Actual source code: ex9.c

petsc-dev 2014-02-02
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  1: static const char help[] = "1D periodic Finite Volume solver in slope-limiter form with semidiscrete time stepping.\n"
  2:   "Solves scalar and vector problems, choose the physical model with -physics\n"
  3:   "  advection   - Constant coefficient scalar advection\n"
  4:   "                u_t       + (a*u)_x               = 0\n"
  5:   "  burgers     - Burgers equation\n"
  6:   "                u_t       + (u^2/2)_x             = 0\n"
  7:   "  traffic     - Traffic equation\n"
  8:   "                u_t       + (u*(1-u))_x           = 0\n"
  9:   "  acoustics   - Acoustic wave propagation\n"
 10:   "                u_t       + (c*z*v)_x             = 0\n"
 11:   "                v_t       + (c/z*u)_x             = 0\n"
 12:   "  isogas      - Isothermal gas dynamics\n"
 13:   "                rho_t     + (rho*u)_x             = 0\n"
 14:   "                (rho*u)_t + (rho*u^2 + c^2*rho)_x = 0\n"
 15:   "  shallow     - Shallow water equations\n"
 16:   "                h_t       + (h*u)_x               = 0\n"
 17:   "                (h*u)_t   + (h*u^2 + g*h^2/2)_x   = 0\n"
 18:   "Some of these physical models have multiple Riemann solvers, select these with -physics_xxx_riemann\n"
 19:   "  exact       - Exact Riemann solver which usually needs to perform a Newton iteration to connect\n"
 20:   "                the states across shocks and rarefactions\n"
 21:   "  roe         - Linearized scheme, usually with an entropy fix inside sonic rarefactions\n"
 22:   "The systems provide a choice of reconstructions with -physics_xxx_reconstruct\n"
 23:   "  characteristic - Limit the characteristic variables, this is usually preferred (default)\n"
 24:   "  conservative   - Limit the conservative variables directly, can cause undesired interaction of waves\n\n"
 25:   "A variety of limiters for high-resolution TVD limiters are available with -limit\n"
 26:   "  upwind,minmod,superbee,mc,vanleer,vanalbada,koren,cada-torillhon (last two are nominally third order)\n"
 27:   "  and non-TVD schemes lax-wendroff,beam-warming,fromm\n\n"
 28:   "To preserve the TVD property, one should time step with a strong stability preserving method.\n"
 29:   "The optimal high order explicit Runge-Kutta methods in TSSSP are recommended for non-stiff problems.\n\n"
 30:   "Several initial conditions can be chosen with -initial N\n\n"
 31:   "The problem size should be set with -da_grid_x M\n\n";

 33: #include <petscts.h>
 34: #include <petscdmda.h>
 35: #include <petscdraw.h>

 37: #include <petsc-private/kernels/blockinvert.h> /* For the Kernel_*_gets_* stuff for BAIJ */

 39: PETSC_STATIC_INLINE PetscReal Sgn(PetscReal a) { return (a<0) ? -1 : 1; }
 40: PETSC_STATIC_INLINE PetscReal Abs(PetscReal a) { return (a<0) ? 0 : a; }
 41: PETSC_STATIC_INLINE PetscReal Sqr(PetscReal a) { return a*a; }
 42: PETSC_STATIC_INLINE PetscReal MaxAbs(PetscReal a,PetscReal b) { return (PetscAbs(a) > PetscAbs(b)) ? a : b; }
 43: PETSC_STATIC_INLINE PetscReal MinAbs(PetscReal a,PetscReal b) { return (PetscAbs(a) < PetscAbs(b)) ? a : b; }
 44: PETSC_STATIC_INLINE PetscReal MinMod2(PetscReal a,PetscReal b) { return (a*b<0) ? 0 : Sgn(a)*PetscMin(PetscAbs(a),PetscAbs(b)); }
 45: PETSC_STATIC_INLINE PetscReal MaxMod2(PetscReal a,PetscReal b) { return (a*b<0) ? 0 : Sgn(a)*PetscMax(PetscAbs(a),PetscAbs(b)); }
 46: 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))); }

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


 51: /* ----------------------- Lots of limiters, these could go in a separate library ------------------------- */
 52: typedef struct _LimitInfo {
 53:   PetscReal hx;
 54:   PetscInt  m;
 55: } *LimitInfo;
 56: static void Limit_Upwind(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
 57: {
 58:   PetscInt i;
 59:   for (i=0; i<info->m; i++) lmt[i] = 0;
 60: }
 61: static void Limit_LaxWendroff(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
 62: {
 63:   PetscInt i;
 64:   for (i=0; i<info->m; i++) lmt[i] = jR[i];
 65: }
 66: static void Limit_BeamWarming(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
 67: {
 68:   PetscInt i;
 69:   for (i=0; i<info->m; i++) lmt[i] = jL[i];
 70: }
 71: static void Limit_Fromm(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
 72: {
 73:   PetscInt i;
 74:   for (i=0; i<info->m; i++) lmt[i] = 0.5*(jL[i] + jR[i]);
 75: }
 76: static void Limit_Minmod(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
 77: {
 78:   PetscInt i;
 79:   for (i=0; i<info->m; i++) lmt[i] = MinMod2(jL[i],jR[i]);
 80: }
 81: static void Limit_Superbee(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
 82: {
 83:   PetscInt i;
 84:   for (i=0; i<info->m; i++) lmt[i] = MaxMod2(MinMod2(jL[i],2*jR[i]),MinMod2(2*jL[i],jR[i]));
 85: }
 86: static void Limit_MC(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
 87: {
 88:   PetscInt i;
 89:   for (i=0; i<info->m; i++) lmt[i] = MinMod3(2*jL[i],0.5*(jL[i]+jR[i]),2*jR[i]);
 90: }
 91: static void Limit_VanLeer(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
 92: { /* phi = (t + abs(t)) / (1 + abs(t)) */
 93:   PetscInt i;
 94:   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);
 95: }
 96: static void Limit_VanAlbada(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt) /* differentiable */
 97: { /* phi = (t + t^2) / (1 + t^2) */
 98:   PetscInt i;
 99:   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);
100: }
101: static void Limit_VanAlbadaTVD(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
102: { /* phi = (t + t^2) / (1 + t^2) */
103:   PetscInt i;
104:   for (i=0; i<info->m; i++) lmt[i] = (jL[i]*jR[i]<0) ? 0
105:                                      : (jL[i]*Sqr(jR[i]) + Sqr(jL[i])*jR[i]) / (Sqr(jL[i]) + Sqr(jR[i]) + 1e-15);
106: }
107: static void Limit_Koren(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt) /* differentiable */
108: { /* phi = (t + 2*t^2) / (2 - t + 2*t^2) */
109:   PetscInt i;
110:   for (i=0; i<info->m; i++) lmt[i] = ((jL[i]*Sqr(jR[i]) + 2*Sqr(jL[i])*jR[i])
111:                                       / (2*Sqr(jL[i]) - jL[i]*jR[i] + 2*Sqr(jR[i]) + 1e-15));
112: }
113: static void Limit_KorenSym(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt) /* differentiable */
114: { /* Symmetric version of above */
115:   PetscInt i;
116:   for (i=0; i<info->m; i++) lmt[i] = (1.5*(jL[i]*Sqr(jR[i]) + Sqr(jL[i])*jR[i])
117:                                       / (2*Sqr(jL[i]) - jL[i]*jR[i] + 2*Sqr(jR[i]) + 1e-15));
118: }
119: static void Limit_Koren3(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
120: { /* Eq 11 of Cada-Torrilhon 2009 */
121:   PetscInt i;
122:   for (i=0; i<info->m; i++) lmt[i] = MinMod3(2*jL[i],(jL[i]+2*jR[i])/3,2*jR[i]);
123: }

125: static PetscReal CadaTorrilhonPhiHatR_Eq13(PetscReal L,PetscReal R)
126: {
127:   return PetscMax(0,PetscMin((L+2*R)/3,
128:                              PetscMax(-0.5*L,
129:                                       PetscMin(2*L,
130:                                                PetscMin((L+2*R)/3,1.6*R)))));
131: }
132: static void Limit_CadaTorrilhon2(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
133: { /* Cada-Torrilhon 2009, Eq 13 */
134:   PetscInt i;
135:   for (i=0; i<info->m; i++) lmt[i] = CadaTorrilhonPhiHatR_Eq13(jL[i],jR[i]);
136: }
137: static void Limit_CadaTorrilhon3R(PetscReal r,LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
138: { /* Cada-Torrilhon 2009, Eq 22 */
139:   /* They recommend 0.001 < r < 1, but larger values are more accurate in smooth regions */
140:   const PetscReal eps = 1e-7,hx = info->hx;
141:   PetscInt i;
142:   for (i=0; i<info->m; i++) {
143:     const PetscReal eta = (Sqr(jL[i]) + Sqr(jR[i])) / Sqr(r*hx);
144:     lmt[i] = ((eta < 1-eps)
145:               ? (jL[i] + 2*jR[i]) / 3
146:               : ((eta > 1+eps)
147:                  ? CadaTorrilhonPhiHatR_Eq13(jL[i],jR[i])
148:                  : 0.5*((1-(eta-1)/eps)*(jL[i]+2*jR[i])/3
149:                         + (1+(eta+1)/eps)*CadaTorrilhonPhiHatR_Eq13(jL[i],jR[i]))));
150:   }
151: }
152: static void Limit_CadaTorrilhon3R0p1(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
153: { Limit_CadaTorrilhon3R(0.1,info,jL,jR,lmt); }
154: static void Limit_CadaTorrilhon3R1(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
155: { Limit_CadaTorrilhon3R(1,info,jL,jR,lmt); }
156: static void Limit_CadaTorrilhon3R10(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
157: { Limit_CadaTorrilhon3R(10,info,jL,jR,lmt); }
158: static void Limit_CadaTorrilhon3R100(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
159: { Limit_CadaTorrilhon3R(100,info,jL,jR,lmt); }


162: /* --------------------------------- Finite Volume data structures ----------------------------------- */

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

169: typedef struct {
170:   PetscErrorCode (*sample)(void*,PetscInt,FVBCType,PetscReal,PetscReal,PetscReal,PetscReal,PetscReal*);
171:   RiemannFunction     riemann;
172:   ReconstructFunction characteristic;
173:   PetscErrorCode (*destroy)(void*);
174:   void     *user;
175:   PetscInt dof;
176:   char     *fieldname[16];
177: } PhysicsCtx;

179: typedef struct {
180:   void (*limit)(LimitInfo,const PetscScalar*,const PetscScalar*,PetscScalar*);
181:   PhysicsCtx physics;

183:   MPI_Comm comm;
184:   char     prefix[256];

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

194:   PetscReal cfl_idt;            /* Max allowable value of 1/Delta t */
195:   PetscReal cfl;
196:   PetscReal xmin,xmax;
197:   PetscInt  initial;
198:   PetscBool exact;
199:   FVBCType  bctype;
200: } FVCtx;


203: /* Utility */

207: PetscErrorCode RiemannListAdd(PetscFunctionList *flist,const char *name,RiemannFunction rsolve)
208: {

212:   PetscFunctionListAdd(flist,name,rsolve);
213:   return(0);
214: }

218: PetscErrorCode RiemannListFind(PetscFunctionList flist,const char *name,RiemannFunction *rsolve)
219: {

223:   PetscFunctionListFind(flist,name,rsolve);
224:   if (!*rsolve) SETERRQ1(PETSC_COMM_SELF,1,"Riemann solver \"%s\" could not be found",name);
225:   return(0);
226: }

230: PetscErrorCode ReconstructListAdd(PetscFunctionList *flist,const char *name,ReconstructFunction r)
231: {

235:   PetscFunctionListAdd(flist,name,r);
236:   return(0);
237: }

241: PetscErrorCode ReconstructListFind(PetscFunctionList flist,const char *name,ReconstructFunction *r)
242: {

246:   PetscFunctionListFind(flist,name,r);
247:   if (!*r) SETERRQ1(PETSC_COMM_SELF,1,"Reconstruction \"%s\" could not be found",name);
248:   return(0);
249: }


252: /* --------------------------------- Physics ----------------------------------- */
253: /**
254: * Each physical model consists of Riemann solver and a function to determine the basis to use for reconstruction.  These
255: * are set with the PhysicsCreate_XXX function which allocates private storage and sets these methods as well as the
256: * number of fields and their names, and a function to deallocate private storage.
257: **/

259: /* First a few functions useful to several different physics */
262: static PetscErrorCode PhysicsCharacteristic_Conservative(void *vctx,PetscInt m,const PetscScalar *u,PetscScalar *X,PetscScalar *Xi,PetscReal *speeds)
263: {
264:   PetscInt i,j;

267:   for (i=0; i<m; i++) {
268:     for (j=0; j<m; j++) Xi[i*m+j] = X[i*m+j] = (PetscScalar)(i==j);
269:     speeds[i] = PETSC_MAX_REAL; /* Indicates invalid */
270:   }
271:   return(0);
272: }

276: static PetscErrorCode PhysicsDestroy_SimpleFree(void *vctx)
277: {

281:   PetscFree(vctx);
282:   return(0);
283: }



287: /* --------------------------------- Advection ----------------------------------- */

289: typedef struct {
290:   PetscReal a;                  /* advective velocity */
291: } AdvectCtx;

295: static PetscErrorCode PhysicsRiemann_Advect(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
296: {
297:   AdvectCtx *ctx = (AdvectCtx*)vctx;
298:   PetscReal speed;

301:   speed     = ctx->a;
302:   flux[0]   = PetscMax(0,speed)*uL[0] + PetscMin(0,speed)*uR[0];
303:   *maxspeed = speed;
304:   return(0);
305: }

309: static PetscErrorCode PhysicsCharacteristic_Advect(void *vctx,PetscInt m,const PetscScalar *u,PetscScalar *X,PetscScalar *Xi,PetscReal *speeds)
310: {
311:   AdvectCtx *ctx = (AdvectCtx*)vctx;

314:   X[0]      = 1.;
315:   Xi[0]     = 1.;
316:   speeds[0] = ctx->a;
317:   return(0);
318: }

322: static PetscErrorCode PhysicsSample_Advect(void *vctx,PetscInt initial,FVBCType bctype,PetscReal xmin,PetscReal xmax,PetscReal t,PetscReal x,PetscReal *u)
323: {
324:   AdvectCtx *ctx = (AdvectCtx*)vctx;
325:   PetscReal a    = ctx->a,x0;

328:   switch (bctype) {
329:     case FVBC_OUTFLOW:   x0 = x-a*t; break;
330:     case FVBC_PERIODIC: x0 = RangeMod(x-a*t,xmin,xmax); break;
331:     default: SETERRQ(PETSC_COMM_SELF,1,"unknown BCType");
332:   }
333:   switch (initial) {
334:     case 0: u[0] = (x0 < 0) ? 1 : -1; break;
335:     case 1: u[0] = (x0 < 0) ? -1 : 1; break;
336:     case 2: u[0] = (0 < x0 && x0 < 1) ? 1 : 0; break;
337:     case 3: u[0] = PetscSinReal(2*PETSC_PI*x0); break;
338:     case 4: u[0] = PetscAbs(x0); break;
339:     case 5: u[0] = (x0 < 0 || x0 > 0.5) ? 0 : PetscSqr(PetscSinReal(2*PETSC_PI*x0)); break;
340:     case 6: u[0] = (x0 < 0) ? 0 : ((x0 < 1) ? x0 : ((x0 < 2) ? 2-x0 : 0)); break;
341:     default: SETERRQ(PETSC_COMM_SELF,1,"unknown initial condition");
342:   }
343:   return(0);
344: }

348: static PetscErrorCode PhysicsCreate_Advect(FVCtx *ctx)
349: {
351:   AdvectCtx      *user;

354:   PetscNew(&user);
355:   ctx->physics.sample         = PhysicsSample_Advect;
356:   ctx->physics.riemann        = PhysicsRiemann_Advect;
357:   ctx->physics.characteristic = PhysicsCharacteristic_Advect;
358:   ctx->physics.destroy        = PhysicsDestroy_SimpleFree;
359:   ctx->physics.user           = user;
360:   ctx->physics.dof            = 1;
361:   PetscStrallocpy("u",&ctx->physics.fieldname[0]);
362:   user->a = 1;
363:   PetscOptionsBegin(ctx->comm,ctx->prefix,"Options for advection","");
364:   {
365:     PetscOptionsReal("-physics_advect_a","Speed","",user->a,&user->a,NULL);
366:   }
367:   PetscOptionsEnd();
368:   return(0);
369: }



373: /* --------------------------------- Burgers ----------------------------------- */

375: typedef struct {
376:   PetscReal lxf_speed;
377: } BurgersCtx;

381: static PetscErrorCode PhysicsSample_Burgers(void *vctx,PetscInt initial,FVBCType bctype,PetscReal xmin,PetscReal xmax,PetscReal t,PetscReal x,PetscReal *u)
382: {

385:   if (bctype == FVBC_PERIODIC && t > 0) SETERRQ(PETSC_COMM_SELF,1,"Exact solution not implemented for periodic");
386:   switch (initial) {
387:     case 0: u[0] = (x < 0) ? 1 : -1; break;
388:     case 1:
389:       if       (x < -t) u[0] = -1;
390:       else if  (x < t)  u[0] = x/t;
391:       else              u[0] = 1;
392:       break;
393:     case 2:
394:       if      (x < 0)       u[0] = 0;
395:       else if (x <= t)      u[0] = x/t;
396:       else if (x < 1+0.5*t) u[0] = 1;
397:       else                  u[0] = 0;
398:       break;
399:     case 3:
400:       if       (x < 0.2*t) u[0] = 0.2;
401:       else if  (x < t) u[0] = x/t;
402:       else             u[0] = 1;
403:       break;
404:     case 4:
405:       if (t > 0) SETERRQ(PETSC_COMM_SELF,1,"Only initial condition available");
406:       u[0] = 0.7 + 0.3*PetscSinReal(2*PETSC_PI*((x-xmin)/(xmax-xmin)));
407:       break;
408:     case 5:                     /* Pure shock solution */
409:       if (x < 0.5*t) u[0] = 1;
410:       else u[0] = 0;
411:       break;
412:     default: SETERRQ(PETSC_COMM_SELF,1,"unknown initial condition");
413:   }
414:   return(0);
415: }

419: static PetscErrorCode PhysicsRiemann_Burgers_Exact(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
420: {

423:   if (uL[0] < uR[0]) {          /* rarefaction */
424:     flux[0] = (uL[0]*uR[0] < 0)
425:       ? 0                       /* sonic rarefaction */
426:       : 0.5*PetscMin(PetscSqr(uL[0]),PetscSqr(uR[0]));
427:   } else {                      /* shock */
428:     flux[0] = 0.5*PetscMax(PetscSqr(uL[0]),PetscSqr(uR[0]));
429:   }
430:   *maxspeed = (PetscAbs(uL[0]) > PetscAbs(uR[0])) ? uL[0] : uR[0];
431:   return(0);
432: }

436: static PetscErrorCode PhysicsRiemann_Burgers_Roe(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
437: {
438:   PetscReal speed;

441:   speed   = 0.5*(uL[0] + uR[0]);
442:   flux[0] = 0.25*(PetscSqr(uL[0]) + PetscSqr(uR[0])) - 0.5*PetscAbs(speed)*(uR[0]-uL[0]);
443:   if (uL[0] <= 0 && 0 <= uR[0]) flux[0] = 0; /* Entropy fix for sonic rarefaction */
444:   *maxspeed = speed;
445:   return(0);
446: }

450: static PetscErrorCode PhysicsRiemann_Burgers_LxF(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
451: {
452:   PetscReal   c;
453:   PetscScalar fL,fR;

456:   c         = ((BurgersCtx*)vctx)->lxf_speed;
457:   fL        = 0.5*PetscSqr(uL[0]);
458:   fR        = 0.5*PetscSqr(uR[0]);
459:   flux[0]   = 0.5*(fL + fR) - 0.5*c*(uR[0] - uL[0]);
460:   *maxspeed = c;
461:   return(0);
462: }

466: static PetscErrorCode PhysicsRiemann_Burgers_Rusanov(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
467: {
468:   PetscReal   c;
469:   PetscScalar fL,fR;

472:   c         = PetscMax(PetscAbs(uL[0]),PetscAbs(uR[0]));
473:   fL        = 0.5*PetscSqr(uL[0]);
474:   fR        = 0.5*PetscSqr(uR[0]);
475:   flux[0]   = 0.5*(fL + fR) - 0.5*c*(uR[0] - uL[0]);
476:   *maxspeed = c;
477:   return(0);
478: }

482: static PetscErrorCode PhysicsCreate_Burgers(FVCtx *ctx)
483: {
484:   BurgersCtx        *user;
485:   PetscErrorCode    ierr;
486:   RiemannFunction   r;
487:   PetscFunctionList rlist      = 0;
488:   char              rname[256] = "exact";

491:   PetscNew(&user);

493:   ctx->physics.sample         = PhysicsSample_Burgers;
494:   ctx->physics.characteristic = PhysicsCharacteristic_Conservative;
495:   ctx->physics.destroy        = PhysicsDestroy_SimpleFree;
496:   ctx->physics.user           = user;
497:   ctx->physics.dof            = 1;

499:   PetscStrallocpy("u",&ctx->physics.fieldname[0]);
500:   RiemannListAdd(&rlist,"exact",  PhysicsRiemann_Burgers_Exact);
501:   RiemannListAdd(&rlist,"roe",    PhysicsRiemann_Burgers_Roe);
502:   RiemannListAdd(&rlist,"lxf",    PhysicsRiemann_Burgers_LxF);
503:   RiemannListAdd(&rlist,"rusanov",PhysicsRiemann_Burgers_Rusanov);
504:   PetscOptionsBegin(ctx->comm,ctx->prefix,"Options for advection","");
505:   {
506:     PetscOptionsFList("-physics_burgers_riemann","Riemann solver","",rlist,rname,rname,sizeof(rname),NULL);
507:   }
508:   PetscOptionsEnd();
509:   RiemannListFind(rlist,rname,&r);
510:   PetscFunctionListDestroy(&rlist);
511:   ctx->physics.riemann = r;

513:   /* *
514:   * Hack to deal with LxF in semi-discrete form
515:   * max speed is 1 for the basic initial conditions (where |u| <= 1)
516:   * */
517:   if (r == PhysicsRiemann_Burgers_LxF) user->lxf_speed = 1;
518:   return(0);
519: }



523: /* --------------------------------- Traffic ----------------------------------- */

525: typedef struct {
526:   PetscReal lxf_speed;
527:   PetscReal a;
528: } TrafficCtx;

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

534: static PetscErrorCode PhysicsSample_Traffic(void *vctx,PetscInt initial,FVBCType bctype,PetscReal xmin,PetscReal xmax,PetscReal t,PetscReal x,PetscReal *u)
535: {
536:   PetscReal a = ((TrafficCtx*)vctx)->a;

539:   if (bctype == FVBC_PERIODIC && t > 0) SETERRQ(PETSC_COMM_SELF,1,"Exact solution not implemented for periodic");
540:   switch (initial) {
541:     case 0:
542:       u[0] = (-a*t < x) ? 2 : 0; break;
543:     case 1:
544:       if      (x < PetscMin(2*a*t,0.5+a*t)) u[0] = -1;
545:       else if (x < 1)                       u[0] = 0;
546:       else                                  u[0] = 1;
547:       break;
548:     case 2:
549:       if (t > 0) SETERRQ(PETSC_COMM_SELF,1,"Only initial condition available");
550:       u[0] = 0.7 + 0.3*PetscSinReal(2*PETSC_PI*((x-xmin)/(xmax-xmin)));
551:       break;
552:     default: SETERRQ(PETSC_COMM_SELF,1,"unknown initial condition");
553:   }
554:   return(0);
555: }

559: static PetscErrorCode PhysicsRiemann_Traffic_Exact(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
560: {
561:   PetscReal a = ((TrafficCtx*)vctx)->a;

564:   if (uL[0] < uR[0]) {
565:     flux[0] = PetscMin(TrafficFlux(a,uL[0]),
566:                        TrafficFlux(a,uR[0]));
567:   } else {
568:     flux[0] = (uR[0] < 0.5 && 0.5 < uL[0])
569:       ? TrafficFlux(a,0.5)
570:       : PetscMax(TrafficFlux(a,uL[0]),
571:                  TrafficFlux(a,uR[0]));
572:   }
573:   *maxspeed = a*MaxAbs(1-2*uL[0],1-2*uR[0]);
574:   return(0);
575: }

579: static PetscErrorCode PhysicsRiemann_Traffic_Roe(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
580: {
581:   PetscReal a = ((TrafficCtx*)vctx)->a;
582:   PetscReal speed;

585:   speed = a*(1 - (uL[0] + uR[0]));
586:   flux[0] = 0.5*(TrafficFlux(a,uL[0]) + TrafficFlux(a,uR[0])) - 0.5*PetscAbs(speed)*(uR[0]-uL[0]);
587:   *maxspeed = speed;
588:   return(0);
589: }

593: static PetscErrorCode PhysicsRiemann_Traffic_LxF(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
594: {
595:   TrafficCtx *phys = (TrafficCtx*)vctx;
596:   PetscReal  a     = phys->a;
597:   PetscReal  speed;

600:   speed     = a*(1 - (uL[0] + uR[0]));
601:   flux[0]   = 0.5*(TrafficFlux(a,uL[0]) + TrafficFlux(a,uR[0])) - 0.5*phys->lxf_speed*(uR[0]-uL[0]);
602:   *maxspeed = speed;
603:   return(0);
604: }

608: static PetscErrorCode PhysicsRiemann_Traffic_Rusanov(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
609: {
610:   PetscReal a = ((TrafficCtx*)vctx)->a;
611:   PetscReal speed;

614:   speed     = a*PetscMax(PetscAbs(1-2*uL[0]),PetscAbs(1-2*uR[0]));
615:   flux[0]   = 0.5*(TrafficFlux(a,uL[0]) + TrafficFlux(a,uR[0])) - 0.5*speed*(uR[0]-uL[0]);
616:   *maxspeed = speed;
617:   return(0);
618: }

622: static PetscErrorCode PhysicsCreate_Traffic(FVCtx *ctx)
623: {
624:   PetscErrorCode    ierr;
625:   TrafficCtx        *user;
626:   RiemannFunction   r;
627:   PetscFunctionList rlist      = 0;
628:   char              rname[256] = "exact";

631:   PetscNew(&user);
632:   ctx->physics.sample         = PhysicsSample_Traffic;
633:   ctx->physics.characteristic = PhysicsCharacteristic_Conservative;
634:   ctx->physics.destroy        = PhysicsDestroy_SimpleFree;
635:   ctx->physics.user           = user;
636:   ctx->physics.dof            = 1;

638:   PetscStrallocpy("density",&ctx->physics.fieldname[0]);
639:   user->a = 0.5;
640:   RiemannListAdd(&rlist,"exact",  PhysicsRiemann_Traffic_Exact);
641:   RiemannListAdd(&rlist,"roe",    PhysicsRiemann_Traffic_Roe);
642:   RiemannListAdd(&rlist,"lxf",    PhysicsRiemann_Traffic_LxF);
643:   RiemannListAdd(&rlist,"rusanov",PhysicsRiemann_Traffic_Rusanov);
644:   PetscOptionsBegin(ctx->comm,ctx->prefix,"Options for Traffic","");
645:   {
646:     PetscOptionsReal("-physics_traffic_a","Flux = a*u*(1-u)","",user->a,&user->a,NULL);
647:     PetscOptionsFList("-physics_traffic_riemann","Riemann solver","",rlist,rname,rname,sizeof(rname),NULL);
648:   }
649:   PetscOptionsEnd();

651:   RiemannListFind(rlist,rname,&r);
652:   PetscFunctionListDestroy(&rlist);

654:   ctx->physics.riemann = r;

656:   /* *
657:   * Hack to deal with LxF in semi-discrete form
658:   * max speed is 3*a for the basic initial conditions (-1 <= u <= 2)
659:   * */
660:   if (r == PhysicsRiemann_Traffic_LxF) user->lxf_speed = 3*user->a;
661:   return(0);
662: }

664: /* --------------------------------- Linear Acoustics ----------------------------------- */

666: /* Flux: u_t + (A u)_x
667:  * z = sqrt(rho*bulk), c = sqrt(rho/bulk)
668:  * Spectral decomposition: A = R * D * Rinv
669:  * [    cz] = [-z   z] [-c    ] [-1/2z  1/2]
670:  * [c/z   ] = [ 1   1] [     c] [ 1/2z  1/2]
671:  *
672:  * We decompose this into the left-traveling waves Al = R * D^- Rinv
673:  * and the right-traveling waves Ar = R * D^+ * Rinv
674:  * Multiplying out these expressions produces the following two matrices
675:  */

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

682: PETSC_STATIC_INLINE void AcousticsFlux(AcousticsCtx *ctx,const PetscScalar *u,PetscScalar *f)
683: {
684:   f[0] = ctx->c*ctx->z*u[1];
685:   f[1] = ctx->c/ctx->z*u[0];
686: }

690: static PetscErrorCode PhysicsCharacteristic_Acoustics(void *vctx,PetscInt m,const PetscScalar *u,PetscScalar *X,PetscScalar *Xi,PetscReal *speeds)
691: {
692:   AcousticsCtx *phys = (AcousticsCtx*)vctx;
693:   PetscReal    z     = phys->z,c = phys->c;

696:   X[0*2+0]  = -z;
697:   X[0*2+1]  = z;
698:   X[1*2+0]  = 1;
699:   X[1*2+1]  = 1;
700:   Xi[0*2+0] = -1./(2*z);
701:   Xi[0*2+1] = 1./2;
702:   Xi[1*2+0] = 1./(2*z);
703:   Xi[1*2+1] = 1./2;
704:   speeds[0] = -c;
705:   speeds[1] = c;
706:   return(0);
707: }

711: static PetscErrorCode PhysicsSample_Acoustics_Initial(AcousticsCtx *phys,PetscInt initial,PetscReal xmin,PetscReal xmax,PetscReal x,PetscReal *u)
712: {
714:   switch (initial) {
715:   case 0:
716:     u[0] = (PetscAbs((x - xmin)/(xmax - xmin) - 0.2) < 0.1) ? 1 : 0.5;
717:     u[1] = (PetscAbs((x - xmin)/(xmax - xmin) - 0.7) < 0.1) ? 1 : -0.5;
718:     break;
719:   case 1:
720:     u[0] = PetscCosReal(3 * 2*PETSC_PI*x/(xmax-xmin));
721:     u[1] = PetscExpReal(-PetscSqr(x - (xmax + xmin)/2) / (2*PetscSqr(0.2*(xmax - xmin)))) - 0.5;
722:     break;
723:   default: SETERRQ(PETSC_COMM_SELF,1,"unknown initial condition");
724:   }
725:   return(0);
726: }

730: static PetscErrorCode PhysicsSample_Acoustics(void *vctx,PetscInt initial,FVBCType bctype,PetscReal xmin,PetscReal xmax,PetscReal t,PetscReal x,PetscReal *u)
731: {
732:   AcousticsCtx   *phys = (AcousticsCtx*)vctx;
733:   PetscReal      c     = phys->c;
734:   PetscReal      x0a,x0b,u0a[2],u0b[2],tmp[2];
735:   PetscReal      X[2][2],Xi[2][2],dummy[2];

739:   switch (bctype) {
740:   case FVBC_OUTFLOW:
741:     x0a = x+c*t;
742:     x0b = x-c*t;
743:     break;
744:   case FVBC_PERIODIC:
745:     x0a = RangeMod(x+c*t,xmin,xmax);
746:     x0b = RangeMod(x-c*t,xmin,xmax);
747:     break;
748:   default: SETERRQ(PETSC_COMM_SELF,1,"unknown BCType");
749:   }
750:   PhysicsSample_Acoustics_Initial(phys,initial,xmin,xmax,x0a,u0a);
751:   PhysicsSample_Acoustics_Initial(phys,initial,xmin,xmax,x0b,u0b);
752:   PhysicsCharacteristic_Acoustics(vctx,2,u,&X[0][0],&Xi[0][0],dummy);
753:   tmp[0] = Xi[0][0]*u0a[0] + Xi[0][1]*u0a[1];
754:   tmp[1] = Xi[1][0]*u0b[0] + Xi[1][1]*u0b[1];
755:   u[0]   = X[0][0]*tmp[0] + X[0][1]*tmp[1];
756:   u[1]   = X[1][0]*tmp[0] + X[1][1]*tmp[1];
757:   return(0);
758: }

762: static PetscErrorCode PhysicsRiemann_Acoustics_Exact(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
763: {
764:   AcousticsCtx *phys = (AcousticsCtx*)vctx;
765:   PetscReal    c     = phys->c,z = phys->z;
766:   PetscReal
767:     Al[2][2] = {{-c/2     , c*z/2  },
768:                 {c/(2*z)  , -c/2   }}, /* Left traveling waves */
769:     Ar[2][2] = {{c/2      , c*z/2  },
770:                 {c/(2*z)  , c/2    }}; /* Right traveling waves */

773:   flux[0]   = Al[0][0]*uR[0] + Al[0][1]*uR[1] + Ar[0][0]*uL[0] + Ar[0][1]*uL[1];
774:   flux[1]   = Al[1][0]*uR[0] + Al[1][1]*uR[1] + Ar[1][0]*uL[0] + Ar[1][1]*uL[1];
775:   *maxspeed = c;
776:   return(0);
777: }

781: static PetscErrorCode PhysicsCreate_Acoustics(FVCtx *ctx)
782: {
783:   PetscErrorCode    ierr;
784:   AcousticsCtx      *user;
785:   PetscFunctionList rlist      = 0,rclist = 0;
786:   char              rname[256] = "exact",rcname[256] = "characteristic";

789:   PetscNew(&user);
790:   ctx->physics.sample         = PhysicsSample_Acoustics;
791:   ctx->physics.destroy        = PhysicsDestroy_SimpleFree;
792:   ctx->physics.user           = user;
793:   ctx->physics.dof            = 2;

795:   PetscStrallocpy("u",&ctx->physics.fieldname[0]);
796:   PetscStrallocpy("v",&ctx->physics.fieldname[1]);

798:   user->c = 1;
799:   user->z = 1;

801:   RiemannListAdd(&rlist,"exact",  PhysicsRiemann_Acoustics_Exact);
802:   ReconstructListAdd(&rclist,"characteristic",PhysicsCharacteristic_Acoustics);
803:   ReconstructListAdd(&rclist,"conservative",PhysicsCharacteristic_Conservative);
804:   PetscOptionsBegin(ctx->comm,ctx->prefix,"Options for linear Acoustics","");
805:   {
806:     PetscOptionsReal("-physics_acoustics_c","c = sqrt(bulk/rho)","",user->c,&user->c,NULL);
807:     PetscOptionsReal("-physics_acoustics_z","z = sqrt(bulk*rho)","",user->z,&user->z,NULL);
808:     PetscOptionsFList("-physics_acoustics_riemann","Riemann solver","",rlist,rname,rname,sizeof(rname),NULL);
809:     PetscOptionsFList("-physics_acoustics_reconstruct","Reconstruction","",rclist,rcname,rcname,sizeof(rcname),NULL);
810:   }
811:   PetscOptionsEnd();
812:   RiemannListFind(rlist,rname,&ctx->physics.riemann);
813:   ReconstructListFind(rclist,rcname,&ctx->physics.characteristic);
814:   PetscFunctionListDestroy(&rlist);
815:   PetscFunctionListDestroy(&rclist);
816:   return(0);
817: }

819: /* --------------------------------- Isothermal Gas Dynamics ----------------------------------- */

821: typedef struct {
822:   PetscReal acoustic_speed;
823: } IsoGasCtx;

825: PETSC_STATIC_INLINE void IsoGasFlux(PetscReal c,const PetscScalar *u,PetscScalar *f)
826: {
827:   f[0] = u[1];
828:   f[1] = PetscSqr(u[1])/u[0] + c*c*u[0];
829: }

833: static PetscErrorCode PhysicsSample_IsoGas(void *vctx,PetscInt initial,FVBCType bctype,PetscReal xmin,PetscReal xmax,PetscReal t,PetscReal x,PetscReal *u)
834: {

837:   if (t > 0) SETERRQ(PETSC_COMM_SELF,1,"Exact solutions not implemented for t > 0");
838:   switch (initial) {
839:     case 0:
840:       u[0] = (x < 0) ? 1 : 0.5;
841:       u[1] = (x < 0) ? 1 : 0.7;
842:       break;
843:     case 1:
844:       u[0] = 1+0.5*PetscSinReal(2*PETSC_PI*x);
845:       u[1] = 1*u[0];
846:       break;
847:     default: SETERRQ(PETSC_COMM_SELF,1,"unknown initial condition");
848:   }
849:   return(0);
850: }

854: static PetscErrorCode PhysicsRiemann_IsoGas_Roe(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
855: {
856:   IsoGasCtx   *phys = (IsoGasCtx*)vctx;
857:   PetscReal   c     = phys->acoustic_speed;
858:   PetscScalar ubar,du[2],a[2],fL[2],fR[2],lam[2],ustar[2],R[2][2];
859:   PetscInt    i;

862:   ubar = (uL[1]/PetscSqrtScalar(uL[0]) + uR[1]/PetscSqrtScalar(uR[0])) / (PetscSqrtScalar(uL[0]) + PetscSqrtScalar(uR[0]));
863:   /* write fluxuations in characteristic basis */
864:   du[0] = uR[0] - uL[0];
865:   du[1] = uR[1] - uL[1];
866:   a[0]  = (1/(2*c)) * ((ubar + c)*du[0] - du[1]);
867:   a[1]  = (1/(2*c)) * ((-ubar + c)*du[0] + du[1]);
868:   /* wave speeds */
869:   lam[0] = ubar - c;
870:   lam[1] = ubar + c;
871:   /* Right eigenvectors */
872:   R[0][0] = 1; R[0][1] = ubar-c;
873:   R[1][0] = 1; R[1][1] = ubar+c;
874:   /* Compute state in star region (between the 1-wave and 2-wave) */
875:   for (i=0; i<2; i++) ustar[i] = uL[i] + a[0]*R[0][i];
876:   if (uL[1]/uL[0] < c && c < ustar[1]/ustar[0]) { /* 1-wave is sonic rarefaction */
877:     PetscScalar ufan[2];
878:     ufan[0] = uL[0]*PetscExpScalar(uL[1]/(uL[0]*c) - 1);
879:     ufan[1] = c*ufan[0];
880:     IsoGasFlux(c,ufan,flux);
881:   } else if (ustar[1]/ustar[0] < -c && -c < uR[1]/uR[0]) { /* 2-wave is sonic rarefaction */
882:     PetscScalar ufan[2];
883:     ufan[0] = uR[0]*PetscExpScalar(-uR[1]/(uR[0]*c) - 1);
884:     ufan[1] = -c*ufan[0];
885:     IsoGasFlux(c,ufan,flux);
886:   } else {                      /* Centered form */
887:     IsoGasFlux(c,uL,fL);
888:     IsoGasFlux(c,uR,fR);
889:     for (i=0; i<2; i++) {
890:       PetscScalar absdu = PetscAbsScalar(lam[0])*a[0]*R[0][i] + PetscAbsScalar(lam[1])*a[1]*R[1][i];
891:       flux[i] = 0.5*(fL[i]+fR[i]) - 0.5*absdu;
892:     }
893:   }
894:   *maxspeed = MaxAbs(lam[0],lam[1]);
895:   return(0);
896: }

900: static PetscErrorCode PhysicsRiemann_IsoGas_Exact(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
901: {
902:   IsoGasCtx                   *phys = (IsoGasCtx*)vctx;
903:   PetscReal                   c     = phys->acoustic_speed;
904:   PetscScalar                 ustar[2];
905:   struct {PetscScalar rho,u;} L = {uL[0],uL[1]/uL[0]},R = {uR[0],uR[1]/uR[0]},star;
906:   PetscInt                    i;
907:   PetscErrorCode              ierr;

910:   if (!(L.rho > 0 && R.rho > 0)) SETERRQ(PETSC_COMM_SELF,1,"Reconstructed density is negative");
911:   {
912:     /* Solve for star state */
913:     PetscScalar res,tmp,rho = 0.5*(L.rho + R.rho); /* initial guess */
914:     for (i=0; i<20; i++) {
915:       PetscScalar fr,fl,dfr,dfl;
916:       fl = (L.rho < rho)
917:         ? (rho-L.rho)/PetscSqrtScalar(L.rho*rho)       /* shock */
918:         : PetscLogScalar(rho) - PetscLogScalar(L.rho); /* rarefaction */
919:       fr = (R.rho < rho)
920:         ? (rho-R.rho)/PetscSqrtScalar(R.rho*rho)       /* shock */
921:         : PetscLogScalar(rho) - PetscLogScalar(R.rho); /* rarefaction */
922:       res = R.u-L.u + c*(fr+fl);
923:       PetscIsInfOrNanScalar(res);
924:       if (PetscAbsScalar(res) < 1e-10) {
925:         star.rho = rho;
926:         star.u   = L.u - c*fl;
927:         goto converged;
928:       }
929:       dfl = (L.rho < rho)
930:         ? 1/PetscSqrtScalar(L.rho*rho)*(1 - 0.5*(rho-L.rho)/rho)
931:         : 1/rho;
932:       dfr = (R.rho < rho)
933:         ? 1/PetscSqrtScalar(R.rho*rho)*(1 - 0.5*(rho-R.rho)/rho)
934:         : 1/rho;
935:       tmp = rho - res/(c*(dfr+dfl));
936:       if (tmp <= 0) rho /= 2;   /* Guard against Newton shooting off to a negative density */
937:       else rho = tmp;
938:       if (!((rho > 0) && PetscIsNormalScalar(rho))) SETERRQ1(PETSC_COMM_SELF,1,"non-normal iterate rho=%g",(double)PetscRealPart(rho));
939:     }
940:     SETERRQ1(PETSC_COMM_SELF,1,"Newton iteration for star.rho diverged after %D iterations",i);
941:   }
942: converged:
943:   if (L.u-c < 0 && 0 < star.u-c) { /* 1-wave is sonic rarefaction */
944:     PetscScalar ufan[2];
945:     ufan[0] = L.rho*PetscExpScalar(L.u/c - 1);
946:     ufan[1] = c*ufan[0];
947:     IsoGasFlux(c,ufan,flux);
948:   } else if (star.u+c < 0 && 0 < R.u+c) { /* 2-wave is sonic rarefaction */
949:     PetscScalar ufan[2];
950:     ufan[0] = R.rho*PetscExpScalar(-R.u/c - 1);
951:     ufan[1] = -c*ufan[0];
952:     IsoGasFlux(c,ufan,flux);
953:   } else if ((L.rho >= star.rho && L.u-c >= 0)
954:              || (L.rho < star.rho && (star.rho*star.u-L.rho*L.u)/(star.rho-L.rho) > 0)) {
955:     /* 1-wave is supersonic rarefaction, or supersonic shock */
956:     IsoGasFlux(c,uL,flux);
957:   } else if ((star.rho <= R.rho && R.u+c <= 0)
958:              || (star.rho > R.rho && (R.rho*R.u-star.rho*star.u)/(R.rho-star.rho) < 0)) {
959:     /* 2-wave is supersonic rarefaction or supersonic shock */
960:     IsoGasFlux(c,uR,flux);
961:   } else {
962:     ustar[0] = star.rho;
963:     ustar[1] = star.rho*star.u;
964:     IsoGasFlux(c,ustar,flux);
965:   }
966:   *maxspeed = MaxAbs(MaxAbs(star.u-c,star.u+c),MaxAbs(L.u-c,R.u+c));
967:   return(0);
968: }

972: static PetscErrorCode PhysicsRiemann_IsoGas_Rusanov(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
973: {
974:   IsoGasCtx   *phys             = (IsoGasCtx*)vctx;
975:   PetscScalar c                 = phys->acoustic_speed,fL[2],fR[2],s;
976:   struct {PetscScalar rho,u;} L = {uL[0],uL[1]/uL[0]},R = {uR[0],uR[1]/uR[0]};

979:   if (!(L.rho > 0 && R.rho > 0)) SETERRQ(PETSC_COMM_SELF,1,"Reconstructed density is negative");
980:   IsoGasFlux(c,uL,fL);
981:   IsoGasFlux(c,uR,fR);
982:   s         = PetscMax(PetscAbs(L.u),PetscAbs(R.u))+c;
983:   flux[0]   = 0.5*(fL[0] + fR[0]) + 0.5*s*(uL[0] - uR[0]);
984:   flux[1]   = 0.5*(fL[1] + fR[1]) + 0.5*s*(uL[1] - uR[1]);
985:   *maxspeed = s;
986:   return(0);
987: }

991: static PetscErrorCode PhysicsCharacteristic_IsoGas(void *vctx,PetscInt m,const PetscScalar *u,PetscScalar *X,PetscScalar *Xi,PetscReal *speeds)
992: {
993:   IsoGasCtx      *phys = (IsoGasCtx*)vctx;
994:   PetscReal      c     = phys->acoustic_speed;

998:   speeds[0] = u[1]/u[0] - c;
999:   speeds[1] = u[1]/u[0] + c;
1000:   X[0*2+0]  = 1;
1001:   X[0*2+1]  = speeds[0];
1002:   X[1*2+0]  = 1;
1003:   X[1*2+1]  = speeds[1];

1005:   PetscMemcpy(Xi,X,4*sizeof(X[0]));
1006:   PetscKernel_A_gets_inverse_A_2(Xi,0);
1007:   return(0);
1008: }

1012: static PetscErrorCode PhysicsCreate_IsoGas(FVCtx *ctx)
1013: {
1014:   PetscErrorCode    ierr;
1015:   IsoGasCtx         *user;
1016:   PetscFunctionList rlist = 0,rclist = 0;
1017:   char              rname[256] = "exact",rcname[256] = "characteristic";

1020:   PetscNew(&user);
1021:   ctx->physics.sample         = PhysicsSample_IsoGas;
1022:   ctx->physics.destroy        = PhysicsDestroy_SimpleFree;
1023:   ctx->physics.user           = user;
1024:   ctx->physics.dof            = 2;

1026:   PetscStrallocpy("density",&ctx->physics.fieldname[0]);
1027:   PetscStrallocpy("momentum",&ctx->physics.fieldname[1]);

1029:   user->acoustic_speed = 1;

1031:   RiemannListAdd(&rlist,"exact",  PhysicsRiemann_IsoGas_Exact);
1032:   RiemannListAdd(&rlist,"roe",    PhysicsRiemann_IsoGas_Roe);
1033:   RiemannListAdd(&rlist,"rusanov",PhysicsRiemann_IsoGas_Rusanov);
1034:   ReconstructListAdd(&rclist,"characteristic",PhysicsCharacteristic_IsoGas);
1035:   ReconstructListAdd(&rclist,"conservative",PhysicsCharacteristic_Conservative);
1036:   PetscOptionsBegin(ctx->comm,ctx->prefix,"Options for IsoGas","");
1037:   {
1038:     PetscOptionsReal("-physics_isogas_acoustic_speed","Acoustic speed","",user->acoustic_speed,&user->acoustic_speed,NULL);
1039:     PetscOptionsFList("-physics_isogas_riemann","Riemann solver","",rlist,rname,rname,sizeof(rname),NULL);
1040:     PetscOptionsFList("-physics_isogas_reconstruct","Reconstruction","",rclist,rcname,rcname,sizeof(rcname),NULL);
1041:   }
1042:   PetscOptionsEnd();
1043:   RiemannListFind(rlist,rname,&ctx->physics.riemann);
1044:   ReconstructListFind(rclist,rcname,&ctx->physics.characteristic);
1045:   PetscFunctionListDestroy(&rlist);
1046:   PetscFunctionListDestroy(&rclist);
1047:   return(0);
1048: }



1052: /* --------------------------------- Shallow Water ----------------------------------- */

1054: typedef struct {
1055:   PetscReal gravity;
1056: } ShallowCtx;

1058: PETSC_STATIC_INLINE void ShallowFlux(ShallowCtx *phys,const PetscScalar *u,PetscScalar *f)
1059: {
1060:   f[0] = u[1];
1061:   f[1] = PetscSqr(u[1])/u[0] + 0.5*phys->gravity*PetscSqr(u[0]);
1062: }

1066: static PetscErrorCode PhysicsRiemann_Shallow_Exact(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
1067: {
1068:   ShallowCtx                *phys = (ShallowCtx*)vctx;
1069:   PetscScalar               g    = phys->gravity,ustar[2],cL,cR,c,cstar;
1070:   struct {PetscScalar h,u;} L = {uL[0],uL[1]/uL[0]},R = {uR[0],uR[1]/uR[0]},star;
1071:   PetscInt                  i;
1072:   PetscErrorCode            ierr;

1075:   if (!(L.h > 0 && R.h > 0)) SETERRQ(PETSC_COMM_SELF,1,"Reconstructed thickness is negative");
1076:   cL = PetscSqrtScalar(g*L.h);
1077:   cR = PetscSqrtScalar(g*R.h);
1078:   c  = PetscMax(cL,cR);
1079:   {
1080:     /* Solve for star state */
1081:     const PetscInt maxits = 50;
1082:     PetscScalar tmp,res,res0=0,h0,h = 0.5*(L.h + R.h); /* initial guess */
1083:     h0 = h;
1084:     for (i=0; i<maxits; i++) {
1085:       PetscScalar fr,fl,dfr,dfl;
1086:       fl = (L.h < h)
1087:         ? PetscSqrtScalar(0.5*g*(h*h - L.h*L.h)*(1/L.h - 1/h)) /* shock */
1088:         : 2*PetscSqrtScalar(g*h) - 2*PetscSqrtScalar(g*L.h);   /* rarefaction */
1089:       fr = (R.h < h)
1090:         ? PetscSqrtScalar(0.5*g*(h*h - R.h*R.h)*(1/R.h - 1/h)) /* shock */
1091:         : 2*PetscSqrtScalar(g*h) - 2*PetscSqrtScalar(g*R.h);   /* rarefaction */
1092:       res = R.u - L.u + fr + fl;
1093:       PetscIsInfOrNanScalar(res);
1094:       if (PetscAbsScalar(res) < 1e-8 || (i > 0 && PetscAbsScalar(h-h0) < 1e-8)) {
1095:         star.h = h;
1096:         star.u = L.u - fl;
1097:         goto converged;
1098:       } else if (i > 0 && PetscAbsScalar(res) >= PetscAbsScalar(res0)) {        /* Line search */
1099:         h = 0.8*h0 + 0.2*h;
1100:         continue;
1101:       }
1102:       /* Accept the last step and take another */
1103:       res0 = res;
1104:       h0 = h;
1105:       dfl = (L.h < h)
1106:         ? 0.5/fl*0.5*g*(-L.h*L.h/(h*h) - 1 + 2*h/L.h)
1107:         : PetscSqrtScalar(g/h);
1108:       dfr = (R.h < h)
1109:         ? 0.5/fr*0.5*g*(-R.h*R.h/(h*h) - 1 + 2*h/R.h)
1110:         : PetscSqrtScalar(g/h);
1111:       tmp = h - res/(dfr+dfl);
1112:       if (tmp <= 0) h /= 2;   /* Guard against Newton shooting off to a negative thickness */
1113:       else h = tmp;
1114:       if (!((h > 0) && PetscIsNormalScalar(h))) SETERRQ1(PETSC_COMM_SELF,1,"non-normal iterate h=%g",(double)h);
1115:     }
1116:     SETERRQ1(PETSC_COMM_SELF,1,"Newton iteration for star.h diverged after %D iterations",i);
1117:   }
1118: converged:
1119:   cstar = PetscSqrtScalar(g*star.h);
1120:   if (L.u-cL < 0 && 0 < star.u-cstar) { /* 1-wave is sonic rarefaction */
1121:     PetscScalar ufan[2];
1122:     ufan[0] = 1/g*PetscSqr(L.u/3 + 2./3*cL);
1123:     ufan[1] = PetscSqrtScalar(g*ufan[0])*ufan[0];
1124:     ShallowFlux(phys,ufan,flux);
1125:   } else if (star.u+cstar < 0 && 0 < R.u+cR) { /* 2-wave is sonic rarefaction */
1126:     PetscScalar ufan[2];
1127:     ufan[0] = 1/g*PetscSqr(R.u/3 - 2./3*cR);
1128:     ufan[1] = -PetscSqrtScalar(g*ufan[0])*ufan[0];
1129:     ShallowFlux(phys,ufan,flux);
1130:   } else if ((L.h >= star.h && L.u-c >= 0)
1131:              || (L.h<star.h && (star.h*star.u-L.h*L.u)/(star.h-L.h) > 0)) {
1132:     /* 1-wave is right-travelling shock (supersonic) */
1133:     ShallowFlux(phys,uL,flux);
1134:   } else if ((star.h <= R.h && R.u+c <= 0)
1135:              || (star.h>R.h && (R.h*R.u-star.h*star.h)/(R.h-star.h) < 0)) {
1136:     /* 2-wave is left-travelling shock (supersonic) */
1137:     ShallowFlux(phys,uR,flux);
1138:   } else {
1139:     ustar[0] = star.h;
1140:     ustar[1] = star.h*star.u;
1141:     ShallowFlux(phys,ustar,flux);
1142:   }
1143:   *maxspeed = MaxAbs(MaxAbs(star.u-cstar,star.u+cstar),MaxAbs(L.u-cL,R.u+cR));
1144:   return(0);
1145: }

1149: static PetscErrorCode PhysicsRiemann_Shallow_Rusanov(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
1150: {
1151:   ShallowCtx *phys = (ShallowCtx*)vctx;
1152:   PetscScalar g    = phys->gravity,fL[2],fR[2],s;
1153:   struct {PetscScalar h,u;} L = {uL[0],uL[1]/uL[0]},R = {uR[0],uR[1]/uR[0]};

1156:   if (!(L.h > 0 && R.h > 0)) SETERRQ(PETSC_COMM_SELF,1,"Reconstructed thickness is negative");
1157:   ShallowFlux(phys,uL,fL);
1158:   ShallowFlux(phys,uR,fR);
1159:   s         = PetscMax(PetscAbs(L.u)+PetscSqrtScalar(g*L.h),PetscAbs(R.u)+PetscSqrtScalar(g*R.h));
1160:   flux[0]   = 0.5*(fL[0] + fR[0]) + 0.5*s*(uL[0] - uR[0]);
1161:   flux[1]   = 0.5*(fL[1] + fR[1]) + 0.5*s*(uL[1] - uR[1]);
1162:   *maxspeed = s;
1163:   return(0);
1164: }

1168: static PetscErrorCode PhysicsCharacteristic_Shallow(void *vctx,PetscInt m,const PetscScalar *u,PetscScalar *X,PetscScalar *Xi,PetscReal *speeds)
1169: {
1170:   ShallowCtx     *phys = (ShallowCtx*)vctx;
1171:   PetscReal      c;

1175:   c         = PetscSqrtScalar(u[0]*phys->gravity);
1176:   speeds[0] = u[1]/u[0] - c;
1177:   speeds[1] = u[1]/u[0] + c;
1178:   X[0*2+0]  = 1;
1179:   X[0*2+1]  = speeds[0];
1180:   X[1*2+0]  = 1;
1181:   X[1*2+1]  = speeds[1];

1183:   PetscMemcpy(Xi,X,4*sizeof(X[0]));
1184:   PetscKernel_A_gets_inverse_A_2(Xi,0);
1185:   return(0);
1186: }

1190: static PetscErrorCode PhysicsCreate_Shallow(FVCtx *ctx)
1191: {
1192:   PetscErrorCode    ierr;
1193:   ShallowCtx        *user;
1194:   PetscFunctionList rlist = 0,rclist = 0;
1195:   char              rname[256] = "exact",rcname[256] = "characteristic";

1198:   PetscNew(&user);
1199:   /* Shallow water and Isothermal Gas dynamics are similar so we reuse initial conditions for now */
1200:   ctx->physics.sample         = PhysicsSample_IsoGas;
1201:   ctx->physics.destroy        = PhysicsDestroy_SimpleFree;
1202:   ctx->physics.user           = user;
1203:   ctx->physics.dof            = 2;

1205:   PetscStrallocpy("density",&ctx->physics.fieldname[0]);
1206:   PetscStrallocpy("momentum",&ctx->physics.fieldname[1]);

1208:   user->gravity = 1;

1210:   RiemannListAdd(&rlist,"exact",  PhysicsRiemann_Shallow_Exact);
1211:   RiemannListAdd(&rlist,"rusanov",PhysicsRiemann_Shallow_Rusanov);
1212:   ReconstructListAdd(&rclist,"characteristic",PhysicsCharacteristic_Shallow);
1213:   ReconstructListAdd(&rclist,"conservative",PhysicsCharacteristic_Conservative);
1214:   PetscOptionsBegin(ctx->comm,ctx->prefix,"Options for Shallow","");
1215:   {
1216:     PetscOptionsReal("-physics_shallow_gravity","Gravity","",user->gravity,&user->gravity,NULL);
1217:     PetscOptionsFList("-physics_shallow_riemann","Riemann solver","",rlist,rname,rname,sizeof(rname),NULL);
1218:     PetscOptionsFList("-physics_shallow_reconstruct","Reconstruction","",rclist,rcname,rcname,sizeof(rcname),NULL);
1219:   }
1220:   PetscOptionsEnd();
1221:   RiemannListFind(rlist,rname,&ctx->physics.riemann);
1222:   ReconstructListFind(rclist,rcname,&ctx->physics.characteristic);
1223:   PetscFunctionListDestroy(&rlist);
1224:   PetscFunctionListDestroy(&rclist);
1225:   return(0);
1226: }



1230: /* --------------------------------- Finite Volume Solver ----------------------------------- */

1234: static PetscErrorCode FVRHSFunction(TS ts,PetscReal time,Vec X,Vec F,void *vctx)
1235: {
1236:   FVCtx          *ctx = (FVCtx*)vctx;
1238:   PetscInt       i,j,k,Mx,dof,xs,xm;
1239:   PetscReal      hx,cfl_idt = 0;
1240:   PetscScalar    *x,*f,*slope;
1241:   Vec            Xloc;
1242:   DM             da;

1245:   TSGetDM(ts,&da);
1246:   DMGetLocalVector(da,&Xloc);
1247:   DMDAGetInfo(da,0, &Mx,0,0, 0,0,0, &dof,0,0,0,0,0);
1248:   hx   = (ctx->xmax - ctx->xmin)/Mx;
1249:   DMGlobalToLocalBegin(da,X,INSERT_VALUES,Xloc);
1250:   DMGlobalToLocalEnd  (da,X,INSERT_VALUES,Xloc);

1252:   VecZeroEntries(F);

1254:   DMDAVecGetArray(da,Xloc,&x);
1255:   DMDAVecGetArray(da,F,&f);
1256:   DMDAGetArray(da,PETSC_TRUE,&slope);

1258:   DMDAGetCorners(da,&xs,0,0,&xm,0,0);

1260:   if (ctx->bctype == FVBC_OUTFLOW) {
1261:     for (i=xs-2; i<0; i++) {
1262:       for (j=0; j<dof; j++) x[i*dof+j] = x[j];
1263:     }
1264:     for (i=Mx; i<xs+xm+2; i++) {
1265:       for (j=0; j<dof; j++) x[i*dof+j] = x[(xs+xm-1)*dof+j];
1266:     }
1267:   }
1268:   for (i=xs-1; i<xs+xm+1; i++) {
1269:     struct _LimitInfo info;
1270:     PetscScalar       *cjmpL,*cjmpR;
1271:     /* Determine the right eigenvectors R, where A = R \Lambda R^{-1} */
1272:     (*ctx->physics.characteristic)(ctx->physics.user,dof,&x[i*dof],ctx->R,ctx->Rinv,ctx->speeds);
1273:     /* Evaluate jumps across interfaces (i-1, i) and (i, i+1), put in characteristic basis */
1274:     PetscMemzero(ctx->cjmpLR,2*dof*sizeof(ctx->cjmpLR[0]));
1275:     cjmpL = &ctx->cjmpLR[0];
1276:     cjmpR = &ctx->cjmpLR[dof];
1277:     for (j=0; j<dof; j++) {
1278:       PetscScalar jmpL,jmpR;
1279:       jmpL = x[(i+0)*dof+j] - x[(i-1)*dof+j];
1280:       jmpR = x[(i+1)*dof+j] - x[(i+0)*dof+j];
1281:       for (k=0; k<dof; k++) {
1282:         cjmpL[k] += ctx->Rinv[k+j*dof] * jmpL;
1283:         cjmpR[k] += ctx->Rinv[k+j*dof] * jmpR;
1284:       }
1285:     }
1286:     /* Apply limiter to the left and right characteristic jumps */
1287:     info.m  = dof;
1288:     info.hx = hx;
1289:     (*ctx->limit)(&info,cjmpL,cjmpR,ctx->cslope);
1290:     for (j=0; j<dof; j++) ctx->cslope[j] /= hx; /* rescale to a slope */
1291:     for (j=0; j<dof; j++) {
1292:       PetscScalar tmp = 0;
1293:       for (k=0; k<dof; k++) tmp += ctx->R[j+k*dof] * ctx->cslope[k];
1294:       slope[i*dof+j] = tmp;
1295:     }
1296:   }

1298:   for (i=xs; i<xs+xm+1; i++) {
1299:     PetscReal   maxspeed;
1300:     PetscScalar *uL,*uR;
1301:     uL = &ctx->uLR[0];
1302:     uR = &ctx->uLR[dof];
1303:     for (j=0; j<dof; j++) {
1304:       uL[j] = x[(i-1)*dof+j] + slope[(i-1)*dof+j]*hx/2;
1305:       uR[j] = x[(i-0)*dof+j] - slope[(i-0)*dof+j]*hx/2;
1306:     }
1307:     (*ctx->physics.riemann)(ctx->physics.user,dof,uL,uR,ctx->flux,&maxspeed);
1308:     cfl_idt = PetscMax(cfl_idt,PetscAbsScalar(maxspeed/hx)); /* Max allowable value of 1/Delta t */

1310:     if (i > xs) {
1311:       for (j=0; j<dof; j++) f[(i-1)*dof+j] -= ctx->flux[j]/hx;
1312:     }
1313:     if (i < xs+xm) {
1314:       for (j=0; j<dof; j++) f[i*dof+j] += ctx->flux[j]/hx;
1315:     }
1316:   }

1318:   DMDAVecRestoreArray(da,Xloc,&x);
1319:   DMDAVecRestoreArray(da,F,&f);
1320:   DMDARestoreArray(da,PETSC_TRUE,&slope);
1321:   DMRestoreLocalVector(da,&Xloc);

1323:   MPI_Allreduce(&cfl_idt,&ctx->cfl_idt,1,MPIU_REAL,MPIU_MAX,PetscObjectComm((PetscObject)da));
1324:   if (0) {
1325:     /* We need to a way to inform the TS of a CFL constraint, this is a debugging fragment */
1326:     PetscReal dt,tnow;
1327:     TSGetTimeStep(ts,&dt);
1328:     TSGetTime(ts,&tnow);
1329:     if (dt > 0.5/ctx->cfl_idt) {
1330:       if (1) {
1331:         PetscPrintf(ctx->comm,"Stability constraint exceeded at t=%g, dt %g > %g\n",(double)tnow,(double)dt,(double)(0.5/ctx->cfl_idt));
1332:       } else SETERRQ2(PETSC_COMM_SELF,1,"Stability constraint exceeded, %g > %g",(double)dt,(double)(ctx->cfl/ctx->cfl_idt));
1333:     }
1334:   }
1335:   return(0);
1336: }

1340: static PetscErrorCode SmallMatMultADB(PetscScalar *C,PetscInt bs,const PetscScalar *A,const PetscReal *D,const PetscScalar *B)
1341: {
1342:   PetscInt i,j,k;

1345:   for (i=0; i<bs; i++) {
1346:     for (j=0; j<bs; j++) {
1347:       PetscScalar tmp = 0;
1348:       for (k=0; k<bs; k++) tmp += A[i*bs+k] * D[k] * B[k*bs+j];
1349:       C[i*bs+j] = tmp;
1350:     }
1351:   }
1352:   return(0);
1353: }


1358: static PetscErrorCode FVIJacobian(TS ts,PetscReal t,Vec X,Vec Xdot,PetscReal shift,Mat *A,Mat *B,MatStructure *flg,void *vctx)
1359: {
1360:   FVCtx          *ctx = (FVCtx*)vctx;
1362:   PetscInt       i,j,dof = ctx->physics.dof;
1363:   PetscScalar    *x,*J;
1364:   PetscReal      hx;
1365:   DM             da;
1366:   DMDALocalInfo  dainfo;

1369:   TSGetDM(ts,&da);
1370:   DMDAVecGetArray(da,X,&x);
1371:   DMDAGetLocalInfo(da,&dainfo);
1372:   hx   = (ctx->xmax - ctx->xmin)/dainfo.mx;
1373:   PetscMalloc1(dof*dof,&J);
1374:   for (i=dainfo.xs; i<dainfo.xs+dainfo.xm; i++) {
1375:     (*ctx->physics.characteristic)(ctx->physics.user,dof,&x[i*dof],ctx->R,ctx->Rinv,ctx->speeds);
1376:     for (j=0; j<dof; j++) ctx->speeds[j] = PetscAbs(ctx->speeds[j]);
1377:     SmallMatMultADB(J,dof,ctx->R,ctx->speeds,ctx->Rinv);
1378:     for (j=0; j<dof*dof; j++) J[j] = J[j]/hx + shift*(j/dof == j%dof);
1379:     MatSetValuesBlocked(*B,1,&i,1,&i,J,INSERT_VALUES);
1380:   }
1381:   PetscFree(J);
1382:   DMDAVecRestoreArray(da,X,&x);

1384:   MatAssemblyBegin(*B,MAT_FINAL_ASSEMBLY);
1385:   MatAssemblyEnd(*B,MAT_FINAL_ASSEMBLY);
1386:   if (*A != *B) {
1387:     MatAssemblyBegin(*A,MAT_FINAL_ASSEMBLY);
1388:     MatAssemblyEnd(*A,MAT_FINAL_ASSEMBLY);
1389:   }
1390:   return(0);
1391: }

1395: static PetscErrorCode FVSample(FVCtx *ctx,DM da,PetscReal time,Vec U)
1396: {
1398:   PetscScalar    *u,*uj;
1399:   PetscInt       i,j,k,dof,xs,xm,Mx;

1402:   if (!ctx->physics.sample) SETERRQ(PETSC_COMM_SELF,1,"Physics has not provided a sampling function");
1403:   DMDAGetInfo(da,0, &Mx,0,0, 0,0,0, &dof,0,0,0,0,0);
1404:   DMDAGetCorners(da,&xs,0,0,&xm,0,0);
1405:   DMDAVecGetArray(da,U,&u);
1406:   PetscMalloc1(dof,&uj);
1407:   for (i=xs; i<xs+xm; i++) {
1408:     const PetscReal h = (ctx->xmax-ctx->xmin)/Mx,xi = ctx->xmin+h/2+i*h;
1409:     const PetscInt  N = 200;
1410:     /* Integrate over cell i using trapezoid rule with N points. */
1411:     for (k=0; k<dof; k++) u[i*dof+k] = 0;
1412:     for (j=0; j<N+1; j++) {
1413:       PetscScalar xj = xi+h*(j-N/2)/(PetscReal)N;
1414:       (*ctx->physics.sample)(ctx->physics.user,ctx->initial,ctx->bctype,ctx->xmin,ctx->xmax,time,xj,uj);
1415:       for (k=0; k<dof; k++) u[i*dof+k] += ((j==0 || j==N) ? 0.5 : 1.0)*uj[k]/N;
1416:     }
1417:   }
1418:   DMDAVecRestoreArray(da,U,&u);
1419:   PetscFree(uj);
1420:   return(0);
1421: }

1425: static PetscErrorCode SolutionStatsView(DM da,Vec X,PetscViewer viewer)
1426: {
1428:   PetscReal      xmin,xmax;
1429:   PetscScalar    sum,*x,tvsum,tvgsum;
1430:   PetscInt       imin,imax,Mx,i,j,xs,xm,dof;
1431:   Vec            Xloc;
1432:   PetscBool      iascii;

1435:   PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&iascii);
1436:   if (iascii) {
1437:     /* PETSc lacks a function to compute total variation norm (difficult in multiple dimensions), we do it here */
1438:     DMGetLocalVector(da,&Xloc);
1439:     DMGlobalToLocalBegin(da,X,INSERT_VALUES,Xloc);
1440:     DMGlobalToLocalEnd  (da,X,INSERT_VALUES,Xloc);
1441:     DMDAVecGetArray(da,Xloc,&x);
1442:     DMDAGetCorners(da,&xs,0,0,&xm,0,0);
1443:     DMDAGetInfo(da,0, &Mx,0,0, 0,0,0, &dof,0,0,0,0,0);
1444:     tvsum = 0;
1445:     for (i=xs; i<xs+xm; i++) {
1446:       for (j=0; j<dof; j++) tvsum += PetscAbsScalar(x[i*dof+j] - x[(i-1)*dof+j]);
1447:     }
1448:     MPI_Allreduce(&tvsum,&tvgsum,1,MPIU_REAL,MPIU_MAX,PetscObjectComm((PetscObject)da));
1449:     DMDAVecRestoreArray(da,Xloc,&x);
1450:     DMRestoreLocalVector(da,&Xloc);

1452:     VecMin(X,&imin,&xmin);
1453:     VecMax(X,&imax,&xmax);
1454:     VecSum(X,&sum);
1455:     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));
1456:   } else SETERRQ(PETSC_COMM_SELF,1,"Viewer type not supported");
1457:   return(0);
1458: }

1462: static PetscErrorCode SolutionErrorNorms(FVCtx *ctx,DM da,PetscReal t,Vec X,PetscReal *nrm1,PetscReal *nrmsup)
1463: {
1465:   Vec            Y;
1466:   PetscInt       Mx;

1469:   VecGetSize(X,&Mx);
1470:   VecDuplicate(X,&Y);
1471:   FVSample(ctx,da,t,Y);
1472:   VecAYPX(Y,-1,X);
1473:   VecNorm(Y,NORM_1,nrm1);
1474:   VecNorm(Y,NORM_INFINITY,nrmsup);
1475:   *nrm1 /= Mx;
1476:   VecDestroy(&Y);
1477:   return(0);
1478: }

1482: int main(int argc,char *argv[])
1483: {
1484:   char              lname[256] = "mc",physname[256] = "advect",final_fname[256] = "solution.m";
1485:   PetscFunctionList limiters   = 0,physics = 0;
1486:   MPI_Comm          comm;
1487:   TS                ts;
1488:   DM                da;
1489:   Vec               X,X0,R;
1490:   Mat               B;
1491:   FVCtx             ctx;
1492:   PetscInt          i,dof,xs,xm,Mx,draw = 0;
1493:   PetscBool         view_final = PETSC_FALSE;
1494:   PetscReal         ptime;
1495:   PetscErrorCode    ierr;

1497:   PetscInitialize(&argc,&argv,0,help);
1498:   comm = PETSC_COMM_WORLD;
1499:   PetscMemzero(&ctx,sizeof(ctx));

1501:   /* Register limiters to be available on the command line */
1502:   PetscFunctionListAdd(&limiters,"upwind"              ,Limit_Upwind);
1503:   PetscFunctionListAdd(&limiters,"lax-wendroff"        ,Limit_LaxWendroff);
1504:   PetscFunctionListAdd(&limiters,"beam-warming"        ,Limit_BeamWarming);
1505:   PetscFunctionListAdd(&limiters,"fromm"               ,Limit_Fromm);
1506:   PetscFunctionListAdd(&limiters,"minmod"              ,Limit_Minmod);
1507:   PetscFunctionListAdd(&limiters,"superbee"            ,Limit_Superbee);
1508:   PetscFunctionListAdd(&limiters,"mc"                  ,Limit_MC);
1509:   PetscFunctionListAdd(&limiters,"vanleer"             ,Limit_VanLeer);
1510:   PetscFunctionListAdd(&limiters,"vanalbada"           ,Limit_VanAlbada);
1511:   PetscFunctionListAdd(&limiters,"vanalbadatvd"        ,Limit_VanAlbadaTVD);
1512:   PetscFunctionListAdd(&limiters,"koren"               ,Limit_Koren);
1513:   PetscFunctionListAdd(&limiters,"korensym"            ,Limit_KorenSym);
1514:   PetscFunctionListAdd(&limiters,"koren3"              ,Limit_Koren3);
1515:   PetscFunctionListAdd(&limiters,"cada-torrilhon2"     ,Limit_CadaTorrilhon2);
1516:   PetscFunctionListAdd(&limiters,"cada-torrilhon3-r0p1",Limit_CadaTorrilhon3R0p1);
1517:   PetscFunctionListAdd(&limiters,"cada-torrilhon3-r1"  ,Limit_CadaTorrilhon3R1);
1518:   PetscFunctionListAdd(&limiters,"cada-torrilhon3-r10" ,Limit_CadaTorrilhon3R10);
1519:   PetscFunctionListAdd(&limiters,"cada-torrilhon3-r100",Limit_CadaTorrilhon3R100);

1521:   /* Register physical models to be available on the command line */
1522:   PetscFunctionListAdd(&physics,"advect"          ,PhysicsCreate_Advect);
1523:   PetscFunctionListAdd(&physics,"burgers"         ,PhysicsCreate_Burgers);
1524:   PetscFunctionListAdd(&physics,"traffic"         ,PhysicsCreate_Traffic);
1525:   PetscFunctionListAdd(&physics,"acoustics"       ,PhysicsCreate_Acoustics);
1526:   PetscFunctionListAdd(&physics,"isogas"          ,PhysicsCreate_IsoGas);
1527:   PetscFunctionListAdd(&physics,"shallow"         ,PhysicsCreate_Shallow);

1529:   ctx.comm = comm;
1530:   ctx.cfl  = 0.9; ctx.bctype = FVBC_PERIODIC;
1531:   ctx.xmin = -1; ctx.xmax = 1;
1532:   PetscOptionsBegin(comm,NULL,"Finite Volume solver options","");
1533:   {
1534:     PetscOptionsReal("-xmin","X min","",ctx.xmin,&ctx.xmin,NULL);
1535:     PetscOptionsReal("-xmax","X max","",ctx.xmax,&ctx.xmax,NULL);
1536:     PetscOptionsFList("-limit","Name of flux limiter to use","",limiters,lname,lname,sizeof(lname),NULL);
1537:     PetscOptionsFList("-physics","Name of physics (Riemann solver and characteristics) to use","",physics,physname,physname,sizeof(physname),NULL);
1538:     PetscOptionsInt("-draw","Draw solution vector, bitwise OR of (1=initial,2=final,4=final error)","",draw,&draw,NULL);
1539:     PetscOptionsString("-view_final","Write final solution in ASCII MATLAB format to given file name","",final_fname,final_fname,sizeof(final_fname),&view_final);
1540:     PetscOptionsInt("-initial","Initial condition (depends on the physics)","",ctx.initial,&ctx.initial,NULL);
1541:     PetscOptionsBool("-exact","Compare errors with exact solution","",ctx.exact,&ctx.exact,NULL);
1542:     PetscOptionsReal("-cfl","CFL number to time step at","",ctx.cfl,&ctx.cfl,NULL);
1543:     PetscOptionsEnum("-bc_type","Boundary condition","",FVBCTypes,(PetscEnum)ctx.bctype,(PetscEnum*)&ctx.bctype,NULL);
1544:   }
1545:   PetscOptionsEnd();

1547:   /* Choose the limiter from the list of registered limiters */
1548:   PetscFunctionListFind(limiters,lname,&ctx.limit);
1549:   if (!ctx.limit) SETERRQ1(PETSC_COMM_SELF,1,"Limiter '%s' not found",lname);

1551:   /* Choose the physics from the list of registered models */
1552:   {
1553:     PetscErrorCode (*r)(FVCtx*);
1554:     PetscFunctionListFind(physics,physname,&r);
1555:     if (!r) SETERRQ1(PETSC_COMM_SELF,1,"Physics '%s' not found",physname);
1556:     /* Create the physics, will set the number of fields and their names */
1557:     (*r)(&ctx);
1558:   }

1560:   /* Create a DMDA to manage the parallel grid */
1561:   DMDACreate1d(comm,DMDA_BOUNDARY_PERIODIC,-50,ctx.physics.dof,2,NULL,&da);
1562:   /* Inform the DMDA of the field names provided by the physics. */
1563:   /* The names will be shown in the title bars when run with -ts_monitor_draw_solution */
1564:   for (i=0; i<ctx.physics.dof; i++) {
1565:     DMDASetFieldName(da,i,ctx.physics.fieldname[i]);
1566:   }
1567:   DMDAGetInfo(da,0, &Mx,0,0, 0,0,0, &dof,0,0,0,0,0);
1568:   DMDAGetCorners(da,&xs,0,0,&xm,0,0);

1570:   /* Set coordinates of cell centers */
1571:   DMDASetUniformCoordinates(da,ctx.xmin+0.5*(ctx.xmax-ctx.xmin)/Mx,ctx.xmax+0.5*(ctx.xmax-ctx.xmin)/Mx,0,0,0,0);

1573:   /* Allocate work space for the Finite Volume solver (so it doesn't have to be reallocated on each function evaluation) */
1574:   PetscMalloc4(dof*dof,&ctx.R,dof*dof,&ctx.Rinv,2*dof,&ctx.cjmpLR,1*dof,&ctx.cslope);
1575:   PetscMalloc3(2*dof,&ctx.uLR,dof,&ctx.flux,dof,&ctx.speeds);

1577:   /* Create a vector to store the solution and to save the initial state */
1578:   DMCreateGlobalVector(da,&X);
1579:   VecDuplicate(X,&X0);
1580:   VecDuplicate(X,&R);

1582:   DMCreateMatrix(da,&B);

1584:   /* Create a time-stepping object */
1585:   TSCreate(comm,&ts);
1586:   TSSetDM(ts,da);
1587:   TSSetRHSFunction(ts,R,FVRHSFunction,&ctx);
1588:   TSSetIJacobian(ts,B,B,FVIJacobian,&ctx);
1589:   TSSetType(ts,TSSSP);
1590:   TSSetDuration(ts,1000,10);

1592:   /* Compute initial conditions and starting time step */
1593:   FVSample(&ctx,da,0,X0);
1594:   FVRHSFunction(ts,0,X0,X,(void*)&ctx); /* Initial function evaluation, only used to determine max speed */
1595:   VecCopy(X0,X);                        /* The function value was not used so we set X=X0 again */
1596:   TSSetInitialTimeStep(ts,0,ctx.cfl/ctx.cfl_idt);

1598:   TSSetFromOptions(ts); /* Take runtime options */

1600:   SolutionStatsView(da,X,PETSC_VIEWER_STDOUT_WORLD);

1602:   {
1603:     PetscReal nrm1,nrmsup;
1604:     PetscInt  steps;

1606:     TSSolve(ts,X);
1607:     TSGetSolveTime(ts,&ptime);
1608:     TSGetTimeStepNumber(ts,&steps);

1610:     PetscPrintf(comm,"Final time %8.5f, steps %D\n",(double)ptime,steps);
1611:     if (ctx.exact) {
1612:       SolutionErrorNorms(&ctx,da,ptime,X,&nrm1,&nrmsup);
1613:       PetscPrintf(comm,"Error ||x-x_e||_1 %8.4e  ||x-x_e||_sup %8.4e\n",(double)nrm1,(double)nrmsup);
1614:     }
1615:   }

1617:   SolutionStatsView(da,X,PETSC_VIEWER_STDOUT_WORLD);

1619:   if (draw & 0x1) {VecView(X0,PETSC_VIEWER_DRAW_WORLD);}
1620:   if (draw & 0x2) {VecView(X,PETSC_VIEWER_DRAW_WORLD);}
1621:   if (draw & 0x4) {
1622:     Vec Y;
1623:     VecDuplicate(X,&Y);
1624:     FVSample(&ctx,da,ptime,Y);
1625:     VecAYPX(Y,-1,X);
1626:     VecView(Y,PETSC_VIEWER_DRAW_WORLD);
1627:     VecDestroy(&Y);
1628:   }

1630:   if (view_final) {
1631:     PetscViewer viewer;
1632:     PetscViewerASCIIOpen(PETSC_COMM_WORLD,final_fname,&viewer);
1633:     PetscViewerSetFormat(viewer,PETSC_VIEWER_ASCII_MATLAB);
1634:     VecView(X,viewer);
1635:     PetscViewerDestroy(&viewer);
1636:   }

1638:   /* Clean up */
1639:   (*ctx.physics.destroy)(ctx.physics.user);
1640:   for (i=0; i<ctx.physics.dof; i++) {PetscFree(ctx.physics.fieldname[i]);}
1641:   PetscFree4(ctx.R,ctx.Rinv,ctx.cjmpLR,ctx.cslope);
1642:   PetscFree3(ctx.uLR,ctx.flux,ctx.speeds);
1643:   VecDestroy(&X);
1644:   VecDestroy(&X0);
1645:   VecDestroy(&R);
1646:   MatDestroy(&B);
1647:   DMDestroy(&da);
1648:   TSDestroy(&ts);
1649:   PetscFunctionListDestroy(&limiters);
1650:   PetscFunctionListDestroy(&physics);
1651:   PetscFinalize();
1652:   return 0;
1653: }