xref: /petsc/src/ts/tutorials/multirate/finitevolume1d.c (revision 834855d6effb0d027771461c8e947ee1ce5a1e17)

#include "finitevolume1d.h"
#include <petscdmda.h>
#include <petscdraw.h>
#include <petsc/private/tsimpl.h>

#include <petsc/private/kernels/blockinvert.h> /* For the Kernel_*_gets_* stuff for BAIJ */
const char *FVBCTypes[] = {"PERIODIC", "OUTFLOW", "INFLOW", "FVBCType", "FVBC_", 0};

static inline PetscReal Sgn(PetscReal a)
{
  return (a < 0) ? -1 : 1;
}
static inline PetscReal Abs(PetscReal a)
{
  return (a < 0) ? 0 : a;
}
static inline PetscReal Sqr(PetscReal a)
{
  return a * a;
}

PETSC_UNUSED static inline PetscReal MinAbs(PetscReal a, PetscReal b)
{
  return (PetscAbs(a) < PetscAbs(b)) ? a : b;
}
static inline PetscReal MinMod2(PetscReal a, PetscReal b)
{
  return (a * b < 0) ? 0 : Sgn(a) * PetscMin(PetscAbs(a), PetscAbs(b));
}
static inline PetscReal MaxMod2(PetscReal a, PetscReal b)
{
  return (a * b < 0) ? 0 : Sgn(a) * PetscMax(PetscAbs(a), PetscAbs(b));
}
static inline PetscReal MinMod3(PetscReal a, PetscReal b, PetscReal c)
{
  return (a * b < 0 || a * c < 0) ? 0 : Sgn(a) * PetscMin(PetscAbs(a), PetscMin(PetscAbs(b), PetscAbs(c)));
}

/* ----------------------- Lots of limiters, these could go in a separate library ------------------------- */
void Limit_Upwind(LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, PetscScalar *lmt)
{
  PetscInt i;
  for (i = 0; i < info->m; i++) lmt[i] = 0;
}
void Limit_LaxWendroff(LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, PetscScalar *lmt)
{
  PetscInt i;
  for (i = 0; i < info->m; i++) lmt[i] = jR[i];
}
void Limit_BeamWarming(LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, PetscScalar *lmt)
{
  PetscInt i;
  for (i = 0; i < info->m; i++) lmt[i] = jL[i];
}
void Limit_Fromm(LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, PetscScalar *lmt)
{
  PetscInt i;
  for (i = 0; i < info->m; i++) lmt[i] = 0.5 * (jL[i] + jR[i]);
}
void Limit_Minmod(LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, PetscScalar *lmt)
{
  PetscInt i;
  for (i = 0; i < info->m; i++) lmt[i] = MinMod2(jL[i], jR[i]);
}
void Limit_Superbee(LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, PetscScalar *lmt)
{
  PetscInt i;
  for (i = 0; i < info->m; i++) lmt[i] = MaxMod2(MinMod2(jL[i], 2 * jR[i]), MinMod2(2 * jL[i], jR[i]));
}
void Limit_MC(LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, PetscScalar *lmt)
{
  PetscInt i;
  for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i], 0.5 * (jL[i] + jR[i]), 2 * jR[i]);
}
void Limit_VanLeer(LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, PetscScalar *lmt)
{ /* phi = (t + abs(t)) / (1 + abs(t)) */
  PetscInt i;
  for (i = 0; i < info->m; i++) lmt[i] = (jL[i] * Abs(jR[i]) + Abs(jL[i]) * jR[i]) / (Abs(jL[i]) + Abs(jR[i]) + 1e-15);
}
void Limit_VanAlbada(LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, PetscScalar *lmt) /* differentiable */
{                                                                                                    /* phi = (t + t^2) / (1 + t^2) */
  PetscInt i;
  for (i = 0; i < info->m; i++) lmt[i] = (jL[i] * Sqr(jR[i]) + Sqr(jL[i]) * jR[i]) / (Sqr(jL[i]) + Sqr(jR[i]) + 1e-15);
}
void Limit_VanAlbadaTVD(LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, PetscScalar *lmt)
{ /* phi = (t + t^2) / (1 + t^2) */
  PetscInt i;
  for (i = 0; i < info->m; i++) lmt[i] = (jL[i] * jR[i] < 0) ? 0 : (jL[i] * Sqr(jR[i]) + Sqr(jL[i]) * jR[i]) / (Sqr(jL[i]) + Sqr(jR[i]) + 1e-15);
}
void Limit_Koren(LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, PetscScalar *lmt) /* differentiable */
{                                                                                                /* phi = (t + 2*t^2) / (2 - t + 2*t^2) */
  PetscInt i;
  for (i = 0; i < info->m; i++) lmt[i] = ((jL[i] * Sqr(jR[i]) + 2 * Sqr(jL[i]) * jR[i]) / (2 * Sqr(jL[i]) - jL[i] * jR[i] + 2 * Sqr(jR[i]) + 1e-15));
}
void Limit_KorenSym(LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, PetscScalar *lmt) /* differentiable */
{                                                                                                   /* Symmetric version of above */
  PetscInt i;
  for (i = 0; i < info->m; i++) lmt[i] = (1.5 * (jL[i] * Sqr(jR[i]) + Sqr(jL[i]) * jR[i]) / (2 * Sqr(jL[i]) - jL[i] * jR[i] + 2 * Sqr(jR[i]) + 1e-15));
}
void Limit_Koren3(LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, PetscScalar *lmt)
{ /* Eq 11 of Cada-Torrilhon 2009 */
  PetscInt i;
  for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i], (jL[i] + 2 * jR[i]) / 3, 2 * jR[i]);
}
static PetscReal CadaTorrilhonPhiHatR_Eq13(PetscReal L, PetscReal R)
{
  return PetscMax(0, PetscMin((L + 2 * R) / 3, PetscMax(-0.5 * L, PetscMin(2 * L, PetscMin((L + 2 * R) / 3, 1.6 * R)))));
}
void Limit_CadaTorrilhon2(LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, PetscScalar *lmt)
{ /* Cada-Torrilhon 2009, Eq 13 */
  PetscInt i;
  for (i = 0; i < info->m; i++) lmt[i] = CadaTorrilhonPhiHatR_Eq13(jL[i], jR[i]);
}
void Limit_CadaTorrilhon3R(PetscReal r, LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, PetscScalar *lmt)
{ /* Cada-Torrilhon 2009, Eq 22 */
  /* They recommend 0.001 < r < 1, but larger values are more accurate in smooth regions */
  const PetscReal eps = 1e-7, hx = info->hx;
  PetscInt        i;
  for (i = 0; i < info->m; i++) {
    const PetscReal eta = (Sqr(jL[i]) + Sqr(jR[i])) / Sqr(r * hx);
    lmt[i] = ((eta < 1 - eps) ? (jL[i] + 2 * jR[i]) / 3 : ((eta > 1 + eps) ? CadaTorrilhonPhiHatR_Eq13(jL[i], jR[i]) : 0.5 * ((1 - (eta - 1) / eps) * (jL[i] + 2 * jR[i]) / 3 + (1 + (eta + 1) / eps) * CadaTorrilhonPhiHatR_Eq13(jL[i], jR[i]))));
  }
}
void Limit_CadaTorrilhon3R0p1(LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, PetscScalar *lmt)
{
  Limit_CadaTorrilhon3R(0.1, info, jL, jR, lmt);
}
void Limit_CadaTorrilhon3R1(LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, PetscScalar *lmt)
{
  Limit_CadaTorrilhon3R(1, info, jL, jR, lmt);
}
void Limit_CadaTorrilhon3R10(LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, PetscScalar *lmt)
{
  Limit_CadaTorrilhon3R(10, info, jL, jR, lmt);
}
void Limit_CadaTorrilhon3R100(LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, PetscScalar *lmt)
{
  Limit_CadaTorrilhon3R(100, info, jL, jR, lmt);
}

/* ----------------------- Limiters for split systems ------------------------- */

void Limit2_Upwind(LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, const PetscInt sf, const PetscInt fs, PetscInt n, PetscScalar *lmt)
{
  PetscInt i;
  for (i = 0; i < info->m; i++) lmt[i] = 0;
}
void Limit2_LaxWendroff(LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, const PetscInt sf, const PetscInt fs, PetscInt n, PetscScalar *lmt)
{
  PetscInt i;
  if (n < sf - 1) { /* slow components */
    for (i = 0; i < info->m; i++) lmt[i] = jR[i] / info->hxs;
  } else if (n == sf - 1) { /* slow component which is next to fast components */
    for (i = 0; i < info->m; i++) lmt[i] = jR[i] / (info->hxs / 2.0 + info->hxf / 2.0);
  } else if (n == sf) { /* fast component which is next to slow components */
    for (i = 0; i < info->m; i++) lmt[i] = jR[i] / info->hxf;
  } else if (n > sf && n < fs - 1) { /* fast components */
    for (i = 0; i < info->m; i++) lmt[i] = jR[i] / info->hxf;
  } else if (n == fs - 1) { /* fast component next to slow components */
    for (i = 0; i < info->m; i++) lmt[i] = jR[i] / (info->hxf / 2.0 + info->hxs / 2.0);
  } else if (n == fs) { /* slow component next to fast components */
    for (i = 0; i < info->m; i++) lmt[i] = jR[i] / info->hxs;
  } else { /* slow components */
    for (i = 0; i < info->m; i++) lmt[i] = jR[i] / info->hxs;
  }
}
void Limit2_BeamWarming(LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, const PetscInt sf, const PetscInt fs, PetscInt n, PetscScalar *lmt)
{
  PetscInt i;
  if (n < sf - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = jL[i] / info->hxs;
  } else if (n == sf - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = jL[i] / info->hxs;
  } else if (n == sf) {
    for (i = 0; i < info->m; i++) lmt[i] = jL[i] / (info->hxs / 2.0 + info->hxf / 2.0);
  } else if (n > sf && n < fs - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = jL[i] / info->hxf;
  } else if (n == fs - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = jL[i] / info->hxf;
  } else if (n == fs) {
    for (i = 0; i < info->m; i++) lmt[i] = jL[i] / (info->hxf / 2.0 + info->hxs / 2.0);
  } else {
    for (i = 0; i < info->m; i++) lmt[i] = jL[i] / info->hxs;
  }
}
void Limit2_Fromm(LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, const PetscInt sf, const PetscInt fs, PetscInt n, PetscScalar *lmt)
{
  PetscInt i;
  if (n < sf - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = 0.5 * (jL[i] + jR[i]) / info->hxs;
  } else if (n == sf - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = 0.5 * (jL[i] / info->hxs + jR[i] / (info->hxs / 2.0 + info->hxf / 2.0));
  } else if (n == sf) {
    for (i = 0; i < info->m; i++) lmt[i] = 0.5 * (jL[i] / (info->hxs / 2.0 + info->hxf / 2.0) + jR[i] / info->hxf);
  } else if (n > sf && n < fs - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = 0.5 * (jL[i] + jR[i]) / info->hxf;
  } else if (n == fs - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = 0.5 * (jL[i] / info->hxf + jR[i] / (info->hxf / 2.0 + info->hxs / 2.0));
  } else if (n == fs) {
    for (i = 0; i < info->m; i++) lmt[i] = 0.5 * (jL[i] / (info->hxf / 2.0 + info->hxs / 2.0) + jR[i] / info->hxs);
  } else {
    for (i = 0; i < info->m; i++) lmt[i] = 0.5 * (jL[i] + jR[i]) / info->hxs;
  }
}
void Limit2_Minmod(LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, const PetscInt sf, const PetscInt fs, PetscInt n, PetscScalar *lmt)
{
  PetscInt i;
  if (n < sf - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod2(jL[i], jR[i]) / info->hxs;
  } else if (n == sf - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod2(jL[i] / info->hxs, jR[i] / (info->hxs / 2.0 + info->hxf / 2.0));
  } else if (n == sf) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod2(jL[i] / (info->hxs / 2.0 + info->hxf / 2.0), jR[i] / info->hxf);
  } else if (n > sf && n < fs - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod2(jL[i], jR[i]) / info->hxf;
  } else if (n == fs - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod2(jL[i] / info->hxf, jR[i] / (info->hxf / 2.0 + info->hxs / 2.0));
  } else if (n == fs) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod2(jL[i] / (info->hxf / 2.0 + info->hxs / 2.0), jR[i] / info->hxs);
  } else {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod2(jL[i], jR[i]) / info->hxs;
  }
}
void Limit2_Superbee(LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, const PetscInt sf, const PetscInt fs, PetscInt n, PetscScalar *lmt)
{
  PetscInt i;
  if (n < sf - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = MaxMod2(MinMod2(jL[i], 2 * jR[i]), MinMod2(2 * jL[i], jR[i])) / info->hxs;
  } else if (n == sf - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = MaxMod2(MinMod2(jL[i] / info->hxs, 2 * jR[i] / (info->hxs / 2.0 + info->hxf / 2.0)), MinMod2(2 * jL[i] / info->hxs, jR[i] / (info->hxs / 2.0 + info->hxf / 2.0)));
  } else if (n == sf) {
    for (i = 0; i < info->m; i++) lmt[i] = MaxMod2(MinMod2(jL[i] / (info->hxs / 2.0 + info->hxf / 2.0), 2 * jR[i] / info->hxf), MinMod2(2 * jL[i] / (info->hxs / 2.0 + info->hxf / 2.0), jR[i] / info->hxf));
  } else if (n > sf && n < fs - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod2(MinMod2(jL[i], 2 * jR[i]), MinMod2(2 * jL[i], jR[i])) / info->hxf;
  } else if (n == fs - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod2(MinMod2(jL[i] / info->hxf, 2 * jR[i] / (info->hxf / 2.0 + info->hxs / 2.0)), MinMod2(2 * jL[i] / info->hxf, jR[i] / (info->hxf / 2.0 + info->hxs / 2.0)));
  } else if (n == fs) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod2(MinMod2(jL[i] / (info->hxf / 2.0 + info->hxs / 2.0), 2 * jR[i] / info->hxs), MinMod2(2 * jL[i] / (info->hxf / 2.0 + info->hxs / 2.0), jR[i] / info->hxs));
  } else {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod2(MinMod2(jL[i], 2 * jR[i]), MinMod2(2 * jL[i], jR[i])) / info->hxs;
  }
}
void Limit2_MC(LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, const PetscInt sf, const PetscInt fs, PetscInt n, PetscScalar *lmt)
{
  PetscInt i;
  if (n < sf - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i], 0.5 * (jL[i] + jR[i]), 2 * jR[i]) / info->hxs;
  } else if (n == sf - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i] / info->hxs, 0.5 * (jL[i] / info->hxs + jR[i] / (info->hxf / 2.0 + info->hxs / 2.0)), 2 * jR[i] / (info->hxf / 2.0 + info->hxs / 2.0));
  } else if (n == sf) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i] / (info->hxs / 2.0 + info->hxf / 2.0), 0.5 * (jL[i] / (info->hxs / 2.0 + info->hxf / 2.0) + jR[i] / info->hxf), 2 * jR[i] / info->hxf);
  } else if (n > sf && n < fs - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i], 0.5 * (jL[i] + jR[i]), 2 * jR[i]) / info->hxf;
  } else if (n == fs - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i] / info->hxf, 0.5 * (jL[i] / info->hxf + jR[i] / (info->hxf / 2.0 + info->hxs / 2.0)), 2 * jR[i] / (info->hxf / 2.0 + info->hxs / 2.0));
  } else if (n == fs) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i] / (info->hxf / 2.0 + info->hxs / 2.0), 0.5 * (jL[i] / (info->hxf / 2.0 + info->hxs / 2.0) + jR[i] / info->hxs), 2 * jR[i] / info->hxs);
  } else {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i], 0.5 * (jL[i] + jR[i]), 2 * jR[i]) / info->hxs;
  }
}
void Limit2_Koren3(LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, const PetscInt sf, const PetscInt fs, PetscInt n, PetscScalar *lmt)
{ /* Eq 11 of Cada-Torrilhon 2009 */
  PetscInt i;
  if (n < sf - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i], (jL[i] + 2 * jR[i]) / 3, 2 * jR[i]) / info->hxs;
  } else if (n == sf - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i] / info->hxs, (jL[i] / info->hxs + 2 * jR[i] / (info->hxf / 2.0 + info->hxs / 2.0)) / 3, 2 * jR[i] / (info->hxf / 2.0 + info->hxs / 2.0));
  } else if (n == sf) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i] / (info->hxs / 2.0 + info->hxf / 2.0), (jL[i] / (info->hxs / 2.0 + info->hxf / 2.0) + 2 * jR[i] / info->hxf) / 3, 2 * jR[i] / info->hxf);
  } else if (n > sf && n < fs - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i], (jL[i] + 2 * jR[i]) / 3, 2 * jR[i]) / info->hxf;
  } else if (n == fs - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i] / info->hxf, (jL[i] / info->hxf + 2 * jR[i] / (info->hxf / 2.0 + info->hxs / 2.0)) / 3, 2 * jR[i] / (info->hxf / 2.0 + info->hxs / 2.0));
  } else if (n == fs) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i] / (info->hxf / 2.0 + info->hxs / 2.0), (jL[i] / (info->hxf / 2.0 + info->hxs / 2.0) + 2 * jR[i] / info->hxs) / 3, 2 * jR[i] / info->hxs);
  } else {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i], (jL[i] + 2 * jR[i]) / 3, 2 * jR[i]) / info->hxs;
  }
}

/* ---- Three-way splitting ---- */
void Limit3_Upwind(LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, const PetscInt sm, const PetscInt mf, const PetscInt fm, const PetscInt ms, PetscInt n, PetscScalar *lmt)
{
  PetscInt i;
  for (i = 0; i < info->m; i++) lmt[i] = 0;
}
void Limit3_LaxWendroff(LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, const PetscInt sm, const PetscInt mf, const PetscInt fm, const PetscInt ms, PetscInt n, PetscScalar *lmt)
{
  PetscInt i;
  if (n < sm - 1 || n > ms) { /* slow components */
    for (i = 0; i < info->m; i++) lmt[i] = jR[i] / info->hxs;
  } else if (n == sm - 1 || n == ms - 1) { /* slow component which is next to medium components */
    for (i = 0; i < info->m; i++) lmt[i] = jR[i] / (info->hxs / 2.0 + info->hxm / 2.0);
  } else if (n < mf - 1 || n > fm) { /* medium components */
    for (i = 0; i < info->m; i++) lmt[i] = jR[i] / info->hxm;
  } else if (n == mf - 1 || n == fm) { /* medium component next to fast components */
    for (i = 0; i < info->m; i++) lmt[i] = jR[i] / (info->hxm / 2.0 + info->hxf / 2.0);
  } else { /* fast components */
    for (i = 0; i < info->m; i++) lmt[i] = jR[i] / info->hxf;
  }
}
void Limit3_BeamWarming(LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, const PetscInt sm, const PetscInt mf, const PetscInt fm, const PetscInt ms, PetscInt n, PetscScalar *lmt)
{
  PetscInt i;
  if (n < sm || n > ms) {
    for (i = 0; i < info->m; i++) lmt[i] = jL[i] / info->hxs;
  } else if (n == sm || n == ms) {
    for (i = 0; i < info->m; i++) lmt[i] = jL[i] / (info->hxs / 2.0 + info->hxf / 2.0);
  } else if (n < mf || n > fm) {
    for (i = 0; i < info->m; i++) lmt[i] = jL[i] / info->hxm;
  } else if (n == mf || n == fm) {
    for (i = 0; i < info->m; i++) lmt[i] = jL[i] / (info->hxm / 2.0 + info->hxf / 2.0);
  } else {
    for (i = 0; i < info->m; i++) lmt[i] = jL[i] / info->hxf;
  }
}
void Limit3_Fromm(LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, const PetscInt sm, const PetscInt mf, const PetscInt fm, const PetscInt ms, PetscInt n, PetscScalar *lmt)
{
  PetscInt i;
  if (n < sm - 1 || n > ms) {
    for (i = 0; i < info->m; i++) lmt[i] = 0.5 * (jL[i] + jR[i]) / info->hxs;
  } else if (n == sm - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = 0.5 * (jL[i] / info->hxs + jR[i] / (info->hxs / 2.0 + info->hxf / 2.0));
  } else if (n == sm) {
    for (i = 0; i < info->m; i++) lmt[i] = 0.5 * (jL[i] / (info->hxs / 2.0 + info->hxm / 2.0) + jR[i] / info->hxm);
  } else if (n == ms - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = 0.5 * (jL[i] / info->hxm + jR[i] / (info->hxs / 2.0 + info->hxf / 2.0));
  } else if (n == ms) {
    for (i = 0; i < info->m; i++) lmt[i] = 0.5 * (jL[i] / (info->hxm / 2.0 + info->hxs / 2.0) + jR[i] / info->hxs);
  } else if (n < mf - 1 || n > fm) {
    for (i = 0; i < info->m; i++) lmt[i] = 0.5 * (jL[i] + jR[i]) / info->hxm;
  } else if (n == mf - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = 0.5 * (jL[i] / info->hxm + jR[i] / (info->hxm / 2.0 + info->hxf / 2.0));
  } else if (n == mf) {
    for (i = 0; i < info->m; i++) lmt[i] = 0.5 * (jL[i] / (info->hxm / 2.0 + info->hxf / 2.0) + jR[i] / info->hxf);
  } else if (n == fm - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = 0.5 * (jL[i] / info->hxf + jR[i] / (info->hxf / 2.0 + info->hxm / 2.0));
  } else if (n == fm) {
    for (i = 0; i < info->m; i++) lmt[i] = 0.5 * (jL[i] / (info->hxf / 2.0 + info->hxm / 2.0) + jR[i] / info->hxm);
  } else {
    for (i = 0; i < info->m; i++) lmt[i] = 0.5 * (jL[i] + jR[i]) / info->hxf;
  }
}
void Limit3_Minmod(LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, const PetscInt sm, const PetscInt mf, const PetscInt fm, const PetscInt ms, PetscInt n, PetscScalar *lmt)
{
  PetscInt i;
  if (n < sm - 1 || n > ms) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod2(jL[i], jR[i]) / info->hxs;
  } else if (n == sm - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod2(jL[i] / info->hxs, jR[i] / (info->hxs / 2.0 + info->hxf / 2.0));
  } else if (n == sm) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod2(jL[i] / (info->hxs / 2.0 + info->hxf / 2.0), jR[i] / info->hxf);
  } else if (n == ms - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod2(jL[i] / info->hxm, jR[i] / (info->hxm / 2.0 + info->hxs / 2.0));
  } else if (n == ms) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod2(jL[i] / (info->hxm / 2.0 + info->hxs / 2.0), jR[i] / info->hxs);
  } else if (n < mf - 1 || n > fm) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod2(jL[i], jR[i]) / info->hxm;
  } else if (n == mf - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod2(jL[i] / info->hxm, jR[i] / (info->hxm / 2.0 + info->hxf / 2.0));
  } else if (n == mf) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod2(jL[i] / (info->hxm / 2.0 + info->hxf / 2.0), jR[i] / info->hxf);
  } else if (n == fm - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod2(jL[i] / info->hxf, jR[i] / (info->hxf / 2.0 + info->hxm / 2.0));
  } else if (n == fm) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod2(jL[i] / (info->hxf / 2.0 + info->hxm / 2.0), jR[i] / info->hxm);
  } else {
    for (i = 0; i < info->m; i++) lmt[i] = 0.5 * (jL[i] + jR[i]) / info->hxf;
  }
}
void Limit3_Superbee(LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, const PetscInt sm, const PetscInt mf, const PetscInt fm, const PetscInt ms, PetscInt n, PetscScalar *lmt)
{
  PetscInt i;
  if (n < sm - 1 || n > ms) {
    for (i = 0; i < info->m; i++) lmt[i] = MaxMod2(MinMod2(jL[i], 2 * jR[i]), MinMod2(2 * jL[i], jR[i])) / info->hxs;
  } else if (n == sm - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = MaxMod2(MinMod2(jL[i] / info->hxs, 2 * jR[i] / (info->hxs / 2.0 + info->hxm / 2.0)), MinMod2(2 * jL[i] / info->hxs, jR[i] / (info->hxs / 2.0 + info->hxm / 2.0)));
  } else if (n == sm) {
    for (i = 0; i < info->m; i++) lmt[i] = MaxMod2(MinMod2(jL[i] / (info->hxs / 2.0 + info->hxm / 2.0), 2 * jR[i] / info->hxm), MinMod2(2 * jL[i] / (info->hxs / 2.0 + info->hxm / 2.0), jR[i] / info->hxm));
  } else if (n == ms - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod2(MinMod2(jL[i] / info->hxm, 2 * jR[i] / (info->hxm / 2.0 + info->hxs / 2.0)), MinMod2(2 * jL[i] / info->hxm, jR[i] / (info->hxm / 2.0 + info->hxs / 2.0)));
  } else if (n == ms) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod2(MinMod2(jL[i] / (info->hxm / 2.0 + info->hxs / 2.0), 2 * jR[i] / info->hxs), MinMod2(2 * jL[i] / (info->hxm / 2.0 + info->hxs / 2.0), jR[i] / info->hxs));
  } else if (n < mf - 1 || n > fm) {
    for (i = 0; i < info->m; i++) lmt[i] = MaxMod2(MinMod2(jL[i], 2 * jR[i]), MinMod2(2 * jL[i], jR[i])) / info->hxm;
  } else if (n == mf - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = MaxMod2(MinMod2(jL[i] / info->hxm, 2 * jR[i] / (info->hxm / 2.0 + info->hxf / 2.0)), MinMod2(2 * jL[i] / info->hxm, jR[i] / (info->hxm / 2.0 + info->hxf / 2.0)));
  } else if (n == mf) {
    for (i = 0; i < info->m; i++) lmt[i] = MaxMod2(MinMod2(jL[i] / (info->hxm / 2.0 + info->hxf / 2.0), 2 * jR[i] / info->hxf), MinMod2(2 * jL[i] / (info->hxm / 2.0 + info->hxf / 2.0), jR[i] / info->hxf));
  } else if (n == fm - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod2(MinMod2(jL[i] / info->hxf, 2 * jR[i] / (info->hxf / 2.0 + info->hxm / 2.0)), MinMod2(2 * jL[i] / info->hxf, jR[i] / (info->hxf / 2.0 + info->hxm / 2.0)));
  } else if (n == fm) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod2(MinMod2(jL[i] / (info->hxf / 2.0 + info->hxm / 2.0), 2 * jR[i] / info->hxm), MinMod2(2 * jL[i] / (info->hxf / 2.0 + info->hxm / 2.0), jR[i] / info->hxm));
  } else {
    for (i = 0; i < info->m; i++) lmt[i] = MaxMod2(MinMod2(jL[i], 2 * jR[i]), MinMod2(2 * jL[i], jR[i])) / info->hxf;
  }
}
void Limit3_MC(LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, const PetscInt sm, const PetscInt mf, const PetscInt fm, const PetscInt ms, PetscInt n, PetscScalar *lmt)
{
  PetscInt i;
  if (n < sm - 1 || n > ms) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i], 0.5 * (jL[i] + jR[i]), 2 * jR[i]) / info->hxs;
  } else if (n == sm - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i] / info->hxs, 0.5 * (jL[i] / info->hxs + jR[i] / (info->hxs / 2.0 + info->hxm / 2.0)), 2 * jR[i] / (info->hxs / 2.0 + info->hxm / 2.0));
  } else if (n == sm) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i] / (info->hxs / 2.0 + info->hxm / 2.0), 0.5 * (jL[i] / (info->hxs / 2.0 + info->hxm / 2.0) + jR[i] / info->hxm), 2 * jR[i] / info->hxm);
  } else if (n == ms - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i] / info->hxm, 0.5 * (jL[i] / info->hxm + jR[i] / (info->hxm / 2.0 + info->hxs / 2.0)), 2 * jR[i] / (info->hxm / 2.0 + info->hxs / 2.0));
  } else if (n == ms) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i] / (info->hxm / 2.0 + info->hxs / 2.0), 0.5 * (jL[i] / (info->hxm / 2.0 + info->hxs / 2.0) + jR[i] / info->hxs), 2 * jR[i] / info->hxs);
  } else if (n < mf - 1 || n > fm) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i], 0.5 * (jL[i] + jR[i]), 2 * jR[i]) / info->hxm;
  } else if (n == mf - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i] / info->hxm, 0.5 * (jL[i] / info->hxm + jR[i] / (info->hxm / 2.0 + info->hxf / 2.0)), 2 * jR[i] / (info->hxm / 2.0 + info->hxf / 2.0));
  } else if (n == mf) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i] / (info->hxm / 2.0 + info->hxf / 2.0), 0.5 * (jL[i] / (info->hxm / 2.0 + info->hxf / 2.0) + jR[i] / info->hxf), 2 * jR[i] / info->hxf);
  } else if (n == fm - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i] / info->hxf, 0.5 * (jL[i] / info->hxf + jR[i] / (info->hxf / 2.0 + info->hxm / 2.0)), 2 * jR[i] / (info->hxf / 2.0 + info->hxm / 2.0));
  } else if (n == fm) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i] / (info->hxf / 2.0 + info->hxm / 2.0), 0.5 * (jL[i] / (info->hxf / 2.0 + info->hxm / 2.0) + jR[i] / info->hxm), 2 * jR[i] / info->hxm);
  } else {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i], 0.5 * (jL[i] + jR[i]), 2 * jR[i]) / info->hxf;
  }
}
void Limit3_Koren3(LimitInfo info, const PetscScalar *jL, const PetscScalar *jR, const PetscInt sm, const PetscInt mf, const PetscInt fm, const PetscInt ms, PetscInt n, PetscScalar *lmt)
{ /* Eq 11 of Cada-Torrilhon 2009 */
  PetscInt i;
  if (n < sm - 1 || n > ms) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i], (jL[i] + 2 * jR[i]) / 3, 2 * jR[i]) / info->hxs;
  } else if (n == sm - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i] / info->hxs, (jL[i] / info->hxs + 2 * jR[i] / (info->hxm / 2.0 + info->hxs / 2.0)) / 3, 2 * jR[i] / (info->hxm / 2.0 + info->hxs / 2.0));
  } else if (n == sm) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i] / (info->hxs / 2.0 + info->hxm / 2.0), (jL[i] / (info->hxs / 2.0 + info->hxm / 2.0) + 2 * jR[i] / info->hxm) / 3, 2 * jR[i] / info->hxm);
  } else if (n == ms - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i] / info->hxm, (jL[i] / info->hxm + 2 * jR[i] / (info->hxm / 2.0 + info->hxs / 2.0)) / 3, 2 * jR[i] / (info->hxm / 2.0 + info->hxs / 2.0));
  } else if (n == ms) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i] / (info->hxm / 2.0 + info->hxs / 2.0), (jL[i] / (info->hxm / 2.0 + info->hxs / 2.0) + 2 * jR[i] / info->hxs) / 3, 2 * jR[i] / info->hxs);
  } else if (n < mf - 1 || n > fm) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i], (jL[i] + 2 * jR[i]) / 3, 2 * jR[i]) / info->hxm;
  } else if (n == mf - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i] / info->hxm, (jL[i] / info->hxm + 2 * jR[i] / (info->hxm / 2.0 + info->hxf / 2.0)) / 3, 2 * jR[i] / (info->hxm / 2.0 + info->hxf / 2.0));
  } else if (n == mf) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i] / (info->hxm / 2.0 + info->hxf / 2.0), (jL[i] / (info->hxm / 2.0 + info->hxf / 2.0) + 2 * jR[i] / info->hxf) / 3, 2 * jR[i] / info->hxf);
  } else if (n == fm - 1) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i] / info->hxf, (jL[i] / info->hxf + 2 * jR[i] / (info->hxf / 2.0 + info->hxm / 2.0)) / 3, 2 * jR[i] / (info->hxf / 2.0 + info->hxm / 2.0));
  } else if (n == fm) {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i] / (info->hxf / 2.0 + info->hxm / 2.0), (jL[i] / (info->hxf / 2.0 + info->hxm / 2.0) + 2 * jR[i] / info->hxm) / 3, 2 * jR[i] / info->hxm);
  } else {
    for (i = 0; i < info->m; i++) lmt[i] = MinMod3(2 * jL[i], (jL[i] + 2 * jR[i]) / 3, 2 * jR[i]) / info->hxs;
  }
}

PetscErrorCode RiemannListAdd(PetscFunctionList *flist, const char *name, RiemannFunction rsolve)
{
  PetscFunctionBeginUser;
  PetscCall(PetscFunctionListAdd(flist, name, rsolve));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode RiemannListFind(PetscFunctionList flist, const char *name, RiemannFunction *rsolve)
{
  PetscFunctionBeginUser;
  PetscCall(PetscFunctionListFind(flist, name, rsolve));
  PetscCheck(*rsolve, PETSC_COMM_SELF, PETSC_ERR_ARG_UNKNOWN_TYPE, "Riemann solver \"%s\" could not be found", name);
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode ReconstructListAdd(PetscFunctionList *flist, const char *name, ReconstructFunction r)
{
  PetscFunctionBeginUser;
  PetscCall(PetscFunctionListAdd(flist, name, r));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode ReconstructListFind(PetscFunctionList flist, const char *name, ReconstructFunction *r)
{
  PetscFunctionBeginUser;
  PetscCall(PetscFunctionListFind(flist, name, r));
  PetscCheck(*r, PETSC_COMM_SELF, PETSC_ERR_ARG_UNKNOWN_TYPE, "Reconstruction \"%s\" could not be found", name);
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode RiemannListAdd_2WaySplit(PetscFunctionList *flist, const char *name, RiemannFunction_2WaySplit rsolve)
{
  PetscFunctionBeginUser;
  PetscCall(PetscFunctionListAdd(flist, name, rsolve));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode RiemannListFind_2WaySplit(PetscFunctionList flist, const char *name, RiemannFunction_2WaySplit *rsolve)
{
  PetscFunctionBeginUser;
  PetscCall(PetscFunctionListFind(flist, name, rsolve));
  PetscCheck(*rsolve, PETSC_COMM_SELF, PETSC_ERR_ARG_UNKNOWN_TYPE, "Riemann solver \"%s\" could not be found", name);
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode ReconstructListAdd_2WaySplit(PetscFunctionList *flist, const char *name, ReconstructFunction_2WaySplit r)
{
  PetscFunctionBeginUser;
  PetscCall(PetscFunctionListAdd(flist, name, r));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode ReconstructListFind_2WaySplit(PetscFunctionList flist, const char *name, ReconstructFunction_2WaySplit *r)
{
  PetscFunctionBeginUser;
  PetscCall(PetscFunctionListFind(flist, name, r));
  PetscCheck(*r, PETSC_COMM_SELF, PETSC_ERR_ARG_UNKNOWN_TYPE, "Reconstruction \"%s\" could not be found", name);
  PetscFunctionReturn(PETSC_SUCCESS);
}

/* --------------------------------- Physics ------- */
PetscErrorCode PhysicsDestroy_SimpleFree(void *vctx)
{
  PetscFunctionBeginUser;
  PetscCall(PetscFree(vctx));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/* --------------------------------- Finite Volume Solver --------------- */
PetscErrorCode FVRHSFunction(TS ts, PetscReal time, Vec X, Vec F, void *vctx)
{
  FVCtx       *ctx = (FVCtx *)vctx;
  PetscInt     i, j, k, Mx, dof, xs, xm;
  PetscReal    hx, 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 */
  hx = (ctx->xmax - ctx->xmin) / Mx;
  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->physics.characteristic)(ctx->physics.user, dof, &x[i * dof], ctx->R, ctx->Rinv, ctx->speeds, ctx->xmin + hx * i));
    /* 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.hx = hx;
    (*ctx->limit)(&info, cjmpL, cjmpR, ctx->cslope);
    for (j = 0; j < dof; j++) ctx->cslope[j] /= hx; /* rescale to a slope */
    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];
    for (j = 0; j < dof; j++) {
      uL[j] = x[(i - 1) * dof + j] + slope[(i - 1) * dof + j] * hx / 2;
      uR[j] = x[(i - 0) * dof + j] - slope[(i - 0) * dof + j] * hx / 2;
    }
    PetscCall((*ctx->physics.riemann)(ctx->physics.user, dof, uL, uR, ctx->flux, &maxspeed, ctx->xmin + hx * i, ctx->xmin, ctx->xmax));
    cfl_idt = PetscMax(cfl_idt, PetscAbsScalar(maxspeed / hx)); /* Max allowable value of 1/Delta t */
    if (i > xs) {
      for (j = 0; j < dof; j++) f[(i - 1) * dof + j] -= ctx->flux[j] / hx;
    }
    if (i < xs + xm) {
      for (j = 0; j < dof; j++) f[i * dof + j] += ctx->flux[j] / hx;
    }
  }
  PetscCall(DMDAVecRestoreArray(da, Xloc, &x));
  PetscCall(DMDAVecRestoreArray(da, F, &f));
  PetscCall(DMDARestoreArray(da, PETSC_TRUE, &slope));
  PetscCall(DMRestoreLocalVector(da, &Xloc));
  PetscCallMPI(MPIU_Allreduce(&cfl_idt, &ctx->cfl_idt, 1, MPIU_REAL, 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) {
      if (1) PetscCall(PetscPrintf(ctx->comm, "Stability constraint exceeded at t=%g, dt %g > %g\n", (double)tnow, (double)dt, (double)(1 / (2 * ctx->cfl_idt))));
      else SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Stability constraint exceeded, %g > %g", (double)dt, (double)(ctx->cfl / ctx->cfl_idt));
    }
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode FVSample(FVCtx *ctx, DM da, PetscReal time, Vec U)
{
  PetscScalar *u, *uj;
  PetscInt     i, j, k, dof, xs, xm, Mx;

  PetscFunctionBeginUser;
  PetscCheck(ctx->physics.sample, 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));
  for (i = xs; i < xs + xm; i++) {
    const PetscReal h = (ctx->xmax - ctx->xmin) / Mx, xi = ctx->xmin + h / 2 + i * h;
    const PetscInt  N = 200;
    /* 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++) {
      PetscScalar xj = xi + h * (j - N / 2) / (PetscReal)N;
      PetscCall((*ctx->physics.sample)(ctx->physics.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);
}

PetscErrorCode SolutionStatsView(DM da, Vec X, PetscViewer viewer)
{
  PetscReal          xmin, xmax;
  PetscScalar        sum, tvsum, tvgsum;
  const PetscScalar *x;
  PetscInt           imin, imax, Mx, i, j, xs, xm, dof;
  Vec                Xloc;
  PetscBool          isascii;

  PetscFunctionBeginUser;
  PetscCall(PetscObjectTypeCompare((PetscObject)viewer, PETSCVIEWERASCII, &isascii));
  PetscCheck(isascii, PETSC_COMM_SELF, PETSC_ERR_SUP, "Viewer type not supported");
  /* PETSc lacks a function to compute total variation norm (difficult in multiple dimensions), we do it here */
  PetscCall(DMGetLocalVector(da, &Xloc));
  PetscCall(DMGlobalToLocalBegin(da, X, INSERT_VALUES, Xloc));
  PetscCall(DMGlobalToLocalEnd(da, X, INSERT_VALUES, Xloc));
  PetscCall(DMDAVecGetArrayRead(da, Xloc, (void *)&x));
  PetscCall(DMDAGetCorners(da, &xs, 0, 0, &xm, 0, 0));
  PetscCall(DMDAGetInfo(da, 0, &Mx, 0, 0, 0, 0, 0, &dof, 0, 0, 0, 0, 0));
  tvsum = 0;
  for (i = xs; i < xs + xm; i++) {
    for (j = 0; j < dof; j++) tvsum += PetscAbsScalar(x[i * dof + j] - x[(i - 1) * dof + j]);
  }
  PetscCallMPI(MPIU_Allreduce(&tvsum, &tvgsum, 1, MPIU_REAL, MPIU_SUM, PetscObjectComm((PetscObject)da)));
  PetscCall(DMDAVecRestoreArrayRead(da, Xloc, (void *)&x));
  PetscCall(DMRestoreLocalVector(da, &Xloc));
  PetscCall(VecMin(X, &imin, &xmin));
  PetscCall(VecMax(X, &imax, &xmax));
  PetscCall(VecSum(X, &sum));
  PetscCall(PetscViewerASCIIPrintf(viewer, "Solution range [%8.5f,%8.5f] with minimum at %" PetscInt_FMT ", mean %8.5f, ||x||_TV %8.5f\n", (double)xmin, (double)xmax, imin, (double)(sum / Mx), (double)(tvgsum / Mx)));
  PetscFunctionReturn(PETSC_SUCCESS);
}