Actual source code: ex9.c

  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 <petscdm.h>
 35: #include <petscdmda.h>
 36: #include <petscdraw.h>

 38: #include <petsc/private/kernels/blockinvert.h>

 40: static inline PetscReal Sgn(PetscReal a) { return (a<0) ? -1 : 1; }
 41: static inline PetscReal Abs(PetscReal a) { return (a<0) ? 0 : a; }
 42: static inline PetscReal Sqr(PetscReal a) { return a*a; }
 43: static inline PetscReal MaxAbs(PetscReal a,PetscReal b) { return (PetscAbs(a) > PetscAbs(b)) ? a : b; }
 44: PETSC_UNUSED static inline PetscReal MinAbs(PetscReal a,PetscReal b) { return (PetscAbs(a) < PetscAbs(b)) ? a : b; }
 45: static inline PetscReal MinMod2(PetscReal a,PetscReal b) { return (a*b<0) ? 0 : Sgn(a)*PetscMin(PetscAbs(a),PetscAbs(b)); }
 46: static inline PetscReal MaxMod2(PetscReal a,PetscReal b) { return (a*b<0) ? 0 : Sgn(a)*PetscMax(PetscAbs(a),PetscAbs(b)); }
 47: 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))); }

 49: static inline PetscReal RangeMod(PetscReal a,PetscReal xmin,PetscReal xmax) { PetscReal range = xmax-xmin; return xmin +PetscFmodReal(range+PetscFmodReal(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 : (jL[i]*Sqr(jR[i]) + Sqr(jL[i])*jR[i]) / (Sqr(jL[i]) + Sqr(jR[i]) + 1e-15);
105: }
106: static void Limit_Koren(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt) /* differentiable */
107: { /* phi = (t + 2*t^2) / (2 - t + 2*t^2) */
108:   PetscInt i;
109:   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));
110: }
111: static void Limit_KorenSym(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt) /* differentiable */
112: { /* Symmetric version of above */
113:   PetscInt i;
114:   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));
115: }
116: static void Limit_Koren3(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
117: { /* Eq 11 of Cada-Torrilhon 2009 */
118:   PetscInt i;
119:   for (i=0; i<info->m; i++) lmt[i] = MinMod3(2*jL[i],(jL[i]+2*jR[i])/3,2*jR[i]);
120: }
121: static PetscReal CadaTorrilhonPhiHatR_Eq13(PetscReal L,PetscReal R)
122: {
123:   return PetscMax(0,PetscMin((L+2*R)/3,PetscMax(-0.5*L,PetscMin(2*L,PetscMin((L+2*R)/3,1.6*R)))));
124: }
125: static void Limit_CadaTorrilhon2(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
126: { /* Cada-Torrilhon 2009, Eq 13 */
127:   PetscInt i;
128:   for (i=0; i<info->m; i++) lmt[i] = CadaTorrilhonPhiHatR_Eq13(jL[i],jR[i]);
129: }
130: static void Limit_CadaTorrilhon3R(PetscReal r,LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
131: { /* Cada-Torrilhon 2009, Eq 22 */
132:   /* They recommend 0.001 < r < 1, but larger values are more accurate in smooth regions */
133:   const PetscReal eps = 1e-7,hx = info->hx;
134:   PetscInt        i;
135:   for (i=0; i<info->m; i++) {
136:     const PetscReal eta = (Sqr(jL[i]) + Sqr(jR[i])) / Sqr(r*hx);
137:     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]))));
138:   }
139: }
140: static void Limit_CadaTorrilhon3R0p1(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
141: {
142:   Limit_CadaTorrilhon3R(0.1,info,jL,jR,lmt);
143: }
144: static void Limit_CadaTorrilhon3R1(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
145: {
146:   Limit_CadaTorrilhon3R(1,info,jL,jR,lmt);
147: }
148: static void Limit_CadaTorrilhon3R10(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
149: {
150:   Limit_CadaTorrilhon3R(10,info,jL,jR,lmt);
151: }
152: static void Limit_CadaTorrilhon3R100(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
153: {
154:   Limit_CadaTorrilhon3R(100,info,jL,jR,lmt);
155: }

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

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

164: typedef struct {
165:   PetscErrorCode      (*sample)(void*,PetscInt,FVBCType,PetscReal,PetscReal,PetscReal,PetscReal,PetscReal*);
166:   RiemannFunction     riemann;
167:   ReconstructFunction characteristic;
168:   PetscErrorCode      (*destroy)(void*);
169:   void                *user;
170:   PetscInt            dof;
171:   char                *fieldname[16];
172: } PhysicsCtx;

174: typedef struct {
175:   void        (*limit)(LimitInfo,const PetscScalar*,const PetscScalar*,PetscScalar*);
176:   PhysicsCtx  physics;
177:   MPI_Comm    comm;
178:   char        prefix[256];

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

188:   PetscReal   cfl_idt;            /* Max allowable value of 1/Delta t */
189:   PetscReal   cfl;
190:   PetscReal   xmin,xmax;
191:   PetscInt    initial;
192:   PetscBool   exact;
193:   FVBCType    bctype;
194: } FVCtx;

196: PetscErrorCode RiemannListAdd(PetscFunctionList *flist,const char *name,RiemannFunction rsolve)
197: {
199:   PetscFunctionListAdd(flist,name,rsolve);
200:   return 0;
201: }

203: PetscErrorCode RiemannListFind(PetscFunctionList flist,const char *name,RiemannFunction *rsolve)
204: {
206:   PetscFunctionListFind(flist,name,rsolve);
208:   return 0;
209: }

211: PetscErrorCode ReconstructListAdd(PetscFunctionList *flist,const char *name,ReconstructFunction r)
212: {
214:   PetscFunctionListAdd(flist,name,r);
215:   return 0;
216: }

218: PetscErrorCode ReconstructListFind(PetscFunctionList flist,const char *name,ReconstructFunction *r)
219: {
221:   PetscFunctionListFind(flist,name,r);
223:   return 0;
224: }

226: /* --------------------------------- Physics ----------------------------------- */
227: /*
228:   Each physical model consists of Riemann solver and a function to determine the basis to use for reconstruction.  These
229:   are set with the PhysicsCreate_XXX function which allocates private storage and sets these methods as well as the
230:   number of fields and their names, and a function to deallocate private storage.
231: */

233: /* First a few functions useful to several different physics */
234: static PetscErrorCode PhysicsCharacteristic_Conservative(void *vctx,PetscInt m,const PetscScalar *u,PetscScalar *X,PetscScalar *Xi,PetscReal *speeds)
235: {
236:   PetscInt i,j;

239:   for (i=0; i<m; i++) {
240:     for (j=0; j<m; j++) Xi[i*m+j] = X[i*m+j] = (PetscScalar)(i==j);
241:     speeds[i] = PETSC_MAX_REAL; /* Indicates invalid */
242:   }
243:   return 0;
244: }

246: static PetscErrorCode PhysicsDestroy_SimpleFree(void *vctx)
247: {
249:   PetscFree(vctx);
250:   return 0;
251: }

253: /* --------------------------------- Advection ----------------------------------- */

255: typedef struct {
256:   PetscReal a;                  /* advective velocity */
257: } AdvectCtx;

259: static PetscErrorCode PhysicsRiemann_Advect(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
260: {
261:   AdvectCtx *ctx = (AdvectCtx*)vctx;
262:   PetscReal speed;

265:   speed     = ctx->a;
266:   flux[0]   = PetscMax(0,speed)*uL[0] + PetscMin(0,speed)*uR[0];
267:   *maxspeed = speed;
268:   return 0;
269: }

271: static PetscErrorCode PhysicsCharacteristic_Advect(void *vctx,PetscInt m,const PetscScalar *u,PetscScalar *X,PetscScalar *Xi,PetscReal *speeds)
272: {
273:   AdvectCtx *ctx = (AdvectCtx*)vctx;

276:   X[0]      = 1.;
277:   Xi[0]     = 1.;
278:   speeds[0] = ctx->a;
279:   return 0;
280: }

282: static PetscErrorCode PhysicsSample_Advect(void *vctx,PetscInt initial,FVBCType bctype,PetscReal xmin,PetscReal xmax,PetscReal t,PetscReal x,PetscReal *u)
283: {
284:   AdvectCtx *ctx = (AdvectCtx*)vctx;
285:   PetscReal a    = ctx->a,x0;

288:   switch (bctype) {
289:     case FVBC_OUTFLOW:   x0 = x-a*t; break;
290:     case FVBC_PERIODIC: x0 = RangeMod(x-a*t,xmin,xmax); break;
291:     default: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_UNKNOWN_TYPE,"unknown BCType");
292:   }
293:   switch (initial) {
294:     case 0: u[0] = (x0 < 0) ? 1 : -1; break;
295:     case 1: u[0] = (x0 < 0) ? -1 : 1; break;
296:     case 2: u[0] = (0 < x0 && x0 < 1) ? 1 : 0; break;
297:     case 3: u[0] = PetscSinReal(2*PETSC_PI*x0); break;
298:     case 4: u[0] = PetscAbs(x0); break;
299:     case 5: u[0] = (x0 < 0 || x0 > 0.5) ? 0 : PetscSqr(PetscSinReal(2*PETSC_PI*x0)); break;
300:     case 6: u[0] = (x0 < 0) ? 0 : ((x0 < 1) ? x0 : ((x0 < 2) ? 2-x0 : 0)); break;
301:     case 7: u[0] = PetscPowReal(PetscSinReal(PETSC_PI*x0),10.0);break;
302:     default: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_UNKNOWN_TYPE,"unknown initial condition");
303:   }
304:   return 0;
305: }

307: static PetscErrorCode PhysicsCreate_Advect(FVCtx *ctx)
308: {
310:   AdvectCtx      *user;

313:   PetscNew(&user);
314:   ctx->physics.sample         = PhysicsSample_Advect;
315:   ctx->physics.riemann        = PhysicsRiemann_Advect;
316:   ctx->physics.characteristic = PhysicsCharacteristic_Advect;
317:   ctx->physics.destroy        = PhysicsDestroy_SimpleFree;
318:   ctx->physics.user           = user;
319:   ctx->physics.dof            = 1;
320:   PetscStrallocpy("u",&ctx->physics.fieldname[0]);
321:   user->a = 1;
322:   PetscOptionsBegin(ctx->comm,ctx->prefix,"Options for advection","");
323:   {
324:     PetscOptionsReal("-physics_advect_a","Speed","",user->a,&user->a,NULL);
325:   }
326:   PetscOptionsEnd();
327:   return 0;
328: }

330: /* --------------------------------- Burgers ----------------------------------- */

332: typedef struct {
333:   PetscReal lxf_speed;
334: } BurgersCtx;

336: static PetscErrorCode PhysicsSample_Burgers(void *vctx,PetscInt initial,FVBCType bctype,PetscReal xmin,PetscReal xmax,PetscReal t,PetscReal x,PetscReal *u)
337: {
340:   switch (initial) {
341:     case 0: u[0] = (x < 0) ? 1 : -1; break;
342:     case 1:
343:       if       (x < -t) u[0] = -1;
344:       else if  (x < t)  u[0] = x/t;
345:       else              u[0] = 1;
346:       break;
347:     case 2:
348:       if      (x <= 0)      u[0] = 0;
349:       else if (x < t)       u[0] = x/t;
350:       else if (x < 1+0.5*t) u[0] = 1;
351:       else                  u[0] = 0;
352:       break;
353:     case 3:
354:       if       (x < 0.2*t) u[0] = 0.2;
355:       else if  (x < t) u[0] = x/t;
356:       else             u[0] = 1;
357:       break;
358:     case 4:
360:       u[0] = 0.7 + 0.3*PetscSinReal(2*PETSC_PI*((x-xmin)/(xmax-xmin)));
361:       break;
362:     case 5:                     /* Pure shock solution */
363:       if (x < 0.5*t) u[0] = 1;
364:       else u[0] = 0;
365:       break;
366:     default: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_UNKNOWN_TYPE,"unknown initial condition");
367:   }
368:   return 0;
369: }

371: static PetscErrorCode PhysicsRiemann_Burgers_Exact(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
372: {
374:   if (uL[0] < uR[0]) {          /* rarefaction */
375:     flux[0] = (uL[0]*uR[0] < 0)
376:       ? 0                       /* sonic rarefaction */
377:       : 0.5*PetscMin(PetscSqr(uL[0]),PetscSqr(uR[0]));
378:   } else {                      /* shock */
379:     flux[0] = 0.5*PetscMax(PetscSqr(uL[0]),PetscSqr(uR[0]));
380:   }
381:   *maxspeed = (PetscAbs(uL[0]) > PetscAbs(uR[0])) ? uL[0] : uR[0];
382:   return 0;
383: }

385: static PetscErrorCode PhysicsRiemann_Burgers_Roe(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
386: {
387:   PetscReal speed;

390:   speed   = 0.5*(uL[0] + uR[0]);
391:   flux[0] = 0.25*(PetscSqr(uL[0]) + PetscSqr(uR[0])) - 0.5*PetscAbs(speed)*(uR[0]-uL[0]);
392:   if (uL[0] <= 0 && 0 <= uR[0]) flux[0] = 0; /* Entropy fix for sonic rarefaction */
393:   *maxspeed = speed;
394:   return 0;
395: }

397: static PetscErrorCode PhysicsRiemann_Burgers_LxF(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
398: {
399:   PetscReal   c;
400:   PetscScalar fL,fR;

403:   c         = ((BurgersCtx*)vctx)->lxf_speed;
404:   fL        = 0.5*PetscSqr(uL[0]);
405:   fR        = 0.5*PetscSqr(uR[0]);
406:   flux[0]   = 0.5*(fL + fR) - 0.5*c*(uR[0] - uL[0]);
407:   *maxspeed = c;
408:   return 0;
409: }

411: static PetscErrorCode PhysicsRiemann_Burgers_Rusanov(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
412: {
413:   PetscReal   c;
414:   PetscScalar fL,fR;

417:   c         = PetscMax(PetscAbs(uL[0]),PetscAbs(uR[0]));
418:   fL        = 0.5*PetscSqr(uL[0]);
419:   fR        = 0.5*PetscSqr(uR[0]);
420:   flux[0]   = 0.5*(fL + fR) - 0.5*c*(uR[0] - uL[0]);
421:   *maxspeed = c;
422:   return 0;
423: }

425: static PetscErrorCode PhysicsCreate_Burgers(FVCtx *ctx)
426: {
427:   BurgersCtx        *user;
428:   PetscErrorCode    ierr;
429:   RiemannFunction   r;
430:   PetscFunctionList rlist      = 0;
431:   char              rname[256] = "exact";

434:   PetscNew(&user);

436:   ctx->physics.sample         = PhysicsSample_Burgers;
437:   ctx->physics.characteristic = PhysicsCharacteristic_Conservative;
438:   ctx->physics.destroy        = PhysicsDestroy_SimpleFree;
439:   ctx->physics.user           = user;
440:   ctx->physics.dof            = 1;

442:   PetscStrallocpy("u",&ctx->physics.fieldname[0]);
443:   RiemannListAdd(&rlist,"exact",  PhysicsRiemann_Burgers_Exact);
444:   RiemannListAdd(&rlist,"roe",    PhysicsRiemann_Burgers_Roe);
445:   RiemannListAdd(&rlist,"lxf",    PhysicsRiemann_Burgers_LxF);
446:   RiemannListAdd(&rlist,"rusanov",PhysicsRiemann_Burgers_Rusanov);
447:   PetscOptionsBegin(ctx->comm,ctx->prefix,"Options for advection","");
448:   {
449:     PetscOptionsFList("-physics_burgers_riemann","Riemann solver","",rlist,rname,rname,sizeof(rname),NULL);
450:   }
451:   PetscOptionsEnd();
452:   RiemannListFind(rlist,rname,&r);
453:   PetscFunctionListDestroy(&rlist);
454:   ctx->physics.riemann = r;

456:   /* *
457:   * Hack to deal with LxF in semi-discrete form
458:   * max speed is 1 for the basic initial conditions (where |u| <= 1)
459:   * */
460:   if (r == PhysicsRiemann_Burgers_LxF) user->lxf_speed = 1;
461:   return 0;
462: }

464: /* --------------------------------- Traffic ----------------------------------- */

466: typedef struct {
467:   PetscReal lxf_speed;
468:   PetscReal a;
469: } TrafficCtx;

471: static inline PetscScalar TrafficFlux(PetscScalar a,PetscScalar u) { return a*u*(1-u); }

473: static PetscErrorCode PhysicsSample_Traffic(void *vctx,PetscInt initial,FVBCType bctype,PetscReal xmin,PetscReal xmax,PetscReal t,PetscReal x,PetscReal *u)
474: {
475:   PetscReal a = ((TrafficCtx*)vctx)->a;

479:   switch (initial) {
480:     case 0:
481:       u[0] = (-a*t < x) ? 2 : 0; break;
482:     case 1:
483:       if      (x < PetscMin(2*a*t,0.5+a*t)) u[0] = -1;
484:       else if (x < 1)                       u[0] = 0;
485:       else                                  u[0] = 1;
486:       break;
487:     case 2:
489:       u[0] = 0.7 + 0.3*PetscSinReal(2*PETSC_PI*((x-xmin)/(xmax-xmin)));
490:       break;
491:     default: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_UNKNOWN_TYPE,"unknown initial condition");
492:   }
493:   return 0;
494: }

496: static PetscErrorCode PhysicsRiemann_Traffic_Exact(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
497: {
498:   PetscReal a = ((TrafficCtx*)vctx)->a;

501:   if (uL[0] < uR[0]) {
502:     flux[0] = PetscMin(TrafficFlux(a,uL[0]),TrafficFlux(a,uR[0]));
503:   } else {
504:     flux[0] = (uR[0] < 0.5 && 0.5 < uL[0]) ? TrafficFlux(a,0.5) : PetscMax(TrafficFlux(a,uL[0]),TrafficFlux(a,uR[0]));
505:   }
506:   *maxspeed = a*MaxAbs(1-2*uL[0],1-2*uR[0]);
507:   return 0;
508: }

510: static PetscErrorCode PhysicsRiemann_Traffic_Roe(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
511: {
512:   PetscReal a = ((TrafficCtx*)vctx)->a;
513:   PetscReal speed;

516:   speed = a*(1 - (uL[0] + uR[0]));
517:   flux[0] = 0.5*(TrafficFlux(a,uL[0]) + TrafficFlux(a,uR[0])) - 0.5*PetscAbs(speed)*(uR[0]-uL[0]);
518:   *maxspeed = speed;
519:   return 0;
520: }

522: static PetscErrorCode PhysicsRiemann_Traffic_LxF(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
523: {
524:   TrafficCtx *phys = (TrafficCtx*)vctx;
525:   PetscReal  a     = phys->a;
526:   PetscReal  speed;

529:   speed     = a*(1 - (uL[0] + uR[0]));
530:   flux[0]   = 0.5*(TrafficFlux(a,uL[0]) + TrafficFlux(a,uR[0])) - 0.5*phys->lxf_speed*(uR[0]-uL[0]);
531:   *maxspeed = speed;
532:   return 0;
533: }

535: static PetscErrorCode PhysicsRiemann_Traffic_Rusanov(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
536: {
537:   PetscReal a = ((TrafficCtx*)vctx)->a;
538:   PetscReal speed;

541:   speed     = a*PetscMax(PetscAbs(1-2*uL[0]),PetscAbs(1-2*uR[0]));
542:   flux[0]   = 0.5*(TrafficFlux(a,uL[0]) + TrafficFlux(a,uR[0])) - 0.5*speed*(uR[0]-uL[0]);
543:   *maxspeed = speed;
544:   return 0;
545: }

547: static PetscErrorCode PhysicsCreate_Traffic(FVCtx *ctx)
548: {
549:   PetscErrorCode    ierr;
550:   TrafficCtx        *user;
551:   RiemannFunction   r;
552:   PetscFunctionList rlist      = 0;
553:   char              rname[256] = "exact";

556:   PetscNew(&user);
557:   ctx->physics.sample         = PhysicsSample_Traffic;
558:   ctx->physics.characteristic = PhysicsCharacteristic_Conservative;
559:   ctx->physics.destroy        = PhysicsDestroy_SimpleFree;
560:   ctx->physics.user           = user;
561:   ctx->physics.dof            = 1;

563:   PetscStrallocpy("density",&ctx->physics.fieldname[0]);
564:   user->a = 0.5;
565:   RiemannListAdd(&rlist,"exact",  PhysicsRiemann_Traffic_Exact);
566:   RiemannListAdd(&rlist,"roe",    PhysicsRiemann_Traffic_Roe);
567:   RiemannListAdd(&rlist,"lxf",    PhysicsRiemann_Traffic_LxF);
568:   RiemannListAdd(&rlist,"rusanov",PhysicsRiemann_Traffic_Rusanov);
569:   PetscOptionsBegin(ctx->comm,ctx->prefix,"Options for Traffic","");
570:     PetscOptionsReal("-physics_traffic_a","Flux = a*u*(1-u)","",user->a,&user->a,NULL);
571:     PetscOptionsFList("-physics_traffic_riemann","Riemann solver","",rlist,rname,rname,sizeof(rname),NULL);
572:   PetscOptionsEnd();

574:   RiemannListFind(rlist,rname,&r);
575:   PetscFunctionListDestroy(&rlist);

577:   ctx->physics.riemann = r;

579:   /* *
580:   * Hack to deal with LxF in semi-discrete form
581:   * max speed is 3*a for the basic initial conditions (-1 <= u <= 2)
582:   * */
583:   if (r == PhysicsRiemann_Traffic_LxF) user->lxf_speed = 3*user->a;
584:   return 0;
585: }

587: /* --------------------------------- Linear Acoustics ----------------------------------- */

589: /* Flux: u_t + (A u)_x
590:  * z = sqrt(rho*bulk), c = sqrt(rho/bulk)
591:  * Spectral decomposition: A = R * D * Rinv
592:  * [    cz] = [-z   z] [-c    ] [-1/2z  1/2]
593:  * [c/z   ] = [ 1   1] [     c] [ 1/2z  1/2]
594:  *
595:  * We decompose this into the left-traveling waves Al = R * D^- Rinv
596:  * and the right-traveling waves Ar = R * D^+ * Rinv
597:  * Multiplying out these expressions produces the following two matrices
598:  */

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

605: PETSC_UNUSED static inline void AcousticsFlux(AcousticsCtx *ctx,const PetscScalar *u,PetscScalar *f)
606: {
607:   f[0] = ctx->c*ctx->z*u[1];
608:   f[1] = ctx->c/ctx->z*u[0];
609: }

611: static PetscErrorCode PhysicsCharacteristic_Acoustics(void *vctx,PetscInt m,const PetscScalar *u,PetscScalar *X,PetscScalar *Xi,PetscReal *speeds)
612: {
613:   AcousticsCtx *phys = (AcousticsCtx*)vctx;
614:   PetscReal    z     = phys->z,c = phys->c;

617:   X[0*2+0]  = -z;
618:   X[0*2+1]  = z;
619:   X[1*2+0]  = 1;
620:   X[1*2+1]  = 1;
621:   Xi[0*2+0] = -1./(2*z);
622:   Xi[0*2+1] = 1./2;
623:   Xi[1*2+0] = 1./(2*z);
624:   Xi[1*2+1] = 1./2;
625:   speeds[0] = -c;
626:   speeds[1] = c;
627:   return 0;
628: }

630: static PetscErrorCode PhysicsSample_Acoustics_Initial(AcousticsCtx *phys,PetscInt initial,PetscReal xmin,PetscReal xmax,PetscReal x,PetscReal *u)
631: {
633:   switch (initial) {
634:   case 0:
635:     u[0] = (PetscAbs((x - xmin)/(xmax - xmin) - 0.2) < 0.1) ? 1 : 0.5;
636:     u[1] = (PetscAbs((x - xmin)/(xmax - xmin) - 0.7) < 0.1) ? 1 : -0.5;
637:     break;
638:   case 1:
639:     u[0] = PetscCosReal(3 * 2*PETSC_PI*x/(xmax-xmin));
640:     u[1] = PetscExpReal(-PetscSqr(x - (xmax + xmin)/2) / (2*PetscSqr(0.2*(xmax - xmin)))) - 0.5;
641:     break;
642:   default: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_UNKNOWN_TYPE,"unknown initial condition");
643:   }
644:   return 0;
645: }

647: static PetscErrorCode PhysicsSample_Acoustics(void *vctx,PetscInt initial,FVBCType bctype,PetscReal xmin,PetscReal xmax,PetscReal t,PetscReal x,PetscReal *u)
648: {
649:   AcousticsCtx   *phys = (AcousticsCtx*)vctx;
650:   PetscReal      c     = phys->c;
651:   PetscReal      x0a,x0b,u0a[2],u0b[2],tmp[2];
652:   PetscReal      X[2][2],Xi[2][2],dummy[2];

655:   switch (bctype) {
656:   case FVBC_OUTFLOW:
657:     x0a = x+c*t;
658:     x0b = x-c*t;
659:     break;
660:   case FVBC_PERIODIC:
661:     x0a = RangeMod(x+c*t,xmin,xmax);
662:     x0b = RangeMod(x-c*t,xmin,xmax);
663:     break;
664:   default: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_UNKNOWN_TYPE,"unknown BCType");
665:   }
666:   PhysicsSample_Acoustics_Initial(phys,initial,xmin,xmax,x0a,u0a);
667:   PhysicsSample_Acoustics_Initial(phys,initial,xmin,xmax,x0b,u0b);
668:   PhysicsCharacteristic_Acoustics(vctx,2,u,&X[0][0],&Xi[0][0],dummy);
669:   tmp[0] = Xi[0][0]*u0a[0] + Xi[0][1]*u0a[1];
670:   tmp[1] = Xi[1][0]*u0b[0] + Xi[1][1]*u0b[1];
671:   u[0]   = X[0][0]*tmp[0] + X[0][1]*tmp[1];
672:   u[1]   = X[1][0]*tmp[0] + X[1][1]*tmp[1];
673:   return 0;
674: }

676: static PetscErrorCode PhysicsRiemann_Acoustics_Exact(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
677: {
678:   AcousticsCtx *phys = (AcousticsCtx*)vctx;
679:   PetscReal    c     = phys->c,z = phys->z;
680:   PetscReal
681:     Al[2][2] = {{-c/2     , c*z/2  },
682:                 {c/(2*z)  , -c/2   }}, /* Left traveling waves */
683:     Ar[2][2] = {{c/2      , c*z/2  },
684:                 {c/(2*z)  , c/2    }}; /* Right traveling waves */

687:   flux[0]   = Al[0][0]*uR[0] + Al[0][1]*uR[1] + Ar[0][0]*uL[0] + Ar[0][1]*uL[1];
688:   flux[1]   = Al[1][0]*uR[0] + Al[1][1]*uR[1] + Ar[1][0]*uL[0] + Ar[1][1]*uL[1];
689:   *maxspeed = c;
690:   return 0;
691: }

693: static PetscErrorCode PhysicsCreate_Acoustics(FVCtx *ctx)
694: {
695:   PetscErrorCode    ierr;
696:   AcousticsCtx      *user;
697:   PetscFunctionList rlist      = 0,rclist = 0;
698:   char              rname[256] = "exact",rcname[256] = "characteristic";

701:   PetscNew(&user);
702:   ctx->physics.sample         = PhysicsSample_Acoustics;
703:   ctx->physics.destroy        = PhysicsDestroy_SimpleFree;
704:   ctx->physics.user           = user;
705:   ctx->physics.dof            = 2;

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

710:   user->c = 1;
711:   user->z = 1;

713:   RiemannListAdd(&rlist,"exact",  PhysicsRiemann_Acoustics_Exact);
714:   ReconstructListAdd(&rclist,"characteristic",PhysicsCharacteristic_Acoustics);
715:   ReconstructListAdd(&rclist,"conservative",PhysicsCharacteristic_Conservative);
716:   PetscOptionsBegin(ctx->comm,ctx->prefix,"Options for linear Acoustics","");
717:   {
718:     PetscOptionsReal("-physics_acoustics_c","c = sqrt(bulk/rho)","",user->c,&user->c,NULL);
719:     PetscOptionsReal("-physics_acoustics_z","z = sqrt(bulk*rho)","",user->z,&user->z,NULL);
720:     PetscOptionsFList("-physics_acoustics_riemann","Riemann solver","",rlist,rname,rname,sizeof(rname),NULL);
721:     PetscOptionsFList("-physics_acoustics_reconstruct","Reconstruction","",rclist,rcname,rcname,sizeof(rcname),NULL);
722:   }
723:   PetscOptionsEnd();
724:   RiemannListFind(rlist,rname,&ctx->physics.riemann);
725:   ReconstructListFind(rclist,rcname,&ctx->physics.characteristic);
726:   PetscFunctionListDestroy(&rlist);
727:   PetscFunctionListDestroy(&rclist);
728:   return 0;
729: }

731: /* --------------------------------- Isothermal Gas Dynamics ----------------------------------- */

733: typedef struct {
734:   PetscReal acoustic_speed;
735: } IsoGasCtx;

737: static inline void IsoGasFlux(PetscReal c,const PetscScalar *u,PetscScalar *f)
738: {
739:   f[0] = u[1];
740:   f[1] = PetscSqr(u[1])/u[0] + c*c*u[0];
741: }

743: static PetscErrorCode PhysicsSample_IsoGas(void *vctx,PetscInt initial,FVBCType bctype,PetscReal xmin,PetscReal xmax,PetscReal t,PetscReal x,PetscReal *u)
744: {
747:   switch (initial) {
748:     case 0:
749:       u[0] = (x < 0) ? 1 : 0.5;
750:       u[1] = (x < 0) ? 1 : 0.7;
751:       break;
752:     case 1:
753:       u[0] = 1+0.5*PetscSinReal(2*PETSC_PI*x);
754:       u[1] = 1*u[0];
755:       break;
756:     default: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_UNKNOWN_TYPE,"unknown initial condition");
757:   }
758:   return 0;
759: }

761: static PetscErrorCode PhysicsRiemann_IsoGas_Roe(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
762: {
763:   IsoGasCtx   *phys = (IsoGasCtx*)vctx;
764:   PetscReal   c     = phys->acoustic_speed;
765:   PetscScalar ubar,du[2],a[2],fL[2],fR[2],lam[2],ustar[2],R[2][2];
766:   PetscInt    i;

769:   ubar = (uL[1]/PetscSqrtScalar(uL[0]) + uR[1]/PetscSqrtScalar(uR[0])) / (PetscSqrtScalar(uL[0]) + PetscSqrtScalar(uR[0]));
770:   /* write fluxuations in characteristic basis */
771:   du[0] = uR[0] - uL[0];
772:   du[1] = uR[1] - uL[1];
773:   a[0]  = (1/(2*c)) * ((ubar + c)*du[0] - du[1]);
774:   a[1]  = (1/(2*c)) * ((-ubar + c)*du[0] + du[1]);
775:   /* wave speeds */
776:   lam[0] = ubar - c;
777:   lam[1] = ubar + c;
778:   /* Right eigenvectors */
779:   R[0][0] = 1; R[0][1] = ubar-c;
780:   R[1][0] = 1; R[1][1] = ubar+c;
781:   /* Compute state in star region (between the 1-wave and 2-wave) */
782:   for (i=0; i<2; i++) ustar[i] = uL[i] + a[0]*R[0][i];
783:   if (uL[1]/uL[0] < c && c < ustar[1]/ustar[0]) { /* 1-wave is sonic rarefaction */
784:     PetscScalar ufan[2];
785:     ufan[0] = uL[0]*PetscExpScalar(uL[1]/(uL[0]*c) - 1);
786:     ufan[1] = c*ufan[0];
787:     IsoGasFlux(c,ufan,flux);
788:   } else if (ustar[1]/ustar[0] < -c && -c < uR[1]/uR[0]) { /* 2-wave is sonic rarefaction */
789:     PetscScalar ufan[2];
790:     ufan[0] = uR[0]*PetscExpScalar(-uR[1]/(uR[0]*c) - 1);
791:     ufan[1] = -c*ufan[0];
792:     IsoGasFlux(c,ufan,flux);
793:   } else {                      /* Centered form */
794:     IsoGasFlux(c,uL,fL);
795:     IsoGasFlux(c,uR,fR);
796:     for (i=0; i<2; i++) {
797:       PetscScalar absdu = PetscAbsScalar(lam[0])*a[0]*R[0][i] + PetscAbsScalar(lam[1])*a[1]*R[1][i];
798:       flux[i] = 0.5*(fL[i]+fR[i]) - 0.5*absdu;
799:     }
800:   }
801:   *maxspeed = MaxAbs(lam[0],lam[1]);
802:   return 0;
803: }

805: static PetscErrorCode PhysicsRiemann_IsoGas_Exact(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
806: {
807:   IsoGasCtx                   *phys = (IsoGasCtx*)vctx;
808:   PetscReal                   c     = phys->acoustic_speed;
809:   PetscScalar                 ustar[2];
810:   struct {PetscScalar rho,u;} L = {uL[0],uL[1]/uL[0]},R = {uR[0],uR[1]/uR[0]},star;
811:   PetscInt                    i;

815:   {
816:     /* Solve for star state */
817:     PetscScalar res,tmp,rho = 0.5*(L.rho + R.rho); /* initial guess */
818:     for (i=0; i<20; i++) {
819:       PetscScalar fr,fl,dfr,dfl;
820:       fl = (L.rho < rho)
821:         ? (rho-L.rho)/PetscSqrtScalar(L.rho*rho)       /* shock */
822:         : PetscLogScalar(rho) - PetscLogScalar(L.rho); /* rarefaction */
823:       fr = (R.rho < rho)
824:         ? (rho-R.rho)/PetscSqrtScalar(R.rho*rho)       /* shock */
825:         : PetscLogScalar(rho) - PetscLogScalar(R.rho); /* rarefaction */
826:       res = R.u-L.u + c*(fr+fl);
828:       if (PetscAbsScalar(res) < 1e-10) {
829:         star.rho = rho;
830:         star.u   = L.u - c*fl;
831:         goto converged;
832:       }
833:       dfl = (L.rho < rho) ? 1/PetscSqrtScalar(L.rho*rho)*(1 - 0.5*(rho-L.rho)/rho) : 1/rho;
834:       dfr = (R.rho < rho) ? 1/PetscSqrtScalar(R.rho*rho)*(1 - 0.5*(rho-R.rho)/rho) : 1/rho;
835:       tmp = rho - res/(c*(dfr+dfl));
836:       if (tmp <= 0) rho /= 2;   /* Guard against Newton shooting off to a negative density */
837:       else rho = tmp;
839:     }
840:     SETERRQ(PETSC_COMM_SELF,PETSC_ERR_CONV_FAILED,"Newton iteration for star.rho diverged after %D iterations",i);
841:   }
842: converged:
843:   if (L.u-c < 0 && 0 < star.u-c) { /* 1-wave is sonic rarefaction */
844:     PetscScalar ufan[2];
845:     ufan[0] = L.rho*PetscExpScalar(L.u/c - 1);
846:     ufan[1] = c*ufan[0];
847:     IsoGasFlux(c,ufan,flux);
848:   } else if (star.u+c < 0 && 0 < R.u+c) { /* 2-wave is sonic rarefaction */
849:     PetscScalar ufan[2];
850:     ufan[0] = R.rho*PetscExpScalar(-R.u/c - 1);
851:     ufan[1] = -c*ufan[0];
852:     IsoGasFlux(c,ufan,flux);
853:   } 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)) {
854:     /* 1-wave is supersonic rarefaction, or supersonic shock */
855:     IsoGasFlux(c,uL,flux);
856:   } 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)) {
857:     /* 2-wave is supersonic rarefaction or supersonic shock */
858:     IsoGasFlux(c,uR,flux);
859:   } else {
860:     ustar[0] = star.rho;
861:     ustar[1] = star.rho*star.u;
862:     IsoGasFlux(c,ustar,flux);
863:   }
864:   *maxspeed = MaxAbs(MaxAbs(star.u-c,star.u+c),MaxAbs(L.u-c,R.u+c));
865:   return 0;
866: }

868: static PetscErrorCode PhysicsRiemann_IsoGas_Rusanov(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
869: {
870:   IsoGasCtx                   *phys = (IsoGasCtx*)vctx;
871:   PetscScalar                 c = phys->acoustic_speed,fL[2],fR[2],s;
872:   struct {PetscScalar rho,u;} L = {uL[0],uL[1]/uL[0]},R = {uR[0],uR[1]/uR[0]};

876:   IsoGasFlux(c,uL,fL);
877:   IsoGasFlux(c,uR,fR);
878:   s         = PetscMax(PetscAbs(L.u),PetscAbs(R.u))+c;
879:   flux[0]   = 0.5*(fL[0] + fR[0]) + 0.5*s*(uL[0] - uR[0]);
880:   flux[1]   = 0.5*(fL[1] + fR[1]) + 0.5*s*(uL[1] - uR[1]);
881:   *maxspeed = s;
882:   return 0;
883: }

885: static PetscErrorCode PhysicsCharacteristic_IsoGas(void *vctx,PetscInt m,const PetscScalar *u,PetscScalar *X,PetscScalar *Xi,PetscReal *speeds)
886: {
887:   IsoGasCtx      *phys = (IsoGasCtx*)vctx;
888:   PetscReal      c     = phys->acoustic_speed;

891:   speeds[0] = u[1]/u[0] - c;
892:   speeds[1] = u[1]/u[0] + c;
893:   X[0*2+0]  = 1;
894:   X[0*2+1]  = speeds[0];
895:   X[1*2+0]  = 1;
896:   X[1*2+1]  = speeds[1];
897:   PetscArraycpy(Xi,X,4);
898:   PetscKernel_A_gets_inverse_A_2(Xi,0,PETSC_FALSE,NULL);
899:   return 0;
900: }

902: static PetscErrorCode PhysicsCreate_IsoGas(FVCtx *ctx)
903: {
904:   PetscErrorCode    ierr;
905:   IsoGasCtx         *user;
906:   PetscFunctionList rlist = 0,rclist = 0;
907:   char              rname[256] = "exact",rcname[256] = "characteristic";

910:   PetscNew(&user);
911:   ctx->physics.sample         = PhysicsSample_IsoGas;
912:   ctx->physics.destroy        = PhysicsDestroy_SimpleFree;
913:   ctx->physics.user           = user;
914:   ctx->physics.dof            = 2;

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

919:   user->acoustic_speed = 1;

921:   RiemannListAdd(&rlist,"exact",  PhysicsRiemann_IsoGas_Exact);
922:   RiemannListAdd(&rlist,"roe",    PhysicsRiemann_IsoGas_Roe);
923:   RiemannListAdd(&rlist,"rusanov",PhysicsRiemann_IsoGas_Rusanov);
924:   ReconstructListAdd(&rclist,"characteristic",PhysicsCharacteristic_IsoGas);
925:   ReconstructListAdd(&rclist,"conservative",PhysicsCharacteristic_Conservative);
926:   PetscOptionsBegin(ctx->comm,ctx->prefix,"Options for IsoGas","");
927:     PetscOptionsReal("-physics_isogas_acoustic_speed","Acoustic speed","",user->acoustic_speed,&user->acoustic_speed,NULL);
928:     PetscOptionsFList("-physics_isogas_riemann","Riemann solver","",rlist,rname,rname,sizeof(rname),NULL);
929:     PetscOptionsFList("-physics_isogas_reconstruct","Reconstruction","",rclist,rcname,rcname,sizeof(rcname),NULL);
930:   PetscOptionsEnd();
931:   RiemannListFind(rlist,rname,&ctx->physics.riemann);
932:   ReconstructListFind(rclist,rcname,&ctx->physics.characteristic);
933:   PetscFunctionListDestroy(&rlist);
934:   PetscFunctionListDestroy(&rclist);
935:   return 0;
936: }

938: /* --------------------------------- Shallow Water ----------------------------------- */
939: typedef struct {
940:   PetscReal gravity;
941: } ShallowCtx;

943: static inline void ShallowFlux(ShallowCtx *phys,const PetscScalar *u,PetscScalar *f)
944: {
945:   f[0] = u[1];
946:   f[1] = PetscSqr(u[1])/u[0] + 0.5*phys->gravity*PetscSqr(u[0]);
947: }

949: static PetscErrorCode PhysicsRiemann_Shallow_Exact(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
950: {
951:   ShallowCtx                *phys = (ShallowCtx*)vctx;
952:   PetscScalar               g    = phys->gravity,ustar[2],cL,cR,c,cstar;
953:   struct {PetscScalar h,u;} L = {uL[0],uL[1]/uL[0]},R = {uR[0],uR[1]/uR[0]},star;
954:   PetscInt                  i;

958:   cL = PetscSqrtScalar(g*L.h);
959:   cR = PetscSqrtScalar(g*R.h);
960:   c  = PetscMax(cL,cR);
961:   {
962:     /* Solve for star state */
963:     const PetscInt maxits = 50;
964:     PetscScalar tmp,res,res0=0,h0,h = 0.5*(L.h + R.h); /* initial guess */
965:     h0 = h;
966:     for (i=0; i<maxits; i++) {
967:       PetscScalar fr,fl,dfr,dfl;
968:       fl = (L.h < h)
969:         ? PetscSqrtScalar(0.5*g*(h*h - L.h*L.h)*(1/L.h - 1/h)) /* shock */
970:         : 2*PetscSqrtScalar(g*h) - 2*PetscSqrtScalar(g*L.h);   /* rarefaction */
971:       fr = (R.h < h)
972:         ? PetscSqrtScalar(0.5*g*(h*h - R.h*R.h)*(1/R.h - 1/h)) /* shock */
973:         : 2*PetscSqrtScalar(g*h) - 2*PetscSqrtScalar(g*R.h);   /* rarefaction */
974:       res = R.u - L.u + fr + fl;
976:       if (PetscAbsScalar(res) < 1e-8 || (i > 0 && PetscAbsScalar(h-h0) < 1e-8)) {
977:         star.h = h;
978:         star.u = L.u - fl;
979:         goto converged;
980:       } else if (i > 0 && PetscAbsScalar(res) >= PetscAbsScalar(res0)) {        /* Line search */
981:         h = 0.8*h0 + 0.2*h;
982:         continue;
983:       }
984:       /* Accept the last step and take another */
985:       res0 = res;
986:       h0 = h;
987:       dfl = (L.h < h) ? 0.5/fl*0.5*g*(-L.h*L.h/(h*h) - 1 + 2*h/L.h) : PetscSqrtScalar(g/h);
988:       dfr = (R.h < h) ? 0.5/fr*0.5*g*(-R.h*R.h/(h*h) - 1 + 2*h/R.h) : PetscSqrtScalar(g/h);
989:       tmp = h - res/(dfr+dfl);
990:       if (tmp <= 0) h /= 2;   /* Guard against Newton shooting off to a negative thickness */
991:       else h = tmp;
993:     }
994:     SETERRQ(PETSC_COMM_SELF,PETSC_ERR_CONV_FAILED,"Newton iteration for star.h diverged after %D iterations",i);
995:   }
996: converged:
997:   cstar = PetscSqrtScalar(g*star.h);
998:   if (L.u-cL < 0 && 0 < star.u-cstar) { /* 1-wave is sonic rarefaction */
999:     PetscScalar ufan[2];
1000:     ufan[0] = 1/g*PetscSqr(L.u/3 + 2./3*cL);
1001:     ufan[1] = PetscSqrtScalar(g*ufan[0])*ufan[0];
1002:     ShallowFlux(phys,ufan,flux);
1003:   } else if (star.u+cstar < 0 && 0 < R.u+cR) { /* 2-wave is sonic rarefaction */
1004:     PetscScalar ufan[2];
1005:     ufan[0] = 1/g*PetscSqr(R.u/3 - 2./3*cR);
1006:     ufan[1] = -PetscSqrtScalar(g*ufan[0])*ufan[0];
1007:     ShallowFlux(phys,ufan,flux);
1008:   } 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)) {
1009:     /* 1-wave is right-travelling shock (supersonic) */
1010:     ShallowFlux(phys,uL,flux);
1011:   } 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)) {
1012:     /* 2-wave is left-travelling shock (supersonic) */
1013:     ShallowFlux(phys,uR,flux);
1014:   } else {
1015:     ustar[0] = star.h;
1016:     ustar[1] = star.h*star.u;
1017:     ShallowFlux(phys,ustar,flux);
1018:   }
1019:   *maxspeed = MaxAbs(MaxAbs(star.u-cstar,star.u+cstar),MaxAbs(L.u-cL,R.u+cR));
1020:   return 0;
1021: }

1023: static PetscErrorCode PhysicsRiemann_Shallow_Rusanov(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
1024: {
1025:   ShallowCtx                *phys = (ShallowCtx*)vctx;
1026:   PetscScalar               g = phys->gravity,fL[2],fR[2],s;
1027:   struct {PetscScalar h,u;} L = {uL[0],uL[1]/uL[0]},R = {uR[0],uR[1]/uR[0]};

1031:   ShallowFlux(phys,uL,fL);
1032:   ShallowFlux(phys,uR,fR);
1033:   s         = PetscMax(PetscAbs(L.u)+PetscSqrtScalar(g*L.h),PetscAbs(R.u)+PetscSqrtScalar(g*R.h));
1034:   flux[0]   = 0.5*(fL[0] + fR[0]) + 0.5*s*(uL[0] - uR[0]);
1035:   flux[1]   = 0.5*(fL[1] + fR[1]) + 0.5*s*(uL[1] - uR[1]);
1036:   *maxspeed = s;
1037:   return 0;
1038: }

1040: static PetscErrorCode PhysicsCharacteristic_Shallow(void *vctx,PetscInt m,const PetscScalar *u,PetscScalar *X,PetscScalar *Xi,PetscReal *speeds)
1041: {
1042:   ShallowCtx     *phys = (ShallowCtx*)vctx;
1043:   PetscReal      c;

1046:   c         = PetscSqrtScalar(u[0]*phys->gravity);
1047:   speeds[0] = u[1]/u[0] - c;
1048:   speeds[1] = u[1]/u[0] + c;
1049:   X[0*2+0]  = 1;
1050:   X[0*2+1]  = speeds[0];
1051:   X[1*2+0]  = 1;
1052:   X[1*2+1]  = speeds[1];
1053:   PetscArraycpy(Xi,X,4);
1054:   PetscKernel_A_gets_inverse_A_2(Xi,0,PETSC_FALSE,NULL);
1055:   return 0;
1056: }

1058: static PetscErrorCode PhysicsCreate_Shallow(FVCtx *ctx)
1059: {
1060:   PetscErrorCode    ierr;
1061:   ShallowCtx        *user;
1062:   PetscFunctionList rlist = 0,rclist = 0;
1063:   char              rname[256] = "exact",rcname[256] = "characteristic";

1066:   PetscNew(&user);
1067:   /* Shallow water and Isothermal Gas dynamics are similar so we reuse initial conditions for now */
1068:   ctx->physics.sample         = PhysicsSample_IsoGas;
1069:   ctx->physics.destroy        = PhysicsDestroy_SimpleFree;
1070:   ctx->physics.user           = user;
1071:   ctx->physics.dof            = 2;

1073:   PetscStrallocpy("density",&ctx->physics.fieldname[0]);
1074:   PetscStrallocpy("momentum",&ctx->physics.fieldname[1]);

1076:   user->gravity = 1;

1078:   RiemannListAdd(&rlist,"exact",  PhysicsRiemann_Shallow_Exact);
1079:   RiemannListAdd(&rlist,"rusanov",PhysicsRiemann_Shallow_Rusanov);
1080:   ReconstructListAdd(&rclist,"characteristic",PhysicsCharacteristic_Shallow);
1081:   ReconstructListAdd(&rclist,"conservative",PhysicsCharacteristic_Conservative);
1082:   PetscOptionsBegin(ctx->comm,ctx->prefix,"Options for Shallow","");
1083:     PetscOptionsReal("-physics_shallow_gravity","Gravity","",user->gravity,&user->gravity,NULL);
1084:     PetscOptionsFList("-physics_shallow_riemann","Riemann solver","",rlist,rname,rname,sizeof(rname),NULL);
1085:     PetscOptionsFList("-physics_shallow_reconstruct","Reconstruction","",rclist,rcname,rcname,sizeof(rcname),NULL);
1086:   PetscOptionsEnd();
1087:   RiemannListFind(rlist,rname,&ctx->physics.riemann);
1088:   ReconstructListFind(rclist,rcname,&ctx->physics.characteristic);
1089:   PetscFunctionListDestroy(&rlist);
1090:   PetscFunctionListDestroy(&rclist);
1091:   return 0;
1092: }

1094: /* --------------------------------- Finite Volume Solver ----------------------------------- */

1096: static PetscErrorCode FVRHSFunction(TS ts,PetscReal time,Vec X,Vec F,void *vctx)
1097: {
1098:   FVCtx             *ctx = (FVCtx*)vctx;
1099:   PetscInt          i,j,k,Mx,dof,xs,xm;
1100:   PetscReal         hx,cfl_idt = 0;
1101:   PetscScalar       *x,*f,*slope;
1102:   Vec               Xloc;
1103:   DM                da;

1106:   TSGetDM(ts,&da);
1107:   DMGetLocalVector(da,&Xloc);
1108:   DMDAGetInfo(da,0, &Mx,0,0, 0,0,0, &dof,0,0,0,0,0);
1109:   hx   = (ctx->xmax - ctx->xmin)/Mx;
1110:   DMGlobalToLocalBegin(da,X,INSERT_VALUES,Xloc);
1111:   DMGlobalToLocalEnd  (da,X,INSERT_VALUES,Xloc);

1113:   VecZeroEntries(F);

1115:   DMDAVecGetArray(da,Xloc,&x);
1116:   DMDAVecGetArray(da,F,&f);
1117:   DMDAGetArray(da,PETSC_TRUE,&slope);

1119:   DMDAGetCorners(da,&xs,0,0,&xm,0,0);

1121:   if (ctx->bctype == FVBC_OUTFLOW) {
1122:     for (i=xs-2; i<0; i++) {
1123:       for (j=0; j<dof; j++) x[i*dof+j] = x[j];
1124:     }
1125:     for (i=Mx; i<xs+xm+2; i++) {
1126:       for (j=0; j<dof; j++) x[i*dof+j] = x[(xs+xm-1)*dof+j];
1127:     }
1128:   }
1129:   for (i=xs-1; i<xs+xm+1; i++) {
1130:     struct _LimitInfo info;
1131:     PetscScalar       *cjmpL,*cjmpR;
1132:     /* Determine the right eigenvectors R, where A = R \Lambda R^{-1} */
1133:     (*ctx->physics.characteristic)(ctx->physics.user,dof,&x[i*dof],ctx->R,ctx->Rinv,ctx->speeds);
1134:     /* Evaluate jumps across interfaces (i-1, i) and (i, i+1), put in characteristic basis */
1135:     PetscArrayzero(ctx->cjmpLR,2*dof);
1136:     cjmpL = &ctx->cjmpLR[0];
1137:     cjmpR = &ctx->cjmpLR[dof];
1138:     for (j=0; j<dof; j++) {
1139:       PetscScalar jmpL,jmpR;
1140:       jmpL = x[(i+0)*dof+j] - x[(i-1)*dof+j];
1141:       jmpR = x[(i+1)*dof+j] - x[(i+0)*dof+j];
1142:       for (k=0; k<dof; k++) {
1143:         cjmpL[k] += ctx->Rinv[k+j*dof] * jmpL;
1144:         cjmpR[k] += ctx->Rinv[k+j*dof] * jmpR;
1145:       }
1146:     }
1147:     /* Apply limiter to the left and right characteristic jumps */
1148:     info.m  = dof;
1149:     info.hx = hx;
1150:     (*ctx->limit)(&info,cjmpL,cjmpR,ctx->cslope);
1151:     for (j=0; j<dof; j++) ctx->cslope[j] /= hx; /* rescale to a slope */
1152:     for (j=0; j<dof; j++) {
1153:       PetscScalar tmp = 0;
1154:       for (k=0; k<dof; k++) tmp += ctx->R[j+k*dof] * ctx->cslope[k];
1155:       slope[i*dof+j] = tmp;
1156:     }
1157:   }

1159:   for (i=xs; i<xs+xm+1; i++) {
1160:     PetscReal   maxspeed;
1161:     PetscScalar *uL,*uR;
1162:     uL = &ctx->uLR[0];
1163:     uR = &ctx->uLR[dof];
1164:     for (j=0; j<dof; j++) {
1165:       uL[j] = x[(i-1)*dof+j] + slope[(i-1)*dof+j]*hx/2;
1166:       uR[j] = x[(i-0)*dof+j] - slope[(i-0)*dof+j]*hx/2;
1167:     }
1168:     (*ctx->physics.riemann)(ctx->physics.user,dof,uL,uR,ctx->flux,&maxspeed);
1169:     cfl_idt = PetscMax(cfl_idt,PetscAbsScalar(maxspeed/hx)); /* Max allowable value of 1/Delta t */

1171:     if (i > xs) {
1172:       for (j=0; j<dof; j++) f[(i-1)*dof+j] -= ctx->flux[j]/hx;
1173:     }
1174:     if (i < xs+xm) {
1175:       for (j=0; j<dof; j++) f[i*dof+j] += ctx->flux[j]/hx;
1176:     }
1177:   }

1179:   DMDAVecRestoreArray(da,Xloc,&x);
1180:   DMDAVecRestoreArray(da,F,&f);
1181:   DMDARestoreArray(da,PETSC_TRUE,&slope);
1182:   DMRestoreLocalVector(da,&Xloc);

1184:   MPI_Allreduce(&cfl_idt,&ctx->cfl_idt,1,MPIU_REAL,MPIU_MAX,PetscObjectComm((PetscObject)da));
1185:   if (0) {
1186:     /* We need to a way to inform the TS of a CFL constraint, this is a debugging fragment */
1187:     PetscReal dt,tnow;
1188:     TSGetTimeStep(ts,&dt);
1189:     TSGetTime(ts,&tnow);
1190:     if (dt > 0.5/ctx->cfl_idt) {
1191:       PetscPrintf(ctx->comm,"Stability constraint exceeded at t=%g, dt %g > %g\n",(double)tnow,(double)dt,(double)(0.5/ctx->cfl_idt));
1192:     }
1193:   }
1194:   return 0;
1195: }

1197: static PetscErrorCode SmallMatMultADB(PetscScalar *C,PetscInt bs,const PetscScalar *A,const PetscReal *D,const PetscScalar *B)
1198: {
1199:   PetscInt i,j,k;

1202:   for (i=0; i<bs; i++) {
1203:     for (j=0; j<bs; j++) {
1204:       PetscScalar tmp = 0;
1205:       for (k=0; k<bs; k++) tmp += A[i*bs+k] * D[k] * B[k*bs+j];
1206:       C[i*bs+j] = tmp;
1207:     }
1208:   }
1209:   return 0;
1210: }

1212: static PetscErrorCode FVIJacobian(TS ts,PetscReal t,Vec X,Vec Xdot,PetscReal shift,Mat A,Mat B,void *vctx)
1213: {
1214:   FVCtx             *ctx = (FVCtx*)vctx;
1215:   PetscInt          i,j,dof = ctx->physics.dof;
1216:   PetscScalar       *J;
1217:   const PetscScalar *x;
1218:   PetscReal         hx;
1219:   DM                da;
1220:   DMDALocalInfo     dainfo;

1223:   TSGetDM(ts,&da);
1224:   DMDAVecGetArrayRead(da,X,(void*)&x);
1225:   DMDAGetLocalInfo(da,&dainfo);
1226:   hx   = (ctx->xmax - ctx->xmin)/dainfo.mx;
1227:   PetscMalloc1(dof*dof,&J);
1228:   for (i=dainfo.xs; i<dainfo.xs+dainfo.xm; i++) {
1229:     (*ctx->physics.characteristic)(ctx->physics.user,dof,&x[i*dof],ctx->R,ctx->Rinv,ctx->speeds);
1230:     for (j=0; j<dof; j++) ctx->speeds[j] = PetscAbs(ctx->speeds[j]);
1231:     SmallMatMultADB(J,dof,ctx->R,ctx->speeds,ctx->Rinv);
1232:     for (j=0; j<dof*dof; j++) J[j] = J[j]/hx + shift*(j/dof == j%dof);
1233:     MatSetValuesBlocked(B,1,&i,1,&i,J,INSERT_VALUES);
1234:   }
1235:   PetscFree(J);
1236:   DMDAVecRestoreArrayRead(da,X,(void*)&x);

1238:   MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);
1239:   MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);
1240:   if (A != B) {
1241:     MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
1242:     MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
1243:   }
1244:   return 0;
1245: }

1247: static PetscErrorCode FVSample(FVCtx *ctx,DM da,PetscReal time,Vec U)
1248: {
1249:   PetscScalar    *u,*uj;
1250:   PetscInt       i,j,k,dof,xs,xm,Mx;

1254:   DMDAGetInfo(da,0, &Mx,0,0, 0,0,0, &dof,0,0,0,0,0);
1255:   DMDAGetCorners(da,&xs,0,0,&xm,0,0);
1256:   DMDAVecGetArray(da,U,&u);
1257:   PetscMalloc1(dof,&uj);
1258:   for (i=xs; i<xs+xm; i++) {
1259:     const PetscReal h = (ctx->xmax-ctx->xmin)/Mx,xi = ctx->xmin+h/2+i*h;
1260:     const PetscInt  N = 200;
1261:     /* Integrate over cell i using trapezoid rule with N points. */
1262:     for (k=0; k<dof; k++) u[i*dof+k] = 0;
1263:     for (j=0; j<N+1; j++) {
1264:       PetscScalar xj = xi+h*(j-N/2)/(PetscReal)N;
1265:       (*ctx->physics.sample)(ctx->physics.user,ctx->initial,ctx->bctype,ctx->xmin,ctx->xmax,time,xj,uj);
1266:       for (k=0; k<dof; k++) u[i*dof+k] += ((j==0 || j==N) ? 0.5 : 1.0)*uj[k]/N;
1267:     }
1268:   }
1269:   DMDAVecRestoreArray(da,U,&u);
1270:   PetscFree(uj);
1271:   return 0;
1272: }

1274: static PetscErrorCode SolutionStatsView(DM da,Vec X,PetscViewer viewer)
1275: {
1276:   PetscReal         xmin,xmax;
1277:   PetscScalar       sum,tvsum,tvgsum;
1278:   const PetscScalar *x;
1279:   PetscInt          imin,imax,Mx,i,j,xs,xm,dof;
1280:   Vec               Xloc;
1281:   PetscBool         iascii;

1284:   PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&iascii);
1285:   if (iascii) {
1286:     /* PETSc lacks a function to compute total variation norm (difficult in multiple dimensions), we do it here */
1287:     DMGetLocalVector(da,&Xloc);
1288:     DMGlobalToLocalBegin(da,X,INSERT_VALUES,Xloc);
1289:     DMGlobalToLocalEnd  (da,X,INSERT_VALUES,Xloc);
1290:     DMDAVecGetArrayRead(da,Xloc,(void*)&x);
1291:     DMDAGetCorners(da,&xs,0,0,&xm,0,0);
1292:     DMDAGetInfo(da,0, &Mx,0,0, 0,0,0, &dof,0,0,0,0,0);
1293:     tvsum = 0;
1294:     for (i=xs; i<xs+xm; i++) {
1295:       for (j=0; j<dof; j++) tvsum += PetscAbsScalar(x[i*dof+j] - x[(i-1)*dof+j]);
1296:     }
1297:     MPI_Allreduce(&tvsum,&tvgsum,1,MPIU_REAL,MPIU_SUM,PetscObjectComm((PetscObject)da));
1298:     DMDAVecRestoreArrayRead(da,Xloc,(void*)&x);
1299:     DMRestoreLocalVector(da,&Xloc);

1301:     VecMin(X,&imin,&xmin);
1302:     VecMax(X,&imax,&xmax);
1303:     VecSum(X,&sum);
1304:     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));
1305:   } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Viewer type not supported");
1306:   return 0;
1307: }

1309: static PetscErrorCode SolutionErrorNorms(FVCtx *ctx,DM da,PetscReal t,Vec X,PetscReal *nrm1,PetscReal *nrmsup)
1310: {
1311:   Vec            Y;
1312:   PetscInt       Mx;

1315:   VecGetSize(X,&Mx);
1316:   VecDuplicate(X,&Y);
1317:   FVSample(ctx,da,t,Y);
1318:   VecAYPX(Y,-1,X);
1319:   VecNorm(Y,NORM_1,nrm1);
1320:   VecNorm(Y,NORM_INFINITY,nrmsup);
1321:   *nrm1 /= Mx;
1322:   VecDestroy(&Y);
1323:   return 0;
1324: }

1326: int main(int argc,char *argv[])
1327: {
1328:   char              lname[256] = "mc",physname[256] = "advect",final_fname[256] = "solution.m";
1329:   PetscFunctionList limiters   = 0,physics = 0;
1330:   MPI_Comm          comm;
1331:   TS                ts;
1332:   DM                da;
1333:   Vec               X,X0,R;
1334:   Mat               B;
1335:   FVCtx             ctx;
1336:   PetscInt          i,dof,xs,xm,Mx,draw = 0;
1337:   PetscBool         view_final = PETSC_FALSE;
1338:   PetscReal         ptime;
1339:   PetscErrorCode    ierr;

1341:   PetscInitialize(&argc,&argv,0,help);
1342:   comm = PETSC_COMM_WORLD;
1343:   PetscMemzero(&ctx,sizeof(ctx));

1345:   /* Register limiters to be available on the command line */
1346:   PetscFunctionListAdd(&limiters,"upwind"              ,Limit_Upwind);
1347:   PetscFunctionListAdd(&limiters,"lax-wendroff"        ,Limit_LaxWendroff);
1348:   PetscFunctionListAdd(&limiters,"beam-warming"        ,Limit_BeamWarming);
1349:   PetscFunctionListAdd(&limiters,"fromm"               ,Limit_Fromm);
1350:   PetscFunctionListAdd(&limiters,"minmod"              ,Limit_Minmod);
1351:   PetscFunctionListAdd(&limiters,"superbee"            ,Limit_Superbee);
1352:   PetscFunctionListAdd(&limiters,"mc"                  ,Limit_MC);
1353:   PetscFunctionListAdd(&limiters,"vanleer"             ,Limit_VanLeer);
1354:   PetscFunctionListAdd(&limiters,"vanalbada"           ,Limit_VanAlbada);
1355:   PetscFunctionListAdd(&limiters,"vanalbadatvd"        ,Limit_VanAlbadaTVD);
1356:   PetscFunctionListAdd(&limiters,"koren"               ,Limit_Koren);
1357:   PetscFunctionListAdd(&limiters,"korensym"            ,Limit_KorenSym);
1358:   PetscFunctionListAdd(&limiters,"koren3"              ,Limit_Koren3);
1359:   PetscFunctionListAdd(&limiters,"cada-torrilhon2"     ,Limit_CadaTorrilhon2);
1360:   PetscFunctionListAdd(&limiters,"cada-torrilhon3-r0p1",Limit_CadaTorrilhon3R0p1);
1361:   PetscFunctionListAdd(&limiters,"cada-torrilhon3-r1"  ,Limit_CadaTorrilhon3R1);
1362:   PetscFunctionListAdd(&limiters,"cada-torrilhon3-r10" ,Limit_CadaTorrilhon3R10);
1363:   PetscFunctionListAdd(&limiters,"cada-torrilhon3-r100",Limit_CadaTorrilhon3R100);

1365:   /* Register physical models to be available on the command line */
1366:   PetscFunctionListAdd(&physics,"advect"          ,PhysicsCreate_Advect);
1367:   PetscFunctionListAdd(&physics,"burgers"         ,PhysicsCreate_Burgers);
1368:   PetscFunctionListAdd(&physics,"traffic"         ,PhysicsCreate_Traffic);
1369:   PetscFunctionListAdd(&physics,"acoustics"       ,PhysicsCreate_Acoustics);
1370:   PetscFunctionListAdd(&physics,"isogas"          ,PhysicsCreate_IsoGas);
1371:   PetscFunctionListAdd(&physics,"shallow"         ,PhysicsCreate_Shallow);

1373:   ctx.comm = comm;
1374:   ctx.cfl  = 0.9; ctx.bctype = FVBC_PERIODIC;
1375:   ctx.xmin = -1; ctx.xmax = 1;
1376:   PetscOptionsBegin(comm,NULL,"Finite Volume solver options","");
1377:     PetscOptionsReal("-xmin","X min","",ctx.xmin,&ctx.xmin,NULL);
1378:     PetscOptionsReal("-xmax","X max","",ctx.xmax,&ctx.xmax,NULL);
1379:     PetscOptionsFList("-limit","Name of flux limiter to use","",limiters,lname,lname,sizeof(lname),NULL);
1380:     PetscOptionsFList("-physics","Name of physics (Riemann solver and characteristics) to use","",physics,physname,physname,sizeof(physname),NULL);
1381:     PetscOptionsInt("-draw","Draw solution vector, bitwise OR of (1=initial,2=final,4=final error)","",draw,&draw,NULL);
1382:     PetscOptionsString("-view_final","Write final solution in ASCII MATLAB format to given file name","",final_fname,final_fname,sizeof(final_fname),&view_final);
1383:     PetscOptionsInt("-initial","Initial condition (depends on the physics)","",ctx.initial,&ctx.initial,NULL);
1384:     PetscOptionsBool("-exact","Compare errors with exact solution","",ctx.exact,&ctx.exact,NULL);
1385:     PetscOptionsReal("-cfl","CFL number to time step at","",ctx.cfl,&ctx.cfl,NULL);
1386:     PetscOptionsEnum("-bc_type","Boundary condition","",FVBCTypes,(PetscEnum)ctx.bctype,(PetscEnum*)&ctx.bctype,NULL);
1387:   PetscOptionsEnd();

1389:   /* Choose the limiter from the list of registered limiters */
1390:   PetscFunctionListFind(limiters,lname,&ctx.limit);

1393:   /* Choose the physics from the list of registered models */
1394:   {
1395:     PetscErrorCode (*r)(FVCtx*);
1396:     PetscFunctionListFind(physics,physname,&r);
1398:     /* Create the physics, will set the number of fields and their names */
1399:     (*r)(&ctx);
1400:   }

1402:   /* Create a DMDA to manage the parallel grid */
1403:   DMDACreate1d(comm,DM_BOUNDARY_PERIODIC,50,ctx.physics.dof,2,NULL,&da);
1404:   DMSetFromOptions(da);
1405:   DMSetUp(da);
1406:   /* Inform the DMDA of the field names provided by the physics. */
1407:   /* The names will be shown in the title bars when run with -ts_monitor_draw_solution */
1408:   for (i=0; i<ctx.physics.dof; i++) {
1409:     DMDASetFieldName(da,i,ctx.physics.fieldname[i]);
1410:   }
1411:   DMDAGetInfo(da,0, &Mx,0,0, 0,0,0, &dof,0,0,0,0,0);
1412:   DMDAGetCorners(da,&xs,0,0,&xm,0,0);

1414:   /* Set coordinates of cell centers */
1415:   DMDASetUniformCoordinates(da,ctx.xmin+0.5*(ctx.xmax-ctx.xmin)/Mx,ctx.xmax+0.5*(ctx.xmax-ctx.xmin)/Mx,0,0,0,0);

1417:   /* Allocate work space for the Finite Volume solver (so it doesn't have to be reallocated on each function evaluation) */
1418:   PetscMalloc4(dof*dof,&ctx.R,dof*dof,&ctx.Rinv,2*dof,&ctx.cjmpLR,1*dof,&ctx.cslope);
1419:   PetscMalloc3(2*dof,&ctx.uLR,dof,&ctx.flux,dof,&ctx.speeds);

1421:   /* Create a vector to store the solution and to save the initial state */
1422:   DMCreateGlobalVector(da,&X);
1423:   VecDuplicate(X,&X0);
1424:   VecDuplicate(X,&R);

1426:   DMCreateMatrix(da,&B);

1428:   /* Create a time-stepping object */
1429:   TSCreate(comm,&ts);
1430:   TSSetDM(ts,da);
1431:   TSSetRHSFunction(ts,R,FVRHSFunction,&ctx);
1432:   TSSetIJacobian(ts,B,B,FVIJacobian,&ctx);
1433:   TSSetType(ts,TSSSP);
1434:   TSSetMaxTime(ts,10);
1435:   TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);

1437:   /* Compute initial conditions and starting time step */
1438:   FVSample(&ctx,da,0,X0);
1439:   FVRHSFunction(ts,0,X0,X,(void*)&ctx); /* Initial function evaluation, only used to determine max speed */
1440:   VecCopy(X0,X);                        /* The function value was not used so we set X=X0 again */
1441:   TSSetTimeStep(ts,ctx.cfl/ctx.cfl_idt);
1442:   TSSetFromOptions(ts); /* Take runtime options */
1443:   SolutionStatsView(da,X,PETSC_VIEWER_STDOUT_WORLD);
1444:   {
1445:     PetscReal nrm1,nrmsup;
1446:     PetscInt  steps;

1448:     TSSolve(ts,X);
1449:     TSGetSolveTime(ts,&ptime);
1450:     TSGetStepNumber(ts,&steps);

1452:     PetscPrintf(comm,"Final time %8.5f, steps %D\n",(double)ptime,steps);
1453:     if (ctx.exact) {
1454:       SolutionErrorNorms(&ctx,da,ptime,X,&nrm1,&nrmsup);
1455:       PetscPrintf(comm,"Error ||x-x_e||_1 %8.4e  ||x-x_e||_sup %8.4e\n",(double)nrm1,(double)nrmsup);
1456:     }
1457:   }

1459:   SolutionStatsView(da,X,PETSC_VIEWER_STDOUT_WORLD);
1460:   if (draw & 0x1) VecView(X0,PETSC_VIEWER_DRAW_WORLD);
1461:   if (draw & 0x2) VecView(X,PETSC_VIEWER_DRAW_WORLD);
1462:   if (draw & 0x4) {
1463:     Vec Y;
1464:     VecDuplicate(X,&Y);
1465:     FVSample(&ctx,da,ptime,Y);
1466:     VecAYPX(Y,-1,X);
1467:     VecView(Y,PETSC_VIEWER_DRAW_WORLD);
1468:     VecDestroy(&Y);
1469:   }

1471:   if (view_final) {
1472:     PetscViewer viewer;
1473:     PetscViewerASCIIOpen(PETSC_COMM_WORLD,final_fname,&viewer);
1474:     PetscViewerPushFormat(viewer,PETSC_VIEWER_ASCII_MATLAB);
1475:     VecView(X,viewer);
1476:     PetscViewerPopFormat(viewer);
1477:     PetscViewerDestroy(&viewer);
1478:   }

1480:   /* Clean up */
1481:   (*ctx.physics.destroy)(ctx.physics.user);
1482:   for (i=0; i<ctx.physics.dof; i++) PetscFree(ctx.physics.fieldname[i]);
1483:   PetscFree4(ctx.R,ctx.Rinv,ctx.cjmpLR,ctx.cslope);
1484:   PetscFree3(ctx.uLR,ctx.flux,ctx.speeds);
1485:   VecDestroy(&X);
1486:   VecDestroy(&X0);
1487:   VecDestroy(&R);
1488:   MatDestroy(&B);
1489:   DMDestroy(&da);
1490:   TSDestroy(&ts);
1491:   PetscFunctionListDestroy(&limiters);
1492:   PetscFunctionListDestroy(&physics);
1493:   PetscFinalize();
1494:   return 0;
1495: }

1497: /*TEST

1499:     build:
1500:       requires: !complex

1502:     test:
1503:       args: -da_grid_x 100 -initial 1 -xmin -2 -xmax 5 -exact -limit mc
1504:       requires: !complex !single

1506:     test:
1507:       suffix: 2
1508:       args: -da_grid_x 100 -initial 2 -xmin -2 -xmax 2 -exact -limit mc -physics burgers -bc_type outflow -ts_max_time 1
1509:       filter:  sed "s/at 48/at 0/g"
1510:       requires: !complex !single

1512:     test:
1513:       suffix: 3
1514:       args: -da_grid_x 100 -initial 2 -xmin -2 -xmax 2 -exact -limit mc -physics burgers -bc_type outflow -ts_max_time 1
1515:       nsize: 3
1516:       filter:  sed "s/at 48/at 0/g"
1517:       requires: !complex !single

1519: TEST*/