xref: /petsc/src/ts/tutorials/ex9.c (revision 0912c85a2fe9dd5dd00b8a78a5de9e5a6dad05cf)

static const char help[] = "1D periodic Finite Volume solver in slope-limiter form with semidiscrete time stepping.\n"
  "Solves scalar and vector problems, choose the physical model with -physics\n"
  "  advection   - Constant coefficient scalar advection\n"
  "                u_t       + (a*u)_x               = 0\n"
  "  burgers     - Burgers equation\n"
  "                u_t       + (u^2/2)_x             = 0\n"
  "  traffic     - Traffic equation\n"
  "                u_t       + (u*(1-u))_x           = 0\n"
  "  acoustics   - Acoustic wave propagation\n"
  "                u_t       + (c*z*v)_x             = 0\n"
  "                v_t       + (c/z*u)_x             = 0\n"
  "  isogas      - Isothermal gas dynamics\n"
  "                rho_t     + (rho*u)_x             = 0\n"
  "                (rho*u)_t + (rho*u^2 + c^2*rho)_x = 0\n"
  "  shallow     - Shallow water equations\n"
  "                h_t       + (h*u)_x               = 0\n"
  "                (h*u)_t   + (h*u^2 + g*h^2/2)_x   = 0\n"
  "Some of these physical models have multiple Riemann solvers, select these with -physics_xxx_riemann\n"
  "  exact       - Exact Riemann solver which usually needs to perform a Newton iteration to connect\n"
  "                the states across shocks and rarefactions\n"
  "  roe         - Linearized scheme, usually with an entropy fix inside sonic rarefactions\n"
  "The systems provide a choice of reconstructions with -physics_xxx_reconstruct\n"
  "  characteristic - Limit the characteristic variables, this is usually preferred (default)\n"
  "  conservative   - Limit the conservative variables directly, can cause undesired interaction of waves\n\n"
  "A variety of limiters for high-resolution TVD limiters are available with -limit\n"
  "  upwind,minmod,superbee,mc,vanleer,vanalbada,koren,cada-torillhon (last two are nominally third order)\n"
  "  and non-TVD schemes lax-wendroff,beam-warming,fromm\n\n"
  "To preserve the TVD property, one should time step with a strong stability preserving method.\n"
  "The optimal high order explicit Runge-Kutta methods in TSSSP are recommended for non-stiff problems.\n\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 <petsc/private/kernels/blockinvert.h> /* For the Kernel_*_gets_* stuff for BAIJ */

PETSC_STATIC_INLINE PetscReal Sgn(PetscReal a) { return (a<0) ? -1 : 1; }
PETSC_STATIC_INLINE PetscReal Abs(PetscReal a) { return (a<0) ? 0 : a; }
PETSC_STATIC_INLINE PetscReal Sqr(PetscReal a) { return a*a; }
PETSC_STATIC_INLINE PetscReal MaxAbs(PetscReal a,PetscReal b) { return (PetscAbs(a) > PetscAbs(b)) ? a : b; }
PETSC_UNUSED PETSC_STATIC_INLINE PetscReal MinAbs(PetscReal a,PetscReal b) { return (PetscAbs(a) < PetscAbs(b)) ? a : b; }
PETSC_STATIC_INLINE PetscReal MinMod2(PetscReal a,PetscReal b) { return (a*b<0) ? 0 : Sgn(a)*PetscMin(PetscAbs(a),PetscAbs(b)); }
PETSC_STATIC_INLINE PetscReal MaxMod2(PetscReal a,PetscReal b) { return (a*b<0) ? 0 : Sgn(a)*PetscMax(PetscAbs(a),PetscAbs(b)); }
PETSC_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))); }

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

/* ----------------------- Lots of limiters, these could go in a separate library ------------------------- */
typedef struct _LimitInfo {
  PetscReal hx;
  PetscInt  m;
} *LimitInfo;
static 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;
}
static 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];
}
static 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];
}
static 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]);
}
static 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]);
}
static 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]));
}
static 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]);
}
static 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);
}
static 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);
}
static 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);
}
static 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));
}
static 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));
}
static 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)))));
}
static 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]);
}
static 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]))));
  }
}
static void Limit_CadaTorrilhon3R0p1(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
{
  Limit_CadaTorrilhon3R(0.1,info,jL,jR,lmt);
}
static void Limit_CadaTorrilhon3R1(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
{
  Limit_CadaTorrilhon3R(1,info,jL,jR,lmt);
}
static void Limit_CadaTorrilhon3R10(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
{
  Limit_CadaTorrilhon3R(10,info,jL,jR,lmt);
}
static void Limit_CadaTorrilhon3R100(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
{
  Limit_CadaTorrilhon3R(100,info,jL,jR,lmt);
}

/* --------------------------------- Finite Volume data structures ----------------------------------- */

typedef enum {FVBC_PERIODIC, FVBC_OUTFLOW} FVBCType;
static const char *FVBCTypes[] = {"PERIODIC","OUTFLOW","FVBCType","FVBC_",0};
typedef PetscErrorCode (*RiemannFunction)(void*,PetscInt,const PetscScalar*,const PetscScalar*,PetscScalar*,PetscReal*);
typedef PetscErrorCode (*ReconstructFunction)(void*,PetscInt,const PetscScalar*,PetscScalar*,PetscScalar*,PetscReal*);

typedef struct {
  PetscErrorCode      (*sample)(void*,PetscInt,FVBCType,PetscReal,PetscReal,PetscReal,PetscReal,PetscReal*);
  RiemannFunction     riemann;
  ReconstructFunction characteristic;
  PetscErrorCode      (*destroy)(void*);
  void                *user;
  PetscInt            dof;
  char                *fieldname[16];
} PhysicsCtx;

typedef struct {
  void        (*limit)(LimitInfo,const PetscScalar*,const PetscScalar*,PetscScalar*);
  PhysicsCtx  physics;
  MPI_Comm    comm;
  char        prefix[256];

  /* Local work arrays */
  PetscScalar *R,*Rinv;         /* Characteristic basis, and it's inverse.  COLUMN-MAJOR */
  PetscScalar *cjmpLR;          /* Jumps at left and right edge of cell, in characteristic basis, len=2*dof */
  PetscScalar *cslope;          /* Limited slope, written in characteristic basis */
  PetscScalar *uLR;             /* Solution at left and right of interface, conservative variables, len=2*dof */
  PetscScalar *flux;            /* Flux across interface */
  PetscReal   *speeds;          /* Speeds of each wave */

  PetscReal   cfl_idt;            /* Max allowable value of 1/Delta t */
  PetscReal   cfl;
  PetscReal   xmin,xmax;
  PetscInt    initial;
  PetscBool   exact;
  FVBCType    bctype;
} FVCtx;

PetscErrorCode RiemannListAdd(PetscFunctionList *flist,const char *name,RiemannFunction rsolve)
{
  PetscErrorCode ierr;

  PetscFunctionBeginUser;
  ierr = PetscFunctionListAdd(flist,name,rsolve);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode RiemannListFind(PetscFunctionList flist,const char *name,RiemannFunction *rsolve)
{
  PetscErrorCode ierr;

  PetscFunctionBeginUser;
  ierr = PetscFunctionListFind(flist,name,rsolve);CHKERRQ(ierr);
  if (!*rsolve) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_ARG_UNKNOWN_TYPE,"Riemann solver \"%s\" could not be found",name);
  PetscFunctionReturn(0);
}

PetscErrorCode ReconstructListAdd(PetscFunctionList *flist,const char *name,ReconstructFunction r)
{
  PetscErrorCode ierr;

  PetscFunctionBeginUser;
  ierr = PetscFunctionListAdd(flist,name,r);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode ReconstructListFind(PetscFunctionList flist,const char *name,ReconstructFunction *r)
{
  PetscErrorCode ierr;

  PetscFunctionBeginUser;
  ierr = PetscFunctionListFind(flist,name,r);CHKERRQ(ierr);
  if (!*r) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_ARG_UNKNOWN_TYPE,"Reconstruction \"%s\" could not be found",name);
  PetscFunctionReturn(0);
}

/* --------------------------------- Physics ----------------------------------- */
/*
  Each physical model consists of Riemann solver and a function to determine the basis to use for reconstruction.  These
  are set with the PhysicsCreate_XXX function which allocates private storage and sets these methods as well as the
  number of fields and their names, and a function to deallocate private storage.
*/

/* First a few functions useful to several different physics */
static PetscErrorCode PhysicsCharacteristic_Conservative(void *vctx,PetscInt m,const PetscScalar *u,PetscScalar *X,PetscScalar *Xi,PetscReal *speeds)
{
  PetscInt i,j;

  PetscFunctionBeginUser;
  for (i=0; i<m; i++) {
    for (j=0; j<m; j++) Xi[i*m+j] = X[i*m+j] = (PetscScalar)(i==j);
    speeds[i] = PETSC_MAX_REAL; /* Indicates invalid */
  }
  PetscFunctionReturn(0);
}

static PetscErrorCode PhysicsDestroy_SimpleFree(void *vctx)
{
  PetscErrorCode ierr;

  PetscFunctionBeginUser;
  ierr = PetscFree(vctx);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/* --------------------------------- Advection ----------------------------------- */

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

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

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

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

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

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

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

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

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

/* --------------------------------- Burgers ----------------------------------- */

typedef struct {
  PetscReal lxf_speed;
} BurgersCtx;

static PetscErrorCode PhysicsSample_Burgers(void *vctx,PetscInt initial,FVBCType bctype,PetscReal xmin,PetscReal xmax,PetscReal t,PetscReal x,PetscReal *u)
{
  PetscFunctionBeginUser;
  if (bctype == FVBC_PERIODIC && t > 0) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Exact solution not implemented for periodic");
  switch (initial) {
    case 0: u[0] = (x < 0) ? 1 : -1; break;
    case 1:
      if       (x < -t) u[0] = -1;
      else if  (x < t)  u[0] = x/t;
      else              u[0] = 1;
      break;
    case 2:
      if      (x <= 0)      u[0] = 0;
      else if (x < t)       u[0] = x/t;
      else if (x < 1+0.5*t) u[0] = 1;
      else                  u[0] = 0;
      break;
    case 3:
      if       (x < 0.2*t) u[0] = 0.2;
      else if  (x < t) u[0] = x/t;
      else             u[0] = 1;
      break;
    case 4:
      if (t > 0) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Only initial condition available");
      u[0] = 0.7 + 0.3*PetscSinReal(2*PETSC_PI*((x-xmin)/(xmax-xmin)));
      break;
    case 5:                     /* Pure shock solution */
      if (x < 0.5*t) u[0] = 1;
      else u[0] = 0;
      break;
    default: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_UNKNOWN_TYPE,"unknown initial condition");
  }
  PetscFunctionReturn(0);
}

static PetscErrorCode PhysicsRiemann_Burgers_Exact(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
{
  PetscFunctionBeginUser;
  if (uL[0] < uR[0]) {          /* rarefaction */
    flux[0] = (uL[0]*uR[0] < 0)
      ? 0                       /* sonic rarefaction */
      : 0.5*PetscMin(PetscSqr(uL[0]),PetscSqr(uR[0]));
  } else {                      /* shock */
    flux[0] = 0.5*PetscMax(PetscSqr(uL[0]),PetscSqr(uR[0]));
  }
  *maxspeed = (PetscAbs(uL[0]) > PetscAbs(uR[0])) ? uL[0] : uR[0];
  PetscFunctionReturn(0);
}

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

  PetscFunctionBeginUser;
  speed   = 0.5*(uL[0] + uR[0]);
  flux[0] = 0.25*(PetscSqr(uL[0]) + PetscSqr(uR[0])) - 0.5*PetscAbs(speed)*(uR[0]-uL[0]);
  if (uL[0] <= 0 && 0 <= uR[0]) flux[0] = 0; /* Entropy fix for sonic rarefaction */
  *maxspeed = speed;
  PetscFunctionReturn(0);
}

static PetscErrorCode PhysicsRiemann_Burgers_LxF(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
{
  PetscReal   c;
  PetscScalar fL,fR;

  PetscFunctionBeginUser;
  c         = ((BurgersCtx*)vctx)->lxf_speed;
  fL        = 0.5*PetscSqr(uL[0]);
  fR        = 0.5*PetscSqr(uR[0]);
  flux[0]   = 0.5*(fL + fR) - 0.5*c*(uR[0] - uL[0]);
  *maxspeed = c;
  PetscFunctionReturn(0);
}

static PetscErrorCode PhysicsRiemann_Burgers_Rusanov(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
{
  PetscReal   c;
  PetscScalar fL,fR;

  PetscFunctionBeginUser;
  c         = PetscMax(PetscAbs(uL[0]),PetscAbs(uR[0]));
  fL        = 0.5*PetscSqr(uL[0]);
  fR        = 0.5*PetscSqr(uR[0]);
  flux[0]   = 0.5*(fL + fR) - 0.5*c*(uR[0] - uL[0]);
  *maxspeed = c;
  PetscFunctionReturn(0);
}

static PetscErrorCode PhysicsCreate_Burgers(FVCtx *ctx)
{
  BurgersCtx        *user;
  PetscErrorCode    ierr;
  RiemannFunction   r;
  PetscFunctionList rlist      = 0;
  char              rname[256] = "exact";

  PetscFunctionBeginUser;
  ierr = PetscNew(&user);CHKERRQ(ierr);

  ctx->physics.sample         = PhysicsSample_Burgers;
  ctx->physics.characteristic = PhysicsCharacteristic_Conservative;
  ctx->physics.destroy        = PhysicsDestroy_SimpleFree;
  ctx->physics.user           = user;
  ctx->physics.dof            = 1;

  ierr = PetscStrallocpy("u",&ctx->physics.fieldname[0]);CHKERRQ(ierr);
  ierr = RiemannListAdd(&rlist,"exact",  PhysicsRiemann_Burgers_Exact);CHKERRQ(ierr);
  ierr = RiemannListAdd(&rlist,"roe",    PhysicsRiemann_Burgers_Roe);CHKERRQ(ierr);
  ierr = RiemannListAdd(&rlist,"lxf",    PhysicsRiemann_Burgers_LxF);CHKERRQ(ierr);
  ierr = RiemannListAdd(&rlist,"rusanov",PhysicsRiemann_Burgers_Rusanov);CHKERRQ(ierr);
  ierr = PetscOptionsBegin(ctx->comm,ctx->prefix,"Options for advection","");CHKERRQ(ierr);
  {
    ierr = PetscOptionsFList("-physics_burgers_riemann","Riemann solver","",rlist,rname,rname,sizeof(rname),NULL);CHKERRQ(ierr);
  }
  ierr = PetscOptionsEnd();CHKERRQ(ierr);
  ierr = RiemannListFind(rlist,rname,&r);CHKERRQ(ierr);
  ierr = PetscFunctionListDestroy(&rlist);CHKERRQ(ierr);
  ctx->physics.riemann = r;

  /* *
  * Hack to deal with LxF in semi-discrete form
  * max speed is 1 for the basic initial conditions (where |u| <= 1)
  * */
  if (r == PhysicsRiemann_Burgers_LxF) user->lxf_speed = 1;
  PetscFunctionReturn(0);
}

/* --------------------------------- Traffic ----------------------------------- */

typedef struct {
  PetscReal lxf_speed;
  PetscReal a;
} TrafficCtx;

PETSC_STATIC_INLINE PetscScalar TrafficFlux(PetscScalar a,PetscScalar u) { return a*u*(1-u); }

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

  PetscFunctionBeginUser;
  if (bctype == FVBC_PERIODIC && t > 0) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Exact solution not implemented for periodic");
  switch (initial) {
    case 0:
      u[0] = (-a*t < x) ? 2 : 0; break;
    case 1:
      if      (x < PetscMin(2*a*t,0.5+a*t)) u[0] = -1;
      else if (x < 1)                       u[0] = 0;
      else                                  u[0] = 1;
      break;
    case 2:
      if (t > 0) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Only initial condition available");
      u[0] = 0.7 + 0.3*PetscSinReal(2*PETSC_PI*((x-xmin)/(xmax-xmin)));
      break;
    default: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_UNKNOWN_TYPE,"unknown initial condition");
  }
  PetscFunctionReturn(0);
}

static PetscErrorCode PhysicsRiemann_Traffic_Exact(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
{
  PetscReal a = ((TrafficCtx*)vctx)->a;

  PetscFunctionBeginUser;
  if (uL[0] < uR[0]) {
    flux[0] = PetscMin(TrafficFlux(a,uL[0]),TrafficFlux(a,uR[0]));
  } else {
    flux[0] = (uR[0] < 0.5 && 0.5 < uL[0]) ? TrafficFlux(a,0.5) : PetscMax(TrafficFlux(a,uL[0]),TrafficFlux(a,uR[0]));
  }
  *maxspeed = a*MaxAbs(1-2*uL[0],1-2*uR[0]);
  PetscFunctionReturn(0);
}

static PetscErrorCode PhysicsRiemann_Traffic_Roe(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
{
  PetscReal a = ((TrafficCtx*)vctx)->a;
  PetscReal speed;

  PetscFunctionBeginUser;
  speed = a*(1 - (uL[0] + uR[0]));
  flux[0] = 0.5*(TrafficFlux(a,uL[0]) + TrafficFlux(a,uR[0])) - 0.5*PetscAbs(speed)*(uR[0]-uL[0]);
  *maxspeed = speed;
  PetscFunctionReturn(0);
}

static PetscErrorCode PhysicsRiemann_Traffic_LxF(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
{
  TrafficCtx *phys = (TrafficCtx*)vctx;
  PetscReal  a     = phys->a;
  PetscReal  speed;

  PetscFunctionBeginUser;
  speed     = a*(1 - (uL[0] + uR[0]));
  flux[0]   = 0.5*(TrafficFlux(a,uL[0]) + TrafficFlux(a,uR[0])) - 0.5*phys->lxf_speed*(uR[0]-uL[0]);
  *maxspeed = speed;
  PetscFunctionReturn(0);
}

static PetscErrorCode PhysicsRiemann_Traffic_Rusanov(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
{
  PetscReal a = ((TrafficCtx*)vctx)->a;
  PetscReal speed;

  PetscFunctionBeginUser;
  speed     = a*PetscMax(PetscAbs(1-2*uL[0]),PetscAbs(1-2*uR[0]));
  flux[0]   = 0.5*(TrafficFlux(a,uL[0]) + TrafficFlux(a,uR[0])) - 0.5*speed*(uR[0]-uL[0]);
  *maxspeed = speed;
  PetscFunctionReturn(0);
}

static PetscErrorCode PhysicsCreate_Traffic(FVCtx *ctx)
{
  PetscErrorCode    ierr;
  TrafficCtx        *user;
  RiemannFunction   r;
  PetscFunctionList rlist      = 0;
  char              rname[256] = "exact";

  PetscFunctionBeginUser;
  ierr = PetscNew(&user);CHKERRQ(ierr);
  ctx->physics.sample         = PhysicsSample_Traffic;
  ctx->physics.characteristic = PhysicsCharacteristic_Conservative;
  ctx->physics.destroy        = PhysicsDestroy_SimpleFree;
  ctx->physics.user           = user;
  ctx->physics.dof            = 1;

  ierr = PetscStrallocpy("density",&ctx->physics.fieldname[0]);CHKERRQ(ierr);
  user->a = 0.5;
  ierr = RiemannListAdd(&rlist,"exact",  PhysicsRiemann_Traffic_Exact);CHKERRQ(ierr);
  ierr = RiemannListAdd(&rlist,"roe",    PhysicsRiemann_Traffic_Roe);CHKERRQ(ierr);
  ierr = RiemannListAdd(&rlist,"lxf",    PhysicsRiemann_Traffic_LxF);CHKERRQ(ierr);
  ierr = RiemannListAdd(&rlist,"rusanov",PhysicsRiemann_Traffic_Rusanov);CHKERRQ(ierr);
  ierr = PetscOptionsBegin(ctx->comm,ctx->prefix,"Options for Traffic","");CHKERRQ(ierr);
    ierr = PetscOptionsReal("-physics_traffic_a","Flux = a*u*(1-u)","",user->a,&user->a,NULL);CHKERRQ(ierr);
    ierr = PetscOptionsFList("-physics_traffic_riemann","Riemann solver","",rlist,rname,rname,sizeof(rname),NULL);CHKERRQ(ierr);
  ierr = PetscOptionsEnd();CHKERRQ(ierr);

  ierr = RiemannListFind(rlist,rname,&r);CHKERRQ(ierr);
  ierr = PetscFunctionListDestroy(&rlist);CHKERRQ(ierr);

  ctx->physics.riemann = r;

  /* *
  * Hack to deal with LxF in semi-discrete form
  * max speed is 3*a for the basic initial conditions (-1 <= u <= 2)
  * */
  if (r == PhysicsRiemann_Traffic_LxF) user->lxf_speed = 3*user->a;
  PetscFunctionReturn(0);
}

/* --------------------------------- Linear Acoustics ----------------------------------- */

/* Flux: u_t + (A u)_x
 * z = sqrt(rho*bulk), c = sqrt(rho/bulk)
 * Spectral decomposition: A = R * D * Rinv
 * [    cz] = [-z   z] [-c    ] [-1/2z  1/2]
 * [c/z   ] = [ 1   1] [     c] [ 1/2z  1/2]
 *
 * We decompose this into the left-traveling waves Al = R * D^- Rinv
 * and the right-traveling waves Ar = R * D^+ * Rinv
 * Multiplying out these expressions produces the following two matrices
 */

typedef struct {
  PetscReal c;                  /* speed of sound: c = sqrt(bulk/rho) */
  PetscReal z;                  /* impedence: z = sqrt(rho*bulk) */
} AcousticsCtx;

PETSC_UNUSED PETSC_STATIC_INLINE void AcousticsFlux(AcousticsCtx *ctx,const PetscScalar *u,PetscScalar *f)
{
  f[0] = ctx->c*ctx->z*u[1];
  f[1] = ctx->c/ctx->z*u[0];
}

static PetscErrorCode PhysicsCharacteristic_Acoustics(void *vctx,PetscInt m,const PetscScalar *u,PetscScalar *X,PetscScalar *Xi,PetscReal *speeds)
{
  AcousticsCtx *phys = (AcousticsCtx*)vctx;
  PetscReal    z     = phys->z,c = phys->c;

  PetscFunctionBeginUser;
  X[0*2+0]  = -z;
  X[0*2+1]  = z;
  X[1*2+0]  = 1;
  X[1*2+1]  = 1;
  Xi[0*2+0] = -1./(2*z);
  Xi[0*2+1] = 1./2;
  Xi[1*2+0] = 1./(2*z);
  Xi[1*2+1] = 1./2;
  speeds[0] = -c;
  speeds[1] = c;
  PetscFunctionReturn(0);
}

static PetscErrorCode PhysicsSample_Acoustics_Initial(AcousticsCtx *phys,PetscInt initial,PetscReal xmin,PetscReal xmax,PetscReal x,PetscReal *u)
{
  PetscFunctionBeginUser;
  switch (initial) {
  case 0:
    u[0] = (PetscAbs((x - xmin)/(xmax - xmin) - 0.2) < 0.1) ? 1 : 0.5;
    u[1] = (PetscAbs((x - xmin)/(xmax - xmin) - 0.7) < 0.1) ? 1 : -0.5;
    break;
  case 1:
    u[0] = PetscCosReal(3 * 2*PETSC_PI*x/(xmax-xmin));
    u[1] = PetscExpReal(-PetscSqr(x - (xmax + xmin)/2) / (2*PetscSqr(0.2*(xmax - xmin)))) - 0.5;
    break;
  default: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_UNKNOWN_TYPE,"unknown initial condition");
  }
  PetscFunctionReturn(0);
}

static PetscErrorCode PhysicsSample_Acoustics(void *vctx,PetscInt initial,FVBCType bctype,PetscReal xmin,PetscReal xmax,PetscReal t,PetscReal x,PetscReal *u)
{
  AcousticsCtx   *phys = (AcousticsCtx*)vctx;
  PetscReal      c     = phys->c;
  PetscReal      x0a,x0b,u0a[2],u0b[2],tmp[2];
  PetscReal      X[2][2],Xi[2][2],dummy[2];
  PetscErrorCode ierr;

  PetscFunctionBeginUser;
  switch (bctype) {
  case FVBC_OUTFLOW:
    x0a = x+c*t;
    x0b = x-c*t;
    break;
  case FVBC_PERIODIC:
    x0a = RangeMod(x+c*t,xmin,xmax);
    x0b = RangeMod(x-c*t,xmin,xmax);
    break;
  default: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_UNKNOWN_TYPE,"unknown BCType");
  }
  ierr   = PhysicsSample_Acoustics_Initial(phys,initial,xmin,xmax,x0a,u0a);CHKERRQ(ierr);
  ierr   = PhysicsSample_Acoustics_Initial(phys,initial,xmin,xmax,x0b,u0b);CHKERRQ(ierr);
  ierr   = PhysicsCharacteristic_Acoustics(vctx,2,u,&X[0][0],&Xi[0][0],dummy);CHKERRQ(ierr);
  tmp[0] = Xi[0][0]*u0a[0] + Xi[0][1]*u0a[1];
  tmp[1] = Xi[1][0]*u0b[0] + Xi[1][1]*u0b[1];
  u[0]   = X[0][0]*tmp[0] + X[0][1]*tmp[1];
  u[1]   = X[1][0]*tmp[0] + X[1][1]*tmp[1];
  PetscFunctionReturn(0);
}

static PetscErrorCode PhysicsRiemann_Acoustics_Exact(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
{
  AcousticsCtx *phys = (AcousticsCtx*)vctx;
  PetscReal    c     = phys->c,z = phys->z;
  PetscReal
    Al[2][2] = {{-c/2     , c*z/2  },
                {c/(2*z)  , -c/2   }}, /* Left traveling waves */
    Ar[2][2] = {{c/2      , c*z/2  },
                {c/(2*z)  , c/2    }}; /* Right traveling waves */

  PetscFunctionBeginUser;
  flux[0]   = Al[0][0]*uR[0] + Al[0][1]*uR[1] + Ar[0][0]*uL[0] + Ar[0][1]*uL[1];
  flux[1]   = Al[1][0]*uR[0] + Al[1][1]*uR[1] + Ar[1][0]*uL[0] + Ar[1][1]*uL[1];
  *maxspeed = c;
  PetscFunctionReturn(0);
}

static PetscErrorCode PhysicsCreate_Acoustics(FVCtx *ctx)
{
  PetscErrorCode    ierr;
  AcousticsCtx      *user;
  PetscFunctionList rlist      = 0,rclist = 0;
  char              rname[256] = "exact",rcname[256] = "characteristic";

  PetscFunctionBeginUser;
  ierr = PetscNew(&user);CHKERRQ(ierr);
  ctx->physics.sample         = PhysicsSample_Acoustics;
  ctx->physics.destroy        = PhysicsDestroy_SimpleFree;
  ctx->physics.user           = user;
  ctx->physics.dof            = 2;

  ierr = PetscStrallocpy("u",&ctx->physics.fieldname[0]);CHKERRQ(ierr);
  ierr = PetscStrallocpy("v",&ctx->physics.fieldname[1]);CHKERRQ(ierr);

  user->c = 1;
  user->z = 1;

  ierr = RiemannListAdd(&rlist,"exact",  PhysicsRiemann_Acoustics_Exact);CHKERRQ(ierr);
  ierr = ReconstructListAdd(&rclist,"characteristic",PhysicsCharacteristic_Acoustics);CHKERRQ(ierr);
  ierr = ReconstructListAdd(&rclist,"conservative",PhysicsCharacteristic_Conservative);CHKERRQ(ierr);
  ierr = PetscOptionsBegin(ctx->comm,ctx->prefix,"Options for linear Acoustics","");CHKERRQ(ierr);
  {
    ierr = PetscOptionsReal("-physics_acoustics_c","c = sqrt(bulk/rho)","",user->c,&user->c,NULL);CHKERRQ(ierr);
    ierr = PetscOptionsReal("-physics_acoustics_z","z = sqrt(bulk*rho)","",user->z,&user->z,NULL);CHKERRQ(ierr);
    ierr = PetscOptionsFList("-physics_acoustics_riemann","Riemann solver","",rlist,rname,rname,sizeof(rname),NULL);CHKERRQ(ierr);
    ierr = PetscOptionsFList("-physics_acoustics_reconstruct","Reconstruction","",rclist,rcname,rcname,sizeof(rcname),NULL);CHKERRQ(ierr);
  }
  ierr = PetscOptionsEnd();CHKERRQ(ierr);
  ierr = RiemannListFind(rlist,rname,&ctx->physics.riemann);CHKERRQ(ierr);
  ierr = ReconstructListFind(rclist,rcname,&ctx->physics.characteristic);CHKERRQ(ierr);
  ierr = PetscFunctionListDestroy(&rlist);CHKERRQ(ierr);
  ierr = PetscFunctionListDestroy(&rclist);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/* --------------------------------- Isothermal Gas Dynamics ----------------------------------- */

typedef struct {
  PetscReal acoustic_speed;
} IsoGasCtx;

PETSC_STATIC_INLINE void IsoGasFlux(PetscReal c,const PetscScalar *u,PetscScalar *f)
{
  f[0] = u[1];
  f[1] = PetscSqr(u[1])/u[0] + c*c*u[0];
}

static PetscErrorCode PhysicsSample_IsoGas(void *vctx,PetscInt initial,FVBCType bctype,PetscReal xmin,PetscReal xmax,PetscReal t,PetscReal x,PetscReal *u)
{
  PetscFunctionBeginUser;
  if (t > 0) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Exact solutions not implemented for t > 0");
  switch (initial) {
    case 0:
      u[0] = (x < 0) ? 1 : 0.5;
      u[1] = (x < 0) ? 1 : 0.7;
      break;
    case 1:
      u[0] = 1+0.5*PetscSinReal(2*PETSC_PI*x);
      u[1] = 1*u[0];
      break;
    default: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_UNKNOWN_TYPE,"unknown initial condition");
  }
  PetscFunctionReturn(0);
}

static PetscErrorCode PhysicsRiemann_IsoGas_Roe(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
{
  IsoGasCtx   *phys = (IsoGasCtx*)vctx;
  PetscReal   c     = phys->acoustic_speed;
  PetscScalar ubar,du[2],a[2],fL[2],fR[2],lam[2],ustar[2],R[2][2];
  PetscInt    i;

  PetscFunctionBeginUser;
  ubar = (uL[1]/PetscSqrtScalar(uL[0]) + uR[1]/PetscSqrtScalar(uR[0])) / (PetscSqrtScalar(uL[0]) + PetscSqrtScalar(uR[0]));
  /* write fluxuations in characteristic basis */
  du[0] = uR[0] - uL[0];
  du[1] = uR[1] - uL[1];
  a[0]  = (1/(2*c)) * ((ubar + c)*du[0] - du[1]);
  a[1]  = (1/(2*c)) * ((-ubar + c)*du[0] + du[1]);
  /* wave speeds */
  lam[0] = ubar - c;
  lam[1] = ubar + c;
  /* Right eigenvectors */
  R[0][0] = 1; R[0][1] = ubar-c;
  R[1][0] = 1; R[1][1] = ubar+c;
  /* Compute state in star region (between the 1-wave and 2-wave) */
  for (i=0; i<2; i++) ustar[i] = uL[i] + a[0]*R[0][i];
  if (uL[1]/uL[0] < c && c < ustar[1]/ustar[0]) { /* 1-wave is sonic rarefaction */
    PetscScalar ufan[2];
    ufan[0] = uL[0]*PetscExpScalar(uL[1]/(uL[0]*c) - 1);
    ufan[1] = c*ufan[0];
    IsoGasFlux(c,ufan,flux);
  } else if (ustar[1]/ustar[0] < -c && -c < uR[1]/uR[0]) { /* 2-wave is sonic rarefaction */
    PetscScalar ufan[2];
    ufan[0] = uR[0]*PetscExpScalar(-uR[1]/(uR[0]*c) - 1);
    ufan[1] = -c*ufan[0];
    IsoGasFlux(c,ufan,flux);
  } else {                      /* Centered form */
    IsoGasFlux(c,uL,fL);
    IsoGasFlux(c,uR,fR);
    for (i=0; i<2; i++) {
      PetscScalar absdu = PetscAbsScalar(lam[0])*a[0]*R[0][i] + PetscAbsScalar(lam[1])*a[1]*R[1][i];
      flux[i] = 0.5*(fL[i]+fR[i]) - 0.5*absdu;
    }
  }
  *maxspeed = MaxAbs(lam[0],lam[1]);
  PetscFunctionReturn(0);
}

static PetscErrorCode PhysicsRiemann_IsoGas_Exact(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
{
  IsoGasCtx                   *phys = (IsoGasCtx*)vctx;
  PetscReal                   c     = phys->acoustic_speed;
  PetscScalar                 ustar[2];
  struct {PetscScalar rho,u;} L = {uL[0],uL[1]/uL[0]},R = {uR[0],uR[1]/uR[0]},star;
  PetscInt                    i;

  PetscFunctionBeginUser;
  if (!(L.rho > 0 && R.rho > 0)) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Reconstructed density is negative");
  {
    /* Solve for star state */
    PetscScalar res,tmp,rho = 0.5*(L.rho + R.rho); /* initial guess */
    for (i=0; i<20; i++) {
      PetscScalar fr,fl,dfr,dfl;
      fl = (L.rho < rho)
        ? (rho-L.rho)/PetscSqrtScalar(L.rho*rho)       /* shock */
        : PetscLogScalar(rho) - PetscLogScalar(L.rho); /* rarefaction */
      fr = (R.rho < rho)
        ? (rho-R.rho)/PetscSqrtScalar(R.rho*rho)       /* shock */
        : PetscLogScalar(rho) - PetscLogScalar(R.rho); /* rarefaction */
      res = R.u-L.u + c*(fr+fl);
      if (PetscIsInfOrNanScalar(res)) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_FP,"Infinity or Not-a-Number generated in computation");
      if (PetscAbsScalar(res) < 1e-10) {
        star.rho = rho;
        star.u   = L.u - c*fl;
        goto converged;
      }
      dfl = (L.rho < rho) ? 1/PetscSqrtScalar(L.rho*rho)*(1 - 0.5*(rho-L.rho)/rho) : 1/rho;
      dfr = (R.rho < rho) ? 1/PetscSqrtScalar(R.rho*rho)*(1 - 0.5*(rho-R.rho)/rho) : 1/rho;
      tmp = rho - res/(c*(dfr+dfl));
      if (tmp <= 0) rho /= 2;   /* Guard against Newton shooting off to a negative density */
      else rho = tmp;
      if (!((rho > 0) && PetscIsNormalScalar(rho))) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_FP,"non-normal iterate rho=%g",(double)PetscRealPart(rho));
    }
    SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_CONV_FAILED,"Newton iteration for star.rho diverged after %D iterations",i);
  }
converged:
  if (L.u-c < 0 && 0 < star.u-c) { /* 1-wave is sonic rarefaction */
    PetscScalar ufan[2];
    ufan[0] = L.rho*PetscExpScalar(L.u/c - 1);
    ufan[1] = c*ufan[0];
    IsoGasFlux(c,ufan,flux);
  } else if (star.u+c < 0 && 0 < R.u+c) { /* 2-wave is sonic rarefaction */
    PetscScalar ufan[2];
    ufan[0] = R.rho*PetscExpScalar(-R.u/c - 1);
    ufan[1] = -c*ufan[0];
    IsoGasFlux(c,ufan,flux);
  } else if ((L.rho >= star.rho && L.u-c >= 0) || (L.rho < star.rho && (star.rho*star.u-L.rho*L.u)/(star.rho-L.rho) > 0)) {
    /* 1-wave is supersonic rarefaction, or supersonic shock */
    IsoGasFlux(c,uL,flux);
  } else if ((star.rho <= R.rho && R.u+c <= 0) || (star.rho > R.rho && (R.rho*R.u-star.rho*star.u)/(R.rho-star.rho) < 0)) {
    /* 2-wave is supersonic rarefaction or supersonic shock */
    IsoGasFlux(c,uR,flux);
  } else {
    ustar[0] = star.rho;
    ustar[1] = star.rho*star.u;
    IsoGasFlux(c,ustar,flux);
  }
  *maxspeed = MaxAbs(MaxAbs(star.u-c,star.u+c),MaxAbs(L.u-c,R.u+c));
  PetscFunctionReturn(0);
}

static PetscErrorCode PhysicsRiemann_IsoGas_Rusanov(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
{
  IsoGasCtx                   *phys = (IsoGasCtx*)vctx;
  PetscScalar                 c = phys->acoustic_speed,fL[2],fR[2],s;
  struct {PetscScalar rho,u;} L = {uL[0],uL[1]/uL[0]},R = {uR[0],uR[1]/uR[0]};

  PetscFunctionBeginUser;
  if (!(L.rho > 0 && R.rho > 0)) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Reconstructed density is negative");
  IsoGasFlux(c,uL,fL);
  IsoGasFlux(c,uR,fR);
  s         = PetscMax(PetscAbs(L.u),PetscAbs(R.u))+c;
  flux[0]   = 0.5*(fL[0] + fR[0]) + 0.5*s*(uL[0] - uR[0]);
  flux[1]   = 0.5*(fL[1] + fR[1]) + 0.5*s*(uL[1] - uR[1]);
  *maxspeed = s;
  PetscFunctionReturn(0);
}

static PetscErrorCode PhysicsCharacteristic_IsoGas(void *vctx,PetscInt m,const PetscScalar *u,PetscScalar *X,PetscScalar *Xi,PetscReal *speeds)
{
  IsoGasCtx      *phys = (IsoGasCtx*)vctx;
  PetscReal      c     = phys->acoustic_speed;
  PetscErrorCode ierr;

  PetscFunctionBeginUser;
  speeds[0] = u[1]/u[0] - c;
  speeds[1] = u[1]/u[0] + c;
  X[0*2+0]  = 1;
  X[0*2+1]  = speeds[0];
  X[1*2+0]  = 1;
  X[1*2+1]  = speeds[1];
  ierr = PetscArraycpy(Xi,X,4);CHKERRQ(ierr);
  ierr = PetscKernel_A_gets_inverse_A_2(Xi,0,PETSC_FALSE,NULL);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode PhysicsCreate_IsoGas(FVCtx *ctx)
{
  PetscErrorCode    ierr;
  IsoGasCtx         *user;
  PetscFunctionList rlist = 0,rclist = 0;
  char              rname[256] = "exact",rcname[256] = "characteristic";

  PetscFunctionBeginUser;
  ierr = PetscNew(&user);CHKERRQ(ierr);
  ctx->physics.sample         = PhysicsSample_IsoGas;
  ctx->physics.destroy        = PhysicsDestroy_SimpleFree;
  ctx->physics.user           = user;
  ctx->physics.dof            = 2;

  ierr = PetscStrallocpy("density",&ctx->physics.fieldname[0]);CHKERRQ(ierr);
  ierr = PetscStrallocpy("momentum",&ctx->physics.fieldname[1]);CHKERRQ(ierr);

  user->acoustic_speed = 1;

  ierr = RiemannListAdd(&rlist,"exact",  PhysicsRiemann_IsoGas_Exact);CHKERRQ(ierr);
  ierr = RiemannListAdd(&rlist,"roe",    PhysicsRiemann_IsoGas_Roe);CHKERRQ(ierr);
  ierr = RiemannListAdd(&rlist,"rusanov",PhysicsRiemann_IsoGas_Rusanov);CHKERRQ(ierr);
  ierr = ReconstructListAdd(&rclist,"characteristic",PhysicsCharacteristic_IsoGas);CHKERRQ(ierr);
  ierr = ReconstructListAdd(&rclist,"conservative",PhysicsCharacteristic_Conservative);CHKERRQ(ierr);
  ierr = PetscOptionsBegin(ctx->comm,ctx->prefix,"Options for IsoGas","");CHKERRQ(ierr);
    ierr = PetscOptionsReal("-physics_isogas_acoustic_speed","Acoustic speed","",user->acoustic_speed,&user->acoustic_speed,NULL);CHKERRQ(ierr);
    ierr = PetscOptionsFList("-physics_isogas_riemann","Riemann solver","",rlist,rname,rname,sizeof(rname),NULL);CHKERRQ(ierr);
    ierr = PetscOptionsFList("-physics_isogas_reconstruct","Reconstruction","",rclist,rcname,rcname,sizeof(rcname),NULL);CHKERRQ(ierr);
  ierr = PetscOptionsEnd();CHKERRQ(ierr);
  ierr = RiemannListFind(rlist,rname,&ctx->physics.riemann);CHKERRQ(ierr);
  ierr = ReconstructListFind(rclist,rcname,&ctx->physics.characteristic);CHKERRQ(ierr);
  ierr = PetscFunctionListDestroy(&rlist);CHKERRQ(ierr);
  ierr = PetscFunctionListDestroy(&rclist);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/* --------------------------------- Shallow Water ----------------------------------- */
typedef struct {
  PetscReal gravity;
} ShallowCtx;

PETSC_STATIC_INLINE void ShallowFlux(ShallowCtx *phys,const PetscScalar *u,PetscScalar *f)
{
  f[0] = u[1];
  f[1] = PetscSqr(u[1])/u[0] + 0.5*phys->gravity*PetscSqr(u[0]);
}

static PetscErrorCode PhysicsRiemann_Shallow_Exact(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
{
  ShallowCtx                *phys = (ShallowCtx*)vctx;
  PetscScalar               g    = phys->gravity,ustar[2],cL,cR,c,cstar;
  struct {PetscScalar h,u;} L = {uL[0],uL[1]/uL[0]},R = {uR[0],uR[1]/uR[0]},star;
  PetscInt                  i;

  PetscFunctionBeginUser;
  if (!(L.h > 0 && R.h > 0)) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Reconstructed thickness is negative");
  cL = PetscSqrtScalar(g*L.h);
  cR = PetscSqrtScalar(g*R.h);
  c  = PetscMax(cL,cR);
  {
    /* Solve for star state */
    const PetscInt maxits = 50;
    PetscScalar tmp,res,res0=0,h0,h = 0.5*(L.h + R.h); /* initial guess */
    h0 = h;
    for (i=0; i<maxits; i++) {
      PetscScalar fr,fl,dfr,dfl;
      fl = (L.h < h)
        ? PetscSqrtScalar(0.5*g*(h*h - L.h*L.h)*(1/L.h - 1/h)) /* shock */
        : 2*PetscSqrtScalar(g*h) - 2*PetscSqrtScalar(g*L.h);   /* rarefaction */
      fr = (R.h < h)
        ? PetscSqrtScalar(0.5*g*(h*h - R.h*R.h)*(1/R.h - 1/h)) /* shock */
        : 2*PetscSqrtScalar(g*h) - 2*PetscSqrtScalar(g*R.h);   /* rarefaction */
      res = R.u - L.u + fr + fl;
      if (PetscIsInfOrNanScalar(res)) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_FP,"Infinity or Not-a-Number generated in computation");
      if (PetscAbsScalar(res) < 1e-8 || (i > 0 && PetscAbsScalar(h-h0) < 1e-8)) {
        star.h = h;
        star.u = L.u - fl;
        goto converged;
      } else if (i > 0 && PetscAbsScalar(res) >= PetscAbsScalar(res0)) {        /* Line search */
        h = 0.8*h0 + 0.2*h;
        continue;
      }
      /* Accept the last step and take another */
      res0 = res;
      h0 = h;
      dfl = (L.h < h) ? 0.5/fl*0.5*g*(-L.h*L.h/(h*h) - 1 + 2*h/L.h) : PetscSqrtScalar(g/h);
      dfr = (R.h < h) ? 0.5/fr*0.5*g*(-R.h*R.h/(h*h) - 1 + 2*h/R.h) : PetscSqrtScalar(g/h);
      tmp = h - res/(dfr+dfl);
      if (tmp <= 0) h /= 2;   /* Guard against Newton shooting off to a negative thickness */
      else h = tmp;
      if (!((h > 0) && PetscIsNormalScalar(h))) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_FP,"non-normal iterate h=%g",(double)h);
    }
    SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_CONV_FAILED,"Newton iteration for star.h diverged after %D iterations",i);
  }
converged:
  cstar = PetscSqrtScalar(g*star.h);
  if (L.u-cL < 0 && 0 < star.u-cstar) { /* 1-wave is sonic rarefaction */
    PetscScalar ufan[2];
    ufan[0] = 1/g*PetscSqr(L.u/3 + 2./3*cL);
    ufan[1] = PetscSqrtScalar(g*ufan[0])*ufan[0];
    ShallowFlux(phys,ufan,flux);
  } else if (star.u+cstar < 0 && 0 < R.u+cR) { /* 2-wave is sonic rarefaction */
    PetscScalar ufan[2];
    ufan[0] = 1/g*PetscSqr(R.u/3 - 2./3*cR);
    ufan[1] = -PetscSqrtScalar(g*ufan[0])*ufan[0];
    ShallowFlux(phys,ufan,flux);
  } else if ((L.h >= star.h && L.u-c >= 0) || (L.h<star.h && (star.h*star.u-L.h*L.u)/(star.h-L.h) > 0)) {
    /* 1-wave is right-travelling shock (supersonic) */
    ShallowFlux(phys,uL,flux);
  } else if ((star.h <= R.h && R.u+c <= 0) || (star.h>R.h && (R.h*R.u-star.h*star.h)/(R.h-star.h) < 0)) {
    /* 2-wave is left-travelling shock (supersonic) */
    ShallowFlux(phys,uR,flux);
  } else {
    ustar[0] = star.h;
    ustar[1] = star.h*star.u;
    ShallowFlux(phys,ustar,flux);
  }
  *maxspeed = MaxAbs(MaxAbs(star.u-cstar,star.u+cstar),MaxAbs(L.u-cL,R.u+cR));
  PetscFunctionReturn(0);
}

static PetscErrorCode PhysicsRiemann_Shallow_Rusanov(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
{
  ShallowCtx                *phys = (ShallowCtx*)vctx;
  PetscScalar               g = phys->gravity,fL[2],fR[2],s;
  struct {PetscScalar h,u;} L = {uL[0],uL[1]/uL[0]},R = {uR[0],uR[1]/uR[0]};

  PetscFunctionBeginUser;
  if (!(L.h > 0 && R.h > 0)) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Reconstructed thickness is negative");
  ShallowFlux(phys,uL,fL);
  ShallowFlux(phys,uR,fR);
  s         = PetscMax(PetscAbs(L.u)+PetscSqrtScalar(g*L.h),PetscAbs(R.u)+PetscSqrtScalar(g*R.h));
  flux[0]   = 0.5*(fL[0] + fR[0]) + 0.5*s*(uL[0] - uR[0]);
  flux[1]   = 0.5*(fL[1] + fR[1]) + 0.5*s*(uL[1] - uR[1]);
  *maxspeed = s;
  PetscFunctionReturn(0);
}

static PetscErrorCode PhysicsCharacteristic_Shallow(void *vctx,PetscInt m,const PetscScalar *u,PetscScalar *X,PetscScalar *Xi,PetscReal *speeds)
{
  ShallowCtx     *phys = (ShallowCtx*)vctx;
  PetscReal      c;
  PetscErrorCode ierr;

  PetscFunctionBeginUser;
  c         = PetscSqrtScalar(u[0]*phys->gravity);
  speeds[0] = u[1]/u[0] - c;
  speeds[1] = u[1]/u[0] + c;
  X[0*2+0]  = 1;
  X[0*2+1]  = speeds[0];
  X[1*2+0]  = 1;
  X[1*2+1]  = speeds[1];
  ierr = PetscArraycpy(Xi,X,4);CHKERRQ(ierr);
  ierr = PetscKernel_A_gets_inverse_A_2(Xi,0,PETSC_FALSE,NULL);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode PhysicsCreate_Shallow(FVCtx *ctx)
{
  PetscErrorCode    ierr;
  ShallowCtx        *user;
  PetscFunctionList rlist = 0,rclist = 0;
  char              rname[256] = "exact",rcname[256] = "characteristic";

  PetscFunctionBeginUser;
  ierr = PetscNew(&user);CHKERRQ(ierr);
  /* Shallow water and Isothermal Gas dynamics are similar so we reuse initial conditions for now */
  ctx->physics.sample         = PhysicsSample_IsoGas;
  ctx->physics.destroy        = PhysicsDestroy_SimpleFree;
  ctx->physics.user           = user;
  ctx->physics.dof            = 2;

  ierr = PetscStrallocpy("density",&ctx->physics.fieldname[0]);CHKERRQ(ierr);
  ierr = PetscStrallocpy("momentum",&ctx->physics.fieldname[1]);CHKERRQ(ierr);

  user->gravity = 1;

  ierr = RiemannListAdd(&rlist,"exact",  PhysicsRiemann_Shallow_Exact);CHKERRQ(ierr);
  ierr = RiemannListAdd(&rlist,"rusanov",PhysicsRiemann_Shallow_Rusanov);CHKERRQ(ierr);
  ierr = ReconstructListAdd(&rclist,"characteristic",PhysicsCharacteristic_Shallow);CHKERRQ(ierr);
  ierr = ReconstructListAdd(&rclist,"conservative",PhysicsCharacteristic_Conservative);CHKERRQ(ierr);
  ierr = PetscOptionsBegin(ctx->comm,ctx->prefix,"Options for Shallow","");CHKERRQ(ierr);
    ierr = PetscOptionsReal("-physics_shallow_gravity","Gravity","",user->gravity,&user->gravity,NULL);CHKERRQ(ierr);
    ierr = PetscOptionsFList("-physics_shallow_riemann","Riemann solver","",rlist,rname,rname,sizeof(rname),NULL);CHKERRQ(ierr);
    ierr = PetscOptionsFList("-physics_shallow_reconstruct","Reconstruction","",rclist,rcname,rcname,sizeof(rcname),NULL);CHKERRQ(ierr);
  ierr = PetscOptionsEnd();CHKERRQ(ierr);
  ierr = RiemannListFind(rlist,rname,&ctx->physics.riemann);CHKERRQ(ierr);
  ierr = ReconstructListFind(rclist,rcname,&ctx->physics.characteristic);CHKERRQ(ierr);
  ierr = PetscFunctionListDestroy(&rlist);CHKERRQ(ierr);
  ierr = PetscFunctionListDestroy(&rclist);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/* --------------------------------- Finite Volume Solver ----------------------------------- */

static PetscErrorCode FVRHSFunction(TS ts,PetscReal time,Vec X,Vec F,void *vctx)
{
  FVCtx             *ctx = (FVCtx*)vctx;
  PetscErrorCode    ierr;
  PetscInt          i,j,k,Mx,dof,xs,xm;
  PetscReal         hx,cfl_idt = 0;
  PetscScalar       *x,*f,*slope;
  Vec               Xloc;
  DM                da;

  PetscFunctionBeginUser;
  ierr = TSGetDM(ts,&da);CHKERRQ(ierr);
  ierr = DMGetLocalVector(da,&Xloc);CHKERRQ(ierr);
  ierr = DMDAGetInfo(da,0, &Mx,0,0, 0,0,0, &dof,0,0,0,0,0);CHKERRQ(ierr);
  hx   = (ctx->xmax - ctx->xmin)/Mx;
  ierr = DMGlobalToLocalBegin(da,X,INSERT_VALUES,Xloc);CHKERRQ(ierr);
  ierr = DMGlobalToLocalEnd  (da,X,INSERT_VALUES,Xloc);CHKERRQ(ierr);

  ierr = VecZeroEntries(F);CHKERRQ(ierr);

  ierr = DMDAVecGetArray(da,Xloc,&x);CHKERRQ(ierr);
  ierr = DMDAVecGetArray(da,F,&f);CHKERRQ(ierr);
  ierr = DMDAGetArray(da,PETSC_TRUE,&slope);CHKERRQ(ierr);

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

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

  ierr = DMDAVecRestoreArray(da,Xloc,&x);CHKERRQ(ierr);
  ierr = DMDAVecRestoreArray(da,F,&f);CHKERRQ(ierr);
  ierr = DMDARestoreArray(da,PETSC_TRUE,&slope);CHKERRQ(ierr);
  ierr = DMRestoreLocalVector(da,&Xloc);CHKERRQ(ierr);

  ierr = MPI_Allreduce(&cfl_idt,&ctx->cfl_idt,1,MPIU_REAL,MPIU_MAX,PetscObjectComm((PetscObject)da));CHKERRMPI(ierr);
  if (0) {
    /* We need to a way to inform the TS of a CFL constraint, this is a debugging fragment */
    PetscReal dt,tnow;
    ierr = TSGetTimeStep(ts,&dt);CHKERRQ(ierr);
    ierr = TSGetTime(ts,&tnow);CHKERRQ(ierr);
    if (dt > 0.5/ctx->cfl_idt) {
      ierr = PetscPrintf(ctx->comm,"Stability constraint exceeded at t=%g, dt %g > %g\n",(double)tnow,(double)dt,(double)(0.5/ctx->cfl_idt));CHKERRQ(ierr);
    }
  }
  PetscFunctionReturn(0);
}

static PetscErrorCode SmallMatMultADB(PetscScalar *C,PetscInt bs,const PetscScalar *A,const PetscReal *D,const PetscScalar *B)
{
  PetscInt i,j,k;

  PetscFunctionBeginUser;
  for (i=0; i<bs; i++) {
    for (j=0; j<bs; j++) {
      PetscScalar tmp = 0;
      for (k=0; k<bs; k++) tmp += A[i*bs+k] * D[k] * B[k*bs+j];
      C[i*bs+j] = tmp;
    }
  }
  PetscFunctionReturn(0);
}

static PetscErrorCode FVIJacobian(TS ts,PetscReal t,Vec X,Vec Xdot,PetscReal shift,Mat A,Mat B,void *vctx)
{
  FVCtx             *ctx = (FVCtx*)vctx;
  PetscErrorCode    ierr;
  PetscInt          i,j,dof = ctx->physics.dof;
  PetscScalar       *J;
  const PetscScalar *x;
  PetscReal         hx;
  DM                da;
  DMDALocalInfo     dainfo;

  PetscFunctionBeginUser;
  ierr = TSGetDM(ts,&da);CHKERRQ(ierr);
  ierr = DMDAVecGetArrayRead(da,X,(void*)&x);CHKERRQ(ierr);
  ierr = DMDAGetLocalInfo(da,&dainfo);CHKERRQ(ierr);
  hx   = (ctx->xmax - ctx->xmin)/dainfo.mx;
  ierr = PetscMalloc1(dof*dof,&J);CHKERRQ(ierr);
  for (i=dainfo.xs; i<dainfo.xs+dainfo.xm; i++) {
    ierr = (*ctx->physics.characteristic)(ctx->physics.user,dof,&x[i*dof],ctx->R,ctx->Rinv,ctx->speeds);CHKERRQ(ierr);
    for (j=0; j<dof; j++) ctx->speeds[j] = PetscAbs(ctx->speeds[j]);
    ierr = SmallMatMultADB(J,dof,ctx->R,ctx->speeds,ctx->Rinv);CHKERRQ(ierr);
    for (j=0; j<dof*dof; j++) J[j] = J[j]/hx + shift*(j/dof == j%dof);
    ierr = MatSetValuesBlocked(B,1,&i,1,&i,J,INSERT_VALUES);CHKERRQ(ierr);
  }
  ierr = PetscFree(J);CHKERRQ(ierr);
  ierr = DMDAVecRestoreArrayRead(da,X,(void*)&x);CHKERRQ(ierr);

  ierr = MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  if (A != B) {
    ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
    ierr = MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  }
  PetscFunctionReturn(0);
}

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

  PetscFunctionBeginUser;
  if (!ctx->physics.sample) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Physics has not provided a sampling function");
  ierr = DMDAGetInfo(da,0, &Mx,0,0, 0,0,0, &dof,0,0,0,0,0);CHKERRQ(ierr);
  ierr = DMDAGetCorners(da,&xs,0,0,&xm,0,0);CHKERRQ(ierr);
  ierr = DMDAVecGetArray(da,U,&u);CHKERRQ(ierr);
  ierr = PetscMalloc1(dof,&uj);CHKERRQ(ierr);
  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;
      ierr = (*ctx->physics.sample)(ctx->physics.user,ctx->initial,ctx->bctype,ctx->xmin,ctx->xmax,time,xj,uj);CHKERRQ(ierr);
      for (k=0; k<dof; k++) u[i*dof+k] += ((j==0 || j==N) ? 0.5 : 1.0)*uj[k]/N;
    }
  }
  ierr = DMDAVecRestoreArray(da,U,&u);CHKERRQ(ierr);
  ierr = PetscFree(uj);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

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

  PetscFunctionBeginUser;
  ierr = PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&iascii);CHKERRQ(ierr);
  if (iascii) {
    /* PETSc lacks a function to compute total variation norm (difficult in multiple dimensions), we do it here */
    ierr  = DMGetLocalVector(da,&Xloc);CHKERRQ(ierr);
    ierr  = DMGlobalToLocalBegin(da,X,INSERT_VALUES,Xloc);CHKERRQ(ierr);
    ierr  = DMGlobalToLocalEnd  (da,X,INSERT_VALUES,Xloc);CHKERRQ(ierr);
    ierr  = DMDAVecGetArrayRead(da,Xloc,(void*)&x);CHKERRQ(ierr);
    ierr  = DMDAGetCorners(da,&xs,0,0,&xm,0,0);CHKERRQ(ierr);
    ierr  = DMDAGetInfo(da,0, &Mx,0,0, 0,0,0, &dof,0,0,0,0,0);CHKERRQ(ierr);
    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]);
    }
    ierr = MPI_Allreduce(&tvsum,&tvgsum,1,MPIU_REAL,MPIU_SUM,PetscObjectComm((PetscObject)da));CHKERRMPI(ierr);
    ierr = DMDAVecRestoreArrayRead(da,Xloc,(void*)&x);CHKERRQ(ierr);
    ierr = DMRestoreLocalVector(da,&Xloc);CHKERRQ(ierr);

    ierr = VecMin(X,&imin,&xmin);CHKERRQ(ierr);
    ierr = VecMax(X,&imax,&xmax);CHKERRQ(ierr);
    ierr = VecSum(X,&sum);CHKERRQ(ierr);
    ierr = PetscViewerASCIIPrintf(viewer,"Solution range [%8.5f,%8.5f] with extrema at %D and %D, mean %8.5f, ||x||_TV %8.5f\n",(double)xmin,(double)xmax,imin,imax,(double)(sum/Mx),(double)(tvgsum/Mx));CHKERRQ(ierr);
  } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Viewer type not supported");
  PetscFunctionReturn(0);
}

static PetscErrorCode SolutionErrorNorms(FVCtx *ctx,DM da,PetscReal t,Vec X,PetscReal *nrm1,PetscReal *nrmsup)
{
  PetscErrorCode ierr;
  Vec            Y;
  PetscInt       Mx;

  PetscFunctionBeginUser;
  ierr   = VecGetSize(X,&Mx);CHKERRQ(ierr);
  ierr   = VecDuplicate(X,&Y);CHKERRQ(ierr);
  ierr   = FVSample(ctx,da,t,Y);CHKERRQ(ierr);
  ierr   = VecAYPX(Y,-1,X);CHKERRQ(ierr);
  ierr   = VecNorm(Y,NORM_1,nrm1);CHKERRQ(ierr);
  ierr   = VecNorm(Y,NORM_INFINITY,nrmsup);CHKERRQ(ierr);
  *nrm1 /= Mx;
  ierr   = VecDestroy(&Y);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

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

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

  /* Register limiters to be available on the command line */
  ierr = PetscFunctionListAdd(&limiters,"upwind"              ,Limit_Upwind);CHKERRQ(ierr);
  ierr = PetscFunctionListAdd(&limiters,"lax-wendroff"        ,Limit_LaxWendroff);CHKERRQ(ierr);
  ierr = PetscFunctionListAdd(&limiters,"beam-warming"        ,Limit_BeamWarming);CHKERRQ(ierr);
  ierr = PetscFunctionListAdd(&limiters,"fromm"               ,Limit_Fromm);CHKERRQ(ierr);
  ierr = PetscFunctionListAdd(&limiters,"minmod"              ,Limit_Minmod);CHKERRQ(ierr);
  ierr = PetscFunctionListAdd(&limiters,"superbee"            ,Limit_Superbee);CHKERRQ(ierr);
  ierr = PetscFunctionListAdd(&limiters,"mc"                  ,Limit_MC);CHKERRQ(ierr);
  ierr = PetscFunctionListAdd(&limiters,"vanleer"             ,Limit_VanLeer);CHKERRQ(ierr);
  ierr = PetscFunctionListAdd(&limiters,"vanalbada"           ,Limit_VanAlbada);CHKERRQ(ierr);
  ierr = PetscFunctionListAdd(&limiters,"vanalbadatvd"        ,Limit_VanAlbadaTVD);CHKERRQ(ierr);
  ierr = PetscFunctionListAdd(&limiters,"koren"               ,Limit_Koren);CHKERRQ(ierr);
  ierr = PetscFunctionListAdd(&limiters,"korensym"            ,Limit_KorenSym);CHKERRQ(ierr);
  ierr = PetscFunctionListAdd(&limiters,"koren3"              ,Limit_Koren3);CHKERRQ(ierr);
  ierr = PetscFunctionListAdd(&limiters,"cada-torrilhon2"     ,Limit_CadaTorrilhon2);CHKERRQ(ierr);
  ierr = PetscFunctionListAdd(&limiters,"cada-torrilhon3-r0p1",Limit_CadaTorrilhon3R0p1);CHKERRQ(ierr);
  ierr = PetscFunctionListAdd(&limiters,"cada-torrilhon3-r1"  ,Limit_CadaTorrilhon3R1);CHKERRQ(ierr);
  ierr = PetscFunctionListAdd(&limiters,"cada-torrilhon3-r10" ,Limit_CadaTorrilhon3R10);CHKERRQ(ierr);
  ierr = PetscFunctionListAdd(&limiters,"cada-torrilhon3-r100",Limit_CadaTorrilhon3R100);CHKERRQ(ierr);

  /* Register physical models to be available on the command line */
  ierr = PetscFunctionListAdd(&physics,"advect"          ,PhysicsCreate_Advect);CHKERRQ(ierr);
  ierr = PetscFunctionListAdd(&physics,"burgers"         ,PhysicsCreate_Burgers);CHKERRQ(ierr);
  ierr = PetscFunctionListAdd(&physics,"traffic"         ,PhysicsCreate_Traffic);CHKERRQ(ierr);
  ierr = PetscFunctionListAdd(&physics,"acoustics"       ,PhysicsCreate_Acoustics);CHKERRQ(ierr);
  ierr = PetscFunctionListAdd(&physics,"isogas"          ,PhysicsCreate_IsoGas);CHKERRQ(ierr);
  ierr = PetscFunctionListAdd(&physics,"shallow"         ,PhysicsCreate_Shallow);CHKERRQ(ierr);

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

  /* Choose the limiter from the list of registered limiters */
  ierr = PetscFunctionListFind(limiters,lname,&ctx.limit);CHKERRQ(ierr);
  if (!ctx.limit) SETERRQ1(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*);
    ierr = PetscFunctionListFind(physics,physname,&r);CHKERRQ(ierr);
    if (!r) SETERRQ1(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 */
    ierr = (*r)(&ctx);CHKERRQ(ierr);
  }

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

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

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

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

  ierr = DMCreateMatrix(da,&B);CHKERRQ(ierr);

  /* Create a time-stepping object */
  ierr = TSCreate(comm,&ts);CHKERRQ(ierr);
  ierr = TSSetDM(ts,da);CHKERRQ(ierr);
  ierr = TSSetRHSFunction(ts,R,FVRHSFunction,&ctx);CHKERRQ(ierr);
  ierr = TSSetIJacobian(ts,B,B,FVIJacobian,&ctx);CHKERRQ(ierr);
  ierr = TSSetType(ts,TSSSP);CHKERRQ(ierr);
  ierr = TSSetMaxTime(ts,10);CHKERRQ(ierr);
  ierr = TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);CHKERRQ(ierr);

  /* Compute initial conditions and starting time step */
  ierr = FVSample(&ctx,da,0,X0);CHKERRQ(ierr);
  ierr = FVRHSFunction(ts,0,X0,X,(void*)&ctx);CHKERRQ(ierr); /* Initial function evaluation, only used to determine max speed */
  ierr = VecCopy(X0,X);CHKERRQ(ierr);                        /* The function value was not used so we set X=X0 again */
  ierr = TSSetTimeStep(ts,ctx.cfl/ctx.cfl_idt);CHKERRQ(ierr);
  ierr = TSSetFromOptions(ts);CHKERRQ(ierr); /* Take runtime options */
  ierr = SolutionStatsView(da,X,PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr);
  {
    PetscReal nrm1,nrmsup;
    PetscInt  steps;

    ierr = TSSolve(ts,X);CHKERRQ(ierr);
    ierr = TSGetSolveTime(ts,&ptime);CHKERRQ(ierr);
    ierr = TSGetStepNumber(ts,&steps);CHKERRQ(ierr);

    ierr = PetscPrintf(comm,"Final time %8.5f, steps %D\n",(double)ptime,steps);CHKERRQ(ierr);
    if (ctx.exact) {
      ierr = SolutionErrorNorms(&ctx,da,ptime,X,&nrm1,&nrmsup);CHKERRQ(ierr);
      ierr = PetscPrintf(comm,"Error ||x-x_e||_1 %8.4e  ||x-x_e||_sup %8.4e\n",(double)nrm1,(double)nrmsup);CHKERRQ(ierr);
    }
  }

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

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

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

/*TEST

    build:
      requires: !complex

    test:
      args: -da_grid_x 100 -initial 1 -xmin -2 -xmax 5 -exact -limit mc
      requires: !complex !single

    test:
      suffix: 2
      args: -da_grid_x 100 -initial 2 -xmin -2 -xmax 2 -exact -limit mc -physics burgers -bc_type outflow -ts_max_time 1
      filter:  sed "s/at 48/at 0/g"
      requires: !complex !single

    test:
      suffix: 3
      args: -da_grid_x 100 -initial 2 -xmin -2 -xmax 2 -exact -limit mc -physics burgers -bc_type outflow -ts_max_time 1
      nsize: 3
      filter:  sed "s/at 48/at 0/g"
      requires: !complex !single

TEST*/