physics2.user, dof, uL, uR, ctx->flux, &maxspeed));
      if (i < xs + xm) {
        for (j = 0; j < dof; j++) f[islow * dof + j] += ctx->flux[j] / hxs;
        islow++;
      }
    }
    if (i > ms + 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;
      }
      PetscCall((*ctx->physics2.riemann2)(ctx->physics2.user, dof, uL, uR, ctx->flux, &maxspeed));
      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++;
      }
    }
  }
  PetscCall(DMDAVecRestoreArray(da, Xloc, &x));
  PetscCall(VecRestoreArray(F, &f));
  PetscCall(DMDARestoreArray(da, PETSC_TRUE, &slope));
  PetscCall(DMRestoreLocalVector(da, &Xloc));
  PetscCall(MPIU_Allreduce(&cfl_idt, &ctx->cfl_idt, 1, MPIU_SCALAR, MPIU_MAX, PetscObjectComm((PetscObject)da)));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode FVRHSFunctionslowbuffer_3WaySplit(TS ts, PetscReal time, Vec X, Vec F, void *vctx)
{
  FVCtx       *ctx = (FVCtx *)vctx;
  PetscInt     i, j, k, Mx, dof, xs, xm, islowbuffer = 0, sm = ctx->sm, ms = ctx->ms, lsbwidth = ctx->lsbwidth, rsbwidth = ctx->rsbwidth;
  PetscReal    hxs, hxm, hxf;
  PetscScalar *x, *f, *slope;
  Vec          Xloc;
  DM           da;

  PetscFunctionBeginUser;
  PetscCall(TSGetDM(ts, &da));
  PetscCall(DMGetLocalVector(da, &Xloc));
  PetscCall(DMDAGetInfo(da, 0, &Mx, 0, 0, 0, 0, 0, &dof, 0, 0, 0, 0, 0));
  hxs = (ctx->xmax - ctx->xmin) / 8.0 / ctx->sm;
  hxm = (ctx->xmax - ctx->xmin) / 4.0 / (ctx->mf - ctx->sm);
  hxf = (ctx->xmax - ctx->xmin) / 4.0 / (ctx->fm - ctx->mf);
  PetscCall(DMGlobalToLocalBegin(da, X, INSERT_VALUES, Xloc));
  PetscCall(DMGlobalToLocalEnd(da, X, INSERT_VALUES, Xloc));
  PetscCall(VecZeroEntries(F));
  PetscCall(DMDAVecGetArray(da, Xloc, &x));
  PetscCall(VecGetArray(F, &f));
  PetscCall(DMDAGetArray(da, PETSC_TRUE, &slope));
  PetscCall(DMDAGetCorners(da, &xs, 0, 0, &xm, 0, 0));

  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 > sm - lsbwidth - 2 && i < sm + 1) || (i > ms - 2 && i < ms + rsbwidth + 1)) {
      /* Determine the right eigenvectors R, where A = R \Lambda R^{-1} */
      PetscCall((*ctx->physics2.characteristic2)(ctx->physics2.user, dof, &x[i * dof], ctx->R, ctx->Rinv, ctx->speeds));
      /* Evaluate jumps across interfaces (i-1, i) and (i, i+1), put in characteristic basis */
      PetscCall(PetscArrayzero(ctx->cjmpLR, 2 * dof));
      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.hxm = hxm;
      info.hxf = hxf;
      (*ctx->limit3)(&info, cjmpL, cjmpR, ctx->sm, ctx->mf, ctx->fm, ctx->ms, 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 == sm - 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;
      }
      PetscCall((*ctx->physics2.riemann2)(ctx->physics2.user, dof, uL, uR, ctx->flux, &maxspeed));
      if (i < xs + xm) {
        for (j = 0; j < dof; j++) f[islowbuffer * dof + j] += ctx->flux[j] / hxs;
        islowbuffer++;
      }
    }
    if (i > sm - lsbwidth && i < sm) {
      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;
      }
      PetscCall((*ctx->physics2.riemann2)(ctx->physics2.user, dof, uL, uR, ctx->flux, &maxspeed));
      if (i > xs) {
        for (j = 0; j < dof; j++) f[(islowbuffer - 1) * dof + j] -= ctx->flux[j] / hxs;
      }
      if (i < xs + xm) {
        for (j = 0; j < dof; j++) f[islowbuffer * dof + j] += ctx->flux[j] / hxs;
        islowbuffer++;
      }
    }
    if (i == sm) { /* interface between the slow region and the medium 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] * hxm / 2;
      }
      PetscCall((*ctx->physics2.riemann2)(ctx->physics2.user, dof, uL, uR, ctx->flux, &maxspeed));
      if (i > xs) {
        for (j = 0; j < dof; j++) f[(islowbuffer - 1) * dof + j] -= ctx->flux[j] / hxs;
      }
    }
    if (i == ms) { /* interface between the medium region and the slow region */
      for (j = 0; j < dof; j++) {
        uL[j] = x[(i - 1) * dof + j] + slope[(i - 1) * dof + j] * hxm / 2;
        uR[j] = x[(i - 0) * dof + j] - slope[(i - 0) * dof + j] * hxs / 2;
      }
      PetscCall((*ctx->physics2.riemann2)(ctx->physics2.user, dof, uL, uR, ctx->flux, &maxspeed));
      if (i < xs + xm) {
        for (j = 0; j < dof; j++) f[islowbuffer * dof + j] += ctx->flux[j] / hxs;
        islowbuffer++;
      }
    }
    if (i > ms && i < ms + 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;
      }
      PetscCall((*ctx->physics2.riemann2)(ctx->physics2.user, dof, uL, uR, ctx->flux, &maxspeed));
      if (i > xs) {
        for (j = 0; j < dof; j++) f[(islowbuffer - 1) * dof + j] -= ctx->flux[j] / hxs;
      }
      if (i < xs + xm) {
        for (j = 0; j < dof; j++) f[islowbuffer * dof + j] += ctx->flux[j] / hxs;
        islowbuffer++;
      }
    }
    if (i == ms + 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;
      }
      PetscCall((*ctx->physics2.riemann2)(ctx->physics2.user, dof, uL, uR, ctx->flux, &maxspeed));
      if (i > xs) {
        for (j = 0; j < dof; j++) f[(islowbuffer - 1) * dof + j] -= ctx->flux[j] / hxs;
      }
    }
  }
  PetscCall(DMDAVecRestoreArray(da, Xloc, &x));
  PetscCall(VecRestoreArray(F, &f));
  PetscCall(DMDARestoreArray(da, PETSC_TRUE, &slope));
  PetscCall(DMRestoreLocalVector(da, &Xloc));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/* --------------------------------- Finite Volume Solver for medium components ----------------------------------- */
PetscErrorCode FVRHSFunctionmedium_3WaySplit(TS ts, PetscReal time, Vec X, Vec F, void *vctx)
{
  FVCtx       *ctx = (FVCtx *)vctx;
  PetscInt     i, j, k, Mx, dof, xs, xm, imedium = 0, sm = ctx->sm, mf = ctx->mf, fm = ctx->fm, ms = ctx->ms, lmbwidth = ctx->lmbwidth, rmbwidth = ctx->rmbwidth;
  PetscReal    hxs, hxm, hxf;
  PetscScalar *x, *f, *slope;
  Vec          Xloc;
  DM           da;

  PetscFunctionBeginUser;
  PetscCall(TSGetDM(ts, &da));
  PetscCall(DMGetLocalVector(da, &Xloc));
  PetscCall(DMDAGetInfo(da, 0, &Mx, 0, 0, 0, 0, 0, &dof, 0, 0, 0, 0, 0));
  hxs = (ctx->xmax - ctx->xmin) / 8.0 / ctx->sm;
  hxm = (ctx->xmax - ctx->xmin) / 4.0 / (ctx->mf - ctx->sm);
  hxf = (ctx->xmax - ctx->xmin) / 4.0 / (ctx->fm - ctx->mf);
  PetscCall(DMGlobalToLocalBegin(da, X, INSERT_VALUES, Xloc));
  PetscCall(DMGlobalToLocalEnd(da, X, INSERT_VALUES, Xloc));
  PetscCall(VecZeroEntries(F));
  PetscCall(DMDAVecGetArray(da, Xloc, &x));
  PetscCall(VecGetArray(F, &f));
  PetscCall(DMDAGetArray(da, PETSC_TRUE, &slope));
  PetscCall(DMDAGetCorners(da, &xs, 0, 0, &xm, 0, 0));

  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 > sm - 2 && i < mf - lmbwidth + 1) || (i > fm + rmbwidth - 2 && i < ms + 1)) { /* slow components and the first and last fast components */
      /* Determine the right eigenvectors R, where A = R \Lambda R^{-1} */
      PetscCall((*ctx->physics2.characteristic2)(ctx->physics2.user, dof, &x[i * dof], ctx->R, ctx->Rinv, ctx->speeds));
      /* Evaluate jumps across interfaces (i-1, i) and (i, i+1), put in characteristic basis */
      PetscCall(PetscArrayzero(ctx->cjmpLR, 2 * dof));
      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.hxm = hxm;
      info.hxf = hxf;
      (*ctx->limit3)(&info, cjmpL, cjmpR, ctx->sm, ctx->mf, ctx->fm, ctx->ms, 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 == sm) { /* interface between slow and medium regions */
      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] * hxm / 2;
      }
      PetscCall((*ctx->physics2.riemann2)(ctx->physics2.user, dof, uL, uR, ctx->flux, &maxspeed));
      if (i < xs + xm) {
        for (j = 0; j < dof; j++) f[imedium * dof + j] += ctx->flux[j] / hxm;
        imedium++;
      }
    }
    if (i > sm && i < mf - lmbwidth) { /* medium region */
      for (j = 0; j < dof; j++) {
        uL[j] = x[(i - 1) * dof + j] + slope[(i - 1) * dof + j] * hxm / 2;
        uR[j] = x[(i - 0) * dof + j] - slope[(i - 0) * dof + j] * hxm / 2;
      }
      PetscCall((*ctx->physics2.riemann2)(ctx->physics2.user, dof, uL, uR, ctx->flux, &maxspeed));
      if (i > xs) {
        for (j = 0; j < dof; j++) f[(imedium - 1) * dof + j] -= ctx->flux[j] / hxm;
      }
      if (i < xs + xm) {
        for (j = 0; j < dof; j++) f[imedium * dof + j] += ctx->flux[j] / hxm;
        imedium++;
      }
    }
    if (i == mf - lmbwidth) { /* interface between medium and fast regions */
      for (j = 0; j < dof; j++) {
        uL[j] = x[(i - 1) * dof + j] + slope[(i - 1) * dof + j] * hxm / 2;
        uR[j] = x[(i - 0) * dof + j] - slope[(i - 0) * dof + j] * hxm / 2;
      }
      PetscCall((*ctx->physics2.riemann2)(ctx->physics2.user, dof, uL, uR, ctx->flux, &maxspeed));
      if (i > xs) {
        for (j = 0; j < dof; j++) f[(imedium - 1) * dof + j] -= ctx->flux[j] / hxm;
      }
    }
    if (i == fm + rmbwidth) { /* interface between fast and medium regions */
      for (j = 0; j < dof; j++) {
        uL[j] = x[(i - 1) * dof + j] + slope[(i - 1) * dof + j] * hxm / 2;
        uR[j] = x[(i - 0) * dof + j] - slope[(i - 0) * dof + j] * hxm / 2;
      }
      PetscCall((*ctx->physics2.riemann2)(ctx->physics2.user, dof, uL, uR, ctx->flux, &maxspeed));
      if (i < xs + xm) {
        for (j = 0; j < dof; j++) f[imedium * dof + j] += ctx->flux[j] / hxm;
        imedium++;
      }
    }
    if (i > fm + rmbwidth && i < ms) { /* medium region */
      for (j = 0; j < dof; j++) {
        uL[j] = x[(i - 1) * dof + j] + slope[(i - 1) * dof + j] * hxm / 2;
        uR[j] = x[(i - 0) * dof + j] - slope[(i - 0) * dof + j] * hxm / 2;
      }
      PetscCall((*ctx->physics2.riemann2)(ctx->physics2.user, dof, uL, uR, ctx->flux, &maxspeed));
      if (i > xs) {
        for (j = 0; j < dof; j++) f[(imedium - 1) * dof + j] -= ctx->flux[j] / hxm;
      }
      if (i < xs + xm) {
        for (j = 0; j < dof; j++) f[imedium * dof + j] += ctx->flux[j] / hxm;
        imedium++;
      }
    }
    if (i == ms) { /* interface between medium and slow regions */
      for (j = 0; j < dof; j++) {
        uL[j] = x[(i - 1) * dof + j] + slope[(i - 1) * dof + j] * hxm / 2;
        uR[j] = x[(i - 0) * dof + j] - slope[(i - 0) * dof + j] * hxs / 2;
      }
      PetscCall((*ctx->physics2.riemann2)(ctx->physics2.user, dof, uL, uR, ctx->flux, &maxspeed));
      if (i > xs) {
        for (j = 0; j < dof; j++) f[(imedium - 1) * dof + j] -= ctx->flux[j] / hxm;
      }
    }
  }
  PetscCall(DMDAVecRestoreArray(da, Xloc, &x));
  PetscCall(VecRestoreArray(F, &f));
  PetscCall(DMDARestoreArray(da, PETSC_TRUE, &slope));
  PetscCall(DMRestoreLocalVector(da, &Xloc));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode FVRHSFunctionmediumbuffer_3WaySplit(TS ts, PetscReal time, Vec X, Vec F, void *vctx)
{
  FVCtx       *ctx = (FVCtx *)vctx;
  PetscInt     i, j, k, Mx, dof, xs, xm, imediumbuffer = 0, mf = ctx->mf, fm = ctx->fm, lmbwidth = ctx->lmbwidth, rmbwidth = ctx->rmbwidth;
  PetscReal    hxs, hxm, hxf;
  PetscScalar *x, *f, *slope;
  Vec          Xloc;
  DM           da;

  PetscFunctionBeginUser;
  PetscCall(TSGetDM(ts, &da));
  PetscCall(DMGetLocalVector(da, &Xloc));
  PetscCall(DMDAGetInfo(da, 0, &Mx, 0, 0, 0, 0, 0, &dof, 0, 0, 0, 0, 0));
  hxs = (ctx->xmax - ctx->xmin) / 8.0 / ctx->sm;
  hxm = (ctx->xmax - ctx->xmin) / 4.0 / (ctx->mf - ctx->sm);
  hxf = (ctx->xmax - ctx->xmin) / 4.0 / (ctx->fm - ctx->mf);
  PetscCall(DMGlobalToLocalBegin(da, X, INSERT_VALUES, Xloc));
  PetscCall(DMGlobalToLocalEnd(da, X, INSERT_VALUES, Xloc));
  PetscCall(VecZeroEntries(F));
  PetscCall(DMDAVecGetArray(da, Xloc, &x));
  PetscCall(VecGetArray(F, &f));
  PetscCall(DMDAGetArray(da, PETSC_TRUE, &slope));
  PetscCall(DMDAGetCorners(da, &xs, 0, 0, &xm, 0, 0));

  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 > mf - lmbwidth - 2 && i < mf + 1) || (i > fm - 2 && i < fm + rmbwidth + 1)) {
      /* Determine the right eigenvectors R, where A = R \Lambda R^{-1} */
      PetscCall((*ctx->physics2.characteristic2)(ctx->physics2.user, dof, &x[i * dof], ctx->R, ctx->Rinv, ctx->speeds));
      /* Evaluate jumps across interfaces (i-1, i) and (i, i+1), put in characteristic basis */
      PetscCall(PetscArrayzero(ctx->cjmpLR, 2 * dof));
      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.hxm = hxm;
      info.hxf = hxf;
      (*ctx->limit3)(&info, cjmpL, cjmpR, ctx->sm, ctx->mf, ctx->fm, ctx->ms, 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 == mf - lmbwidth) {
      for (j = 0; j < dof; j++) {
        uL[j] = x[(i - 1) * dof + j] + slope[(i - 1) * dof + j] * hxm / 2;
        uR[j] = x[(i - 0) * dof + j] - slope[(i - 0) * dof + j] * hxm / 2;
      }
      PetscCall((*ctx->physics2.riemann2)(ctx->physics2.user, dof, uL, uR, ctx->flux, &maxspeed));
      if (i < xs + xm) {
        for (j = 0; j < dof; j++) f[imediumbuffer * dof + j] += ctx->flux[j] / hxm;
        imediumbuffer++;
      }
    }
    if (i > mf - lmbwidth && i < mf) {
      for (j = 0; j < dof; j++) {
        uL[j] = x[(i - 1) * dof + j] + slope[(i - 1) * dof + j] * hxm / 2;
        uR[j] = x[(i - 0) * dof + j] - slope[(i - 0) * dof + j] * hxm / 2;
      }
      PetscCall((*ctx->physics2.riemann2)(ctx->physics2.user, dof, uL, uR, ctx->flux, &maxspeed));
      if (i > xs) {
        for (j = 0; j < dof; j++) f[(imediumbuffer - 1) * dof + j] -= ctx->flux[j] / hxm;
      }
      if (i < xs + xm) {
        for (j = 0; j < dof; j++) f[imediumbuffer * dof + j] += ctx->flux[j] / hxm;
        imediumbuffer++;
      }
    }
    if (i == mf) { /* interface between the medium region and the fast region */
      for (j = 0; j < dof; j++) {
        uL[j] = x[(i - 1) * dof + j] + slope[(i - 1) * dof + j] * hxm / 2;
        uR[j] = x[(i - 0) * dof + j] - slope[(i - 0) * dof + j] * hxf / 2;
      }
      PetscCall((*ctx->physics2.riemann2)(ctx->physics2.user, dof, uL, uR, ctx->flux, &maxspeed));
      if (i > xs) {
        for (j = 0; j < dof; j++) f[(imediumbuffer - 1) * dof + j] -= ctx->flux[j] / hxm;
      }
    }
    if (i == fm) { /* interface between the fast region and the medium 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] * hxm / 2;
      }
      PetscCall((*ctx->physics2.riemann2)(ctx->physics2.user, dof, uL, uR, ctx->flux, &maxspeed));
      if (i < xs + xm) {
        for (j = 0; j < dof; j++) f[imediumbuffer * dof + j] += ctx->flux[j] / hxm;
        imediumbuffer++;
      }
    }
    if (i > fm && i < fm + rmbwidth) {
      for (j = 0; j < dof; j++) {
        uL[j] = x[(i - 1) * dof + j] + slope[(i - 1) * dof + j] * hxm / 2;
        uR[j] = x[(i - 0) * dof + j] - slope[(i - 0) * dof + j] * hxm / 2;
      }
      PetscCall((*ctx->physics2.riemann2)(ctx->physics2.user, dof, uL, uR, ctx->flux, &maxspeed));
      if (i > xs) {
        for (j = 0; j < dof; j++) f[(imediumbuffer - 1) * dof + j] -= ctx->flux[j] / hxm;
      }
      if (i < xs + xm) {
        for (j = 0; j < dof; j++) f[imediumbuffer * dof + j] += ctx->flux[j] / hxm;
        imediumbuffer++;
      }
    }
    if (i == fm + rmbwidth) {
      for (j = 0; j < dof; j++) {
        uL[j] = x[(i - 1) * dof + j] + slope[(i - 1) * dof + j] * hxm / 2;
        uR[j] = x[(i - 0) * dof + j] - slope[(i - 0) * dof + j] * hxm / 2;
      }
      PetscCall((*ctx->physics2.riemann2)(ctx->physics2.user, dof, uL, uR, ctx->flux, &maxspeed));
      if (i > xs) {
        for (j = 0; j < dof; j++) f[(imediumbuffer - 1) * dof + j] -= ctx->flux[j] / hxm;
      }
    }
  }
  PetscCall(DMDAVecRestoreArray(da, Xloc, &x));
  PetscCall(VecRestoreArray(F, &f));
  PetscCall(DMDARestoreArray(da, PETSC_TRUE, &slope));
  PetscCall(DMRestoreLocalVector(da, &Xloc));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/* --------------------------------- Finite Volume Solver for fast  parts ----------------------------------- */
PetscErrorCode FVRHSFunctionfast_3WaySplit(TS ts, PetscReal time, Vec X, Vec F, void *vctx)
{
  FVCtx       *ctx = (FVCtx *)vctx;
  PetscInt     i, j, k, Mx, dof, xs, xm, ifast = 0, mf = ctx->mf, fm = ctx->fm;
  PetscReal    hxs, hxm, hxf;
  PetscScalar *x, *f, *slope;
  Vec          Xloc;
  DM           da;

  PetscFunctionBeginUser;
  PetscCall(TSGetDM(ts, &da));
  PetscCall(DMGetLocalVector(da, &Xloc));
  PetscCall(DMDAGetInfo(da, 0, &Mx, 0, 0, 0, 0, 0, &dof, 0, 0, 0, 0, 0));
  hxs = (ctx->xmax - ctx->xmin) / 8.0 / ctx->sm;
  hxm = (ctx->xmax - ctx->xmin) / 4.0 / (ctx->mf - ctx->sm);
  hxf = (ctx->xmax - ctx->xmin) / 4.0 / (ctx->fm - ctx->mf);
  PetscCall(DMGlobalToLocalBegin(da, X, INSERT_VALUES, Xloc));
  PetscCall(DMGlobalToLocalEnd(da, X, INSERT_VALUES, Xloc));
  PetscCall(VecZeroEntries(F));
  PetscCall(DMDAVecGetArray(da, Xloc, &x));
  PetscCall(VecGetArray(F, &f));
  PetscCall(DMDAGetArray(da, PETSC_TRUE, &slope));
  PetscCall(DMDAGetCorners(da, &xs, 0, 0, &xm, 0, 0));

  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++) { /* fast components and the last slow components before fast components and the first slow component after fast components */
    struct _LimitInfo info;
    PetscScalar      *cjmpL, *cjmpR;
    if (i > mf - 2 && i < fm + 1) {
      PetscCall((*ctx->physics2.characteristic2)(ctx->physics2.user, dof, &x[i * dof], ctx->R, ctx->Rinv, ctx->speeds));
      PetscCall(PetscArrayzero(ctx->cjmpLR, 2 * dof));
      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.hxm = hxm;
      info.hxf = hxf;
      (*ctx->limit3)(&info, cjmpL, cjmpR, ctx->sm, ctx->mf, ctx->fm, ctx->ms, 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 == mf) { /* interface between medium and fast regions */
      for (j = 0; j < dof; j++) {
        uL[j] = x[(i - 1) * dof + j] + slope[(i - 1) * dof + j] * hxm / 2;
        uR[j] = x[(i - 0) * dof + j] - slope[(i - 0) * dof + j] * hxf / 2;
      }
      PetscCall((*ctx->physics2.riemann2)(ctx->physics2.user, dof, uL, uR, ctx->flux, &maxspeed));
      if (i < xs + xm) {
        for (j = 0; j < dof; j++) f[ifast * dof + j] += ctx->flux[j] / hxf;
        ifast++;
      }
    }
    if (i > mf && i < fm) { /* 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;
      }
      PetscCall((*ctx->physics2.riemann2)(ctx->physics2.user, dof, uL, uR, ctx->flux, &maxspeed));
      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 == fm) { /* interface between fast and medium regions */
      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] * hxm / 2;
      }
      PetscCall((*ctx->physics2.riemann2)(ctx->physics2.user, dof, uL, uR, ctx->flux, &maxspeed));
      if (i > xs) {
        for (j = 0; j < dof; j++) f[(ifast - 1) * dof + j] -= ctx->flux[j] / hxf;
      }
    }
  }
  PetscCall(DMDAVecRestoreArray(da, Xloc, &x));
  PetscCall(VecRestoreArray(F, &f));
  PetscCall(DMDARestoreArray(da, PETSC_TRUE, &slope));
  PetscCall(DMRestoreLocalVector(da, &Xloc));
  PetscFunctionReturn(PETSC_SUCCESS);
}

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_medium, count_fast, islow = 0, imedium = 0, ifast = 0, *index_slow, *index_medium, *index_fast, islowbuffer = 0, *index_slowbuffer, imediumbuffer = 0, *index_mediumbuffer;
  PetscBool         view_final = PETSC_FALSE;
  PetscReal         ptime;

  PetscFunctionBeginUser;
  PetscCall(PetscInitialize(&argc, &argv, 0, help));
  comm = PETSC_COMM_WORLD;
  PetscCall(PetscMemzero(&ctx, sizeof(ctx)));

  /* Register limiters to be available on the command line */
  PetscCall(PetscFunctionListAdd(&limiters, "upwind", Limit3_Upwind));
  PetscCall(PetscFunctionListAdd(&limiters, "lax-wendroff", Limit3_LaxWendroff));
  PetscCall(PetscFunctionListAdd(&limiters, "beam-warming", Limit3_BeamWarming));
  PetscCall(PetscFunctionListAdd(&limiters, "fromm", Limit3_Fromm));
  PetscCall(PetscFunctionListAdd(&limiters, "minmod", Limit3_Minmod));
  PetscCall(PetscFunctionListAdd(&limiters, "superbee", Limit3_Superbee));
  PetscCall(PetscFunctionListAdd(&limiters, "mc", Limit3_MC));
  PetscCall(PetscFunctionListAdd(&limiters, "koren3", Limit3_Koren3));

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

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

  /* Choose the limiter from the list of registered limiters */
  PetscCall(PetscFunctionListFind(limiters, lname, &ctx.limit3));
  PetscCheck(ctx.limit3, PETSC_COMM_SELF, PETSC_ERR_ARG_UNKNOWN_TYPE, "Limiter '%s' not found", lname);

  /* Choose the physics from the list of registered models */
  {
    PetscErrorCode (*r)(FVCtx *);
    PetscCall(PetscFunctionListFind(physics, physname, &r));
    PetscCheck(r, PETSC_COMM_SELF, PETSC_ERR_ARG_UNKNOWN_TYPE, "Physics '%s' not found", physname);
    /* Create the physics, will set the number of fields and their names */
    PetscCall((*r)(&ctx));
  }

  /* Create a DMDA to manage the parallel grid */
  PetscCall(DMDACreate1d(comm, DM_BOUNDARY_PERIODIC, 50, ctx.physics2.dof, 2, NULL, &da));
  PetscCall(DMSetFromOptions(da));
  PetscCall(DMSetUp(da));
  /* 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++) PetscCall(DMDASetFieldName(da, i, ctx.physics2.fieldname[i]));
  PetscCall(DMDAGetInfo(da, 0, &Mx, 0, 0, 0, 0, 0, &dof, 0, 0, 0, 0, 0));
  PetscCall(DMDAGetCorners(da, &xs, 0, 0, &xm, 0, 0));

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

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

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

  /* 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 + ctx.hratio) / (1 + ctx.hratio);
  count_medium = 2 * ctx.hratio * count_slow;
  PetscCheck((count_slow % 2) == 0 && (count_medium % 2) == 0, PETSC_COMM_WORLD, PETSC_ERR_USER, "Please adjust grid size Mx (-da_grid_x) and hratio (-hratio) so that Mx/(1+hartio)^2 and Mx*2*hratio/(1+hratio)^2 is even");
  count_fast = ctx.hratio * ctx.hratio * count_slow;
  ctx.sm     = count_slow / 2;
  ctx.mf     = ctx.sm + count_medium / 2;
  ctx.fm     = ctx.mf + count_fast;
  ctx.ms     = ctx.fm + count_medium / 2;
  PetscCall(PetscMalloc1(xm * dof, &index_slow));
  PetscCall(PetscMalloc1(xm * dof, &index_medium));
  PetscCall(PetscMalloc1(xm * dof, &index_fast));
  PetscCall(PetscMalloc1(6 * dof, &index_slowbuffer));
  PetscCall(PetscMalloc1(6 * dof, &index_mediumbuffer));
  if (((AdvectCtx *)ctx.physics2.user)->a > 0) {
    ctx.lsbwidth = 2;
    ctx.rsbwidth = 4;
    ctx.lmbwidth = 2;
    ctx.rmbwidth = 4;
  } else {
    ctx.lsbwidth = 4;
    ctx.rsbwidth = 2;
    ctx.lmbwidth = 4;
    ctx.rmbwidth = 2;
  }

  for (i = xs; i < xs + xm; i++) {
    if (i < ctx.sm - ctx.lsbwidth || i > ctx.ms + ctx.rsbwidth - 1)
      for (k = 0; k < dof; k++) index_slow[islow++] = i * dof + k;
    else if ((i >= ctx.sm - ctx.lsbwidth && i < ctx.sm) || (i > ctx.ms - 1 && i <= ctx.ms + ctx.rsbwidth - 1))
      for (k = 0; k < dof; k++) index_slowbuffer[islowbuffer++] = i * dof + k;
    else if (i < ctx.mf - ctx.lmbwidth || i > ctx.fm + ctx.rmbwidth - 1)
      for (k = 0; k < dof; k++) index_medium[imedium++] = i * dof + k;
    else if ((i >= ctx.mf - ctx.lmbwidth && i < ctx.mf) || (i > ctx.fm - 1 && i <= ctx.fm + ctx.rmbwidth - 1))
      for (k = 0; k < dof; k++) index_mediumbuffer[imediumbuffer++] = i * dof + k;
    else
      for (k = 0; k < dof; k++) index_fast[ifast++] = i * dof + k;
  }
  PetscCall(ISCreateGeneral(PETSC_COMM_WORLD, islow, index_slow, PETSC_COPY_VALUES, &ctx.iss));
  PetscCall(ISCreateGeneral(PETSC_COMM_WORLD, imedium, index_medium, PETSC_COPY_VALUES, &ctx.ism));
  PetscCall(ISCreateGeneral(PETSC_COMM_WORLD, ifast, index_fast, PETSC_COPY_VALUES, &ctx.isf));
  PetscCall(ISCreateGeneral(PETSC_COMM_WORLD, islowbuffer, index_slowbuffer, PETSC_COPY_VALUES, &ctx.issb));
  PetscCall(ISCreateGeneral(PETSC_COMM_WORLD, imediumbuffer, index_mediumbuffer, PETSC_COPY_VALUES, &ctx.ismb));

  /* Create a time-stepping object */
  PetscCall(TSCreate(comm, &ts));
  PetscCall(TSSetDM(ts, da));
  PetscCall(TSSetRHSFunction(ts, R, FVRHSFunction_3WaySplit, &ctx));
  PetscCall(TSRHSSplitSetIS(ts, "slow", ctx.iss));
  PetscCall(TSRHSSplitSetIS(ts, "medium", ctx.ism));
  PetscCall(TSRHSSplitSetIS(ts, "fast", ctx.isf));
  PetscCall(TSRHSSplitSetIS(ts, "slowbuffer", ctx.issb));
  PetscCall(TSRHSSplitSetIS(ts, "mediumbuffer", ctx.ismb));
  PetscCall(TSRHSSplitSetRHSFunction(ts, "slow", NULL, FVRHSFunctionslow_3WaySplit, &ctx));
  PetscCall(TSRHSSplitSetRHSFunction(ts, "medium", NULL, FVRHSFunctionmedium_3WaySplit, &ctx));
  PetscCall(TSRHSSplitSetRHSFunction(ts, "fast", NULL, FVRHSFunctionfast_3WaySplit, &ctx));
  PetscCall(TSRHSSplitSetRHSFunction(ts, "slowbuffer", NULL, FVRHSFunctionslowbuffer_3WaySplit, &ctx));
  PetscCall(TSRHSSplitSetRHSFunction(ts, "mediumbuffer", NULL, FVRHSFunctionmediumbuffer_3WaySplit, &ctx));

  PetscCall(TSSetType(ts, TSSSP));
  /*PetscCall(TSSetType(ts,TSMPRK));*/
  PetscCall(TSSetMaxTime(ts, 10));
  PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER));

  /* Compute initial conditions and starting time step */
  PetscCall(FVSample_3WaySplit(&ctx, da, 0, X0));
  PetscCall(FVRHSFunction_3WaySplit(ts, 0, X0, X, (void *)&ctx)); /* Initial function evaluation, only used to determine max speed */
  PetscCall(VecCopy(X0, X));                                      /* The function value was not used so we set X=X0 again */
  PetscCall(TSSetTimeStep(ts, ctx.cfl / ctx.cfl_idt));
  PetscCall(TSSetFromOptions(ts)); /* Take runtime options */
  PetscCall(SolutionStatsView(da, X, PETSC_VIEWER_STDOUT_WORLD));
  {
    PetscInt           steps;
    PetscScalar        mass_initial, mass_final, mass_difference, mass_differenceg;
    const PetscScalar *ptr_X, *ptr_X0;
    const PetscReal    hs = (ctx.xmax - ctx.xmin) / 4.0 / count_slow;
    const PetscReal    hm = (ctx.xmax - ctx.xmin) / 2.0 / count_medium;
    const PetscReal    hf = (ctx.xmax - ctx.xmin) / 4.0 / count_fast;

    PetscCall(TSSolve(ts, X));
    PetscCall(TSGetSolveTime(ts, &ptime));
    PetscCall(TSGetStepNumber(ts, &steps));
    /* calculate the total mass at initial time and final time */
    mass_initial = 0.0;
    mass_final   = 0.0;
    PetscCall(DMDAVecGetArrayRead(da, X0, (void *)&ptr_X0));
    PetscCall(DMDAVecGetArrayRead(da, X, (void *)&ptr_X));
    for (i = xs; i < xs + xm; i++) {
      if (i < ctx.sm || i > ctx.ms - 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 if (i < ctx.mf || i > ctx.fm - 1)
        for (k = 0; k < dof; k++) {
          mass_initial = mass_initial + hm * ptr_X0[i * dof + k];
          mass_final   = mass_final + hm * 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];
        }
      }
    }
    PetscCall(DMDAVecRestoreArrayRead(da, X0, (void *)&ptr_X0));
    PetscCall(DMDAVecRestoreArrayRead(da, X, (void *)&ptr_X));
    mass_difference = mass_final - mass_initial;
    PetscCall(MPIU_Allreduce(&mass_difference, &mass_differenceg, 1, MPIU_SCALAR, MPIU_SUM, comm));
    PetscCall(PetscPrintf(comm, "Mass difference %g\n", (double)mass_differenceg));
    PetscCall(PetscPrintf(comm, "Final time %g, steps %" PetscInt_FMT "\n", (double)ptime, steps));
    PetscCall(PetscPrintf(comm, "Maximum allowable stepsize according to CFL %g\n", (double)(1.0 / ctx.cfl_idt)));
    if (ctx.exact) {
      PetscReal nrm1 = 0;
      PetscCall(SolutionErrorNorms_3WaySplit(&ctx, da, ptime, X, &nrm1));
      PetscCall(PetscPrintf(comm, "Error ||x-x_e||_1 %g\n", (double)nrm1));
    }
    if (ctx.simulation) {
      PetscReal          nrm1 = 0;
      PetscViewer        fd;
      char               filename[PETSC_MAX_PATH_LEN] = "binaryoutput";
      Vec                XR;
      PetscBool          flg;
      const PetscScalar *ptr_XR;
      PetscCall(PetscOptionsGetString(NULL, NULL, "-f", filename, sizeof(filename), &flg));
      PetscCheck(flg, PETSC_COMM_WORLD, PETSC_ERR_USER, "Must indicate binary file with the -f option");
      PetscCall(PetscViewerBinaryOpen(PETSC_COMM_WORLD, filename, FILE_MODE_READ, &fd));
      PetscCall(VecDuplicate(X0, &XR));
      PetscCall(VecLoad(XR, fd));
      PetscCall(PetscViewerDestroy(&fd));
      PetscCall(VecGetArrayRead(X, &ptr_X));
      PetscCall(VecGetArrayRead(XR, &ptr_XR));
      for (i = xs; i < xs + xm; i++) {
        if (i < ctx.sm || i > ctx.ms - 1)
          for (k = 0; k < dof; k++) nrm1 = nrm1 + hs * PetscAbs(ptr_X[i * dof + k] - ptr_XR[i * dof + k]);
        else if (i < ctx.mf || i < ctx.fm - 1)
          for (k = 0; k < dof; k++) nrm1 = nrm1 + hm * 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]);
      }
      PetscCall(VecRestoreArrayRead(X, &ptr_X));
      PetscCall(VecRestoreArrayRead(XR, &ptr_XR));
      PetscCall(PetscPrintf(comm, "Error ||x-x_e||_1 %g\n", (double)nrm1));
      PetscCall(VecDestroy(&XR));
    }
  }

  PetscCall(SolutionStatsView(da, X, PETSC_VIEWER_STDOUT_WORLD));
  if (draw & 0x1) PetscCall(VecView(X0, PETSC_VIEWER_DRAW_WORLD));
  if (draw & 0x2) PetscCall(VecView(X, PETSC_VIEWER_DRAW_WORLD));
  if (draw & 0x4) {
    Vec Y;
    PetscCall(VecDuplicate(X, &Y));
    PetscCall(FVSample_3WaySplit(&ctx, da, ptime, Y));
    PetscCall(VecAYPX(Y, -1, X));
    PetscCall(VecView(Y, PETSC_VIEWER_DRAW_WORLD));
    PetscCall(VecDestroy(&Y));
  }

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

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

/*TEST

    build:
      requires: !complex
      depends: finitevolume1d.c

    test:
      suffix: 1
      args: -da_grid_x 90 -initial 7 -xmin -1 -xmax 1 -hratio 2 -limit mc -ts_dt 0.025 -ts_max_steps 24 -ts_type mprk -ts_mprk_type 2a23 -ts_use_splitrhsfunction 0

    test:
      suffix: 2
      args: -da_grid_x 90 -initial 7 -xmin -1 -xmax 1 -hratio 2 -limit mc -ts_dt 0.025 -ts_max_steps 24 -ts_type mprk -ts_mprk_type 2a23 -ts_use_splitrhsfunction 1

TEST*/