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: " isogas - Isothermal gas dynamics\n"
10: " rho_t + (rho*u)_x = 0\n"
11: " (rho*u)_t + (rho*u^2 + c^2*rho)_x = 0\n"
12: " shallow - Shallow water equations\n"
13: " h_t + (h*u)_x = 0\n"
14: " (h*u)_t + (h*u^2 + g*h^2/2)_x = 0\n"
15: "Some of these physical models have multiple Riemann solvers, select these with -physics_xxx_riemann\n"
16: " exact - Exact Riemann solver which usually needs to perform a Newton iteration to connect\n"
17: " the states across shocks and rarefactions\n"
18: " roe - Linearized scheme, usually with an entropy fix inside sonic rarefactions\n"
19: "The systems provide a choice of reconstructions with -physics_xxx_reconstruct\n"
20: " characteristic - Limit the characteristic variables, this is usually preferred (default)\n"
21: " conservative - Limit the conservative variables directly, can cause undesired interaction of waves\n\n"
22: "A variety of limiters for high-resolution TVD limiters are available with -limit\n"
23: " upwind,minmod,superbee,mc,vanleer,vanalbada,koren,cada-torillhon (last two are nominally third order)\n"
24: " and non-TVD schemes lax-wendroff,beam-warming,fromm\n\n"
25: "To preserve the TVD property, one should time step with a strong stability preserving method.\n"
26: "The optimal high order explicit Runge-Kutta methods in TSSSP are recommended for non-stiff problems.\n\n"
27: "Several initial conditions can be chosen with -initial N\n\n"
28: "The problem size should be set with -da_grid_x M\n\n";
30: /* To get isfinite in math.h */
31: #define _XOPEN_SOURCE 600
33: #include <petscts.h>
34: #include <petscdmda.h>
36: #include <../src/mat/blockinvert.h> /* For the Kernel_*_gets_* stuff for BAIJ */
38: static inline PetscReal Sgn(PetscReal a) { return (a<0) ? -1 : 1; }
39: static inline PetscReal Abs(PetscReal a) { return (a<0) ? 0 : a; }
40: static inline PetscReal Sqr(PetscReal a) { return a*a; }
41: static inline PetscReal MaxAbs(PetscReal a,PetscReal b) { return (PetscAbs(a) > PetscAbs(b)) ? a : b; }
42: static inline PetscReal MinAbs(PetscReal a,PetscReal b) { return (PetscAbs(a) < PetscAbs(b)) ? a : b; }
43: static inline PetscReal MinMod2(PetscReal a,PetscReal b)
44: { return (a*b<0) ? 0 : Sgn(a)*PetscMin(PetscAbs(a),PetscAbs(b)); }
45: static inline PetscReal MaxMod2(PetscReal a,PetscReal b)
46: { return (a*b<0) ? 0 : Sgn(a)*PetscMax(PetscAbs(a),PetscAbs(b)); }
47: static inline PetscReal MinMod3(PetscReal a,PetscReal b,PetscReal c)
48: {return (a*b<0 || a*c<0) ? 0 : Sgn(a)*PetscMin(PetscAbs(a),PetscMin(PetscAbs(b),PetscAbs(c))); }
50: static inline PetscReal RangeMod(PetscReal a,PetscReal xmin,PetscReal xmax)
51: { PetscReal range = xmax-xmin; return xmin + fmod(range+fmod(a,range),range); }
54: /* ----------------------- Lots of limiters, these could go in a separate library ------------------------- */
55: typedef struct _LimitInfo {
56: PetscReal hx;
57: PetscInt m;
58: } *LimitInfo;
59: static void Limit_Upwind(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
60: {
61: PetscInt i;
62: for (i=0; i<info->m; i++) lmt[i] = 0;
63: }
64: static void Limit_LaxWendroff(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
65: {
66: PetscInt i;
67: for (i=0; i<info->m; i++) lmt[i] = jR[i];
68: }
69: static void Limit_BeamWarming(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
70: {
71: PetscInt i;
72: for (i=0; i<info->m; i++) lmt[i] = jL[i];
73: }
74: static void Limit_Fromm(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
75: {
76: PetscInt i;
77: for (i=0; i<info->m; i++) lmt[i] = 0.5*(jL[i] + jR[i]);
78: }
79: static void Limit_Minmod(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
80: {
81: PetscInt i;
82: for (i=0; i<info->m; i++) lmt[i] = MinMod2(jL[i],jR[i]);
83: }
84: static void Limit_Superbee(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
85: {
86: PetscInt i;
87: for (i=0; i<info->m; i++) lmt[i] = MaxMod2(MinMod2(jL[i],2*jR[i]),MinMod2(2*jL[i],jR[i]));
88: }
89: static void Limit_MC(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
90: {
91: PetscInt i;
92: for (i=0; i<info->m; i++) lmt[i] = MinMod3(2*jL[i],0.5*(jL[i]+jR[i]),2*jR[i]);
93: }
94: static void Limit_VanLeer(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
95: { /* phi = (t + abs(t)) / (1 + abs(t)) */
96: PetscInt i;
97: 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);
98: }
99: static void Limit_VanAlbada(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt) /* differentiable */
100: { /* phi = (t + t^2) / (1 + t^2) */
101: PetscInt i;
102: 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);
103: }
104: static void Limit_VanAlbadaTVD(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
105: { /* phi = (t + t^2) / (1 + t^2) */
106: PetscInt i;
107: for (i=0; i<info->m; i++) lmt[i] = (jL[i]*jR[i]<0) ? 0
108: : (jL[i]*Sqr(jR[i]) + Sqr(jL[i])*jR[i]) / (Sqr(jL[i]) + Sqr(jR[i]) + 1e-15);
109: }
110: static void Limit_Koren(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt) /* differentiable */
111: { /* phi = (t + 2*t^2) / (2 - t + 2*t^2) */
112: PetscInt i;
113: for (i=0; i<info->m; i++) lmt[i] = ((jL[i]*Sqr(jR[i]) + 2*Sqr(jL[i])*jR[i])
114: / (2*Sqr(jL[i]) - jL[i]*jR[i] + 2*Sqr(jR[i]) + 1e-15));
115: }
116: static void Limit_KorenSym(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt) /* differentiable */
117: { /* Symmetric version of above */
118: PetscInt i;
119: for (i=0; i<info->m; i++) lmt[i] = (1.5*(jL[i]*Sqr(jR[i]) + Sqr(jL[i])*jR[i])
120: / (2*Sqr(jL[i]) - jL[i]*jR[i] + 2*Sqr(jR[i]) + 1e-15));
121: }
122: static void Limit_Koren3(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
123: { /* Eq 11 of Cada-Torrilhon 2009 */
124: PetscInt i;
125: for (i=0; i<info->m; i++) lmt[i] = MinMod3(2*jL[i],(jL[i]+2*jR[i])/3,2*jR[i]);
126: }
128: static PetscReal CadaTorrilhonPhiHatR_Eq13(PetscReal L,PetscReal R)
129: { return PetscMax(0,PetscMin((L+2*R)/3,
130: PetscMax(-0.5*L,
131: PetscMin(2*L,
132: PetscMin((L+2*R)/3,1.6*R)))));
133: }
134: static void Limit_CadaTorrilhon2(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
135: { /* Cada-Torrilhon 2009, Eq 13 */
136: PetscInt i;
137: for (i=0; i<info->m; i++) lmt[i] = CadaTorrilhonPhiHatR_Eq13(jL[i],jR[i]);
138: }
139: static void Limit_CadaTorrilhon3R(PetscReal r,LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
140: { /* Cada-Torrilhon 2009, Eq 22 */
141: /* They recommend 0.001 < r < 1, but larger values are more accurate in smooth regions */
142: const PetscReal eps = 1e-7,hx = info->hx;
143: PetscInt i;
144: for (i=0; i<info->m; i++) {
145: const PetscReal eta = (Sqr(jL[i]) + Sqr(jR[i])) / Sqr(r*hx);
146: lmt[i] = ((eta < 1-eps)
147: ? (jL[i] + 2*jR[i]) / 3
148: : ((eta > 1+eps)
149: ? CadaTorrilhonPhiHatR_Eq13(jL[i],jR[i])
150: : 0.5*((1-(eta-1)/eps)*(jL[i]+2*jR[i])/3
151: + (1+(eta+1)/eps)*CadaTorrilhonPhiHatR_Eq13(jL[i],jR[i]))));
152: }
153: }
154: static void Limit_CadaTorrilhon3R0p1(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
155: { Limit_CadaTorrilhon3R(0.1,info,jL,jR,lmt); }
156: static void Limit_CadaTorrilhon3R1(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
157: { Limit_CadaTorrilhon3R(1,info,jL,jR,lmt); }
158: static void Limit_CadaTorrilhon3R10(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
159: { Limit_CadaTorrilhon3R(10,info,jL,jR,lmt); }
160: static void Limit_CadaTorrilhon3R100(LimitInfo info,const PetscScalar *jL,const PetscScalar *jR,PetscScalar *lmt)
161: { Limit_CadaTorrilhon3R(100,info,jL,jR,lmt); }
164: /* --------------------------------- Finite Volume data structures ----------------------------------- */
166: typedef enum {FVBC_PERIODIC, FVBC_OUTFLOW} FVBCType;
167: static const char *FVBCTypes[] = {"PERIODIC","OUTFLOW","FVBCType","FVBC_",0};
168: typedef PetscErrorCode (*RiemannFunction)(void*,PetscInt,const PetscScalar*,const PetscScalar*,PetscScalar*,PetscReal*);
169: typedef PetscErrorCode (*ReconstructFunction)(void*,PetscInt,const PetscScalar*,PetscScalar*,PetscScalar*);
171: typedef struct {
172: PetscErrorCode (*sample)(void*,PetscInt,FVBCType,PetscReal,PetscReal,PetscReal,PetscReal,PetscReal*);
173: RiemannFunction riemann;
174: ReconstructFunction characteristic;
175: PetscErrorCode (*destroy)(void*);
176: void *user;
177: PetscInt dof;
178: char *fieldname[16];
179: } PhysicsCtx;
181: typedef struct {
182: void (*limit)(LimitInfo,const PetscScalar*,const PetscScalar*,PetscScalar*);
183: PhysicsCtx physics;
185: MPI_Comm comm;
186: char prefix[256];
187: DM da;
188: /* Local work arrays */
189: PetscScalar *R,*Rinv; /* Characteristic basis, and it's inverse. COLUMN-MAJOR */
190: PetscScalar *cjmpLR; /* Jumps at left and right edge of cell, in characteristic basis, len=2*dof */
191: PetscScalar *cslope; /* Limited slope, written in characteristic basis */
192: PetscScalar *uLR; /* Solution at left and right of interface, conservative variables, len=2*dof */
193: PetscScalar *flux; /* Flux across interface */
195: PetscReal cfl_idt; /* Max allowable value of 1/Delta t */
196: PetscReal cfl;
197: PetscReal xmin,xmax;
198: PetscInt initial;
199: PetscBool exact;
200: FVBCType bctype;
201: } FVCtx;
204: /* Utility */
208: PetscErrorCode RiemannListAdd(PetscFList *flist,const char *name,RiemannFunction rsolve)
209: {
213: PetscFListAdd(flist,name,"",(void(*)(void))rsolve);
214: return(0);
215: }
219: PetscErrorCode RiemannListFind(PetscFList flist,const char *name,RiemannFunction *rsolve)
220: {
224: PetscFListFind(flist,PETSC_COMM_WORLD,name,PETSC_FALSE,(void(**)(void))rsolve);
225: if (!*rsolve) SETERRQ1(PETSC_COMM_SELF,1,"Riemann solver \"%s\" could not be found",name);
226: return(0);
227: }
231: PetscErrorCode ReconstructListAdd(PetscFList *flist,const char *name,ReconstructFunction r)
232: {
236: PetscFListAdd(flist,name,"",(void(*)(void))r);
237: return(0);
238: }
242: PetscErrorCode ReconstructListFind(PetscFList flist,const char *name,ReconstructFunction *r)
243: {
247: PetscFListFind(flist,PETSC_COMM_WORLD,name,PETSC_FALSE,(void(**)(void))r);
248: if (!*r) SETERRQ1(PETSC_COMM_SELF,1,"Reconstruction \"%s\" could not be found",name);
249: return(0);
250: }
253: /* --------------------------------- Physics ----------------------------------- */
254: /**
255: * Each physical model consists of Riemann solver and a function to determine the basis to use for reconstruction. These
256: * are set with the PhysicsCreate_XXX function which allocates private storage and sets these methods as well as the
257: * number of fields and their names, and a function to deallocate private storage.
258: **/
260: /* First a few functions useful to several different physics */
263: static PetscErrorCode PhysicsCharacteristic_Conservative(void *vctx,PetscInt m,const PetscScalar *u,PetscScalar *X,PetscScalar *Xi)
264: {
265: PetscInt i,j;
268: for (i=0; i<m; i++) {
269: for (j=0; j<m; j++) {
270: Xi[i*m+j] = X[i*m+j] = (PetscScalar)(i==j);
271: }
272: }
273: return(0);
274: }
278: static PetscErrorCode PhysicsDestroy_SimpleFree(void *vctx)
279: {
283: PetscFree(vctx);
284: return(0);
285: }
289: /* --------------------------------- Advection ----------------------------------- */
291: typedef struct {
292: PetscReal a; /* advective velocity */
293: } AdvectCtx;
297: static PetscErrorCode PhysicsRiemann_Advect(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
298: {
299: AdvectCtx *ctx = (AdvectCtx*)vctx;
300: PetscReal speed;
303: speed = ctx->a;
304: flux[0] = PetscMax(0,speed)*uL[0] + PetscMin(0,speed)*uR[0];
305: *maxspeed = speed;
306: return(0);
307: }
311: static PetscErrorCode PhysicsSample_Advect(void *vctx,PetscInt initial,FVBCType bctype,PetscReal xmin,PetscReal xmax,PetscReal t,PetscReal x,PetscReal *u)
312: {
313: AdvectCtx *ctx = (AdvectCtx*)vctx;
314: PetscReal a = ctx->a,x0;
317: switch (bctype) {
318: case FVBC_OUTFLOW: x0 = x-a*t; break;
319: case FVBC_PERIODIC: x0 = RangeMod(x-a*t,xmin,xmax); break;
320: default: SETERRQ(PETSC_COMM_SELF,1,"unknown BCType");
321: }
322: switch (initial) {
323: case 0: u[0] = (x0 < 0) ? 1 : -1; break;
324: case 1: u[0] = (x0 < 0) ? -1 : 1; break;
325: case 2: u[0] = (0 < x0 && x0 < 1) ? 1 : 0; break;
326: case 3: u[0] = sin(2*M_PI*x0); break;
327: case 4: u[0] = PetscAbs(x0); break;
328: default: SETERRQ(PETSC_COMM_SELF,1,"unknown initial condition");
329: }
330: return(0);
331: }
335: static PetscErrorCode PhysicsCreate_Advect(FVCtx *ctx)
336: {
338: AdvectCtx *user;
341: PetscNew(*user,&user);
342: ctx->physics.sample = PhysicsSample_Advect;
343: ctx->physics.riemann = PhysicsRiemann_Advect;
344: ctx->physics.characteristic = PhysicsCharacteristic_Conservative;
345: ctx->physics.destroy = PhysicsDestroy_SimpleFree;
346: ctx->physics.user = user;
347: ctx->physics.dof = 1;
348: PetscStrallocpy("u",&ctx->physics.fieldname[0]);
349: user->a = 1;
350: PetscOptionsBegin(ctx->comm,ctx->prefix,"Options for advection","");
351: {
352: PetscOptionsReal("-physics_advect_a","Speed","",user->a,&user->a,PETSC_NULL);
353: }
354: PetscOptionsEnd();
355: return(0);
356: }
360: /* --------------------------------- Burgers ----------------------------------- */
362: typedef struct {
363: PetscReal lxf_speed;
364: } BurgersCtx;
368: static PetscErrorCode PhysicsSample_Burgers(void *vctx,PetscInt initial,FVBCType bctype,PetscReal xmin,PetscReal xmax,PetscReal t,PetscReal x,PetscReal *u)
369: {
372: if (bctype == FVBC_PERIODIC && t > 0) SETERRQ(PETSC_COMM_SELF,1,"Exact solution not implemented for periodic");
373: switch (initial) {
374: case 0: u[0] = (x < 0) ? 1 : -1; break;
375: case 1:
376: if (x < -t) u[0] = -1;
377: else if (x < t) u[0] = x/t;
378: else u[0] = 1;
379: break;
380: case 2:
381: if (x < 0) u[0] = 0;
382: else if (x <= t) u[0] = x/t;
383: else if (x < 1+0.5*t) u[0] = 1;
384: else u[0] = 0;
385: break;
386: case 3:
387: if (x < 0.2*t) u[0] = 0.2;
388: else if (x < t) u[0] = x/t;
389: else u[0] = 1;
390: break;
391: case 4:
392: if (t > 0) SETERRQ(PETSC_COMM_SELF,1,"Only initial condition available");
393: u[0] = 0.7 + 0.3*sin(2*M_PI*((x-xmin)/(xmax-xmin)));
394: break;
395: case 5: /* Pure shock solution */
396: if (x < 0.5*t) u[0] = 1;
397: else u[0] = 0;
398: break;
399: default: SETERRQ(PETSC_COMM_SELF,1,"unknown initial condition");
400: }
401: return(0);
402: }
406: static PetscErrorCode PhysicsRiemann_Burgers_Exact(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
407: {
410: if (uL[0] < uR[0]) { /* rarefaction */
411: flux[0] = (uL[0]*uR[0] < 0)
412: ? 0 /* sonic rarefaction */
413: : 0.5*PetscMin(PetscSqr(uL[0]),PetscSqr(uR[0]));
414: } else { /* shock */
415: flux[0] = 0.5*PetscMax(PetscSqr(uL[0]),PetscSqr(uR[0]));
416: }
417: *maxspeed = (PetscAbs(uL[0]) > PetscAbs(uR[0])) ? uL[0] : uR[0];
418: return(0);
419: }
423: static PetscErrorCode PhysicsRiemann_Burgers_Roe(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
424: {
425: PetscReal speed;
428: speed = 0.5*(uL[0] + uR[0]);
429: flux[0] = 0.25*(PetscSqr(uL[0]) + PetscSqr(uR[0])) - 0.5*PetscAbs(speed)*(uR[0]-uL[0]);
430: if (uL[0] <= 0 && 0 <= uR[0]) flux[0] = 0; /* Entropy fix for sonic rarefaction */
431: *maxspeed = speed;
432: return(0);
433: }
437: static PetscErrorCode PhysicsRiemann_Burgers_LxF(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
438: {
439: PetscReal c;
440: PetscScalar fL,fR;
443: c = ((BurgersCtx*)vctx)->lxf_speed;
444: fL = 0.5*PetscSqr(uL[0]);
445: fR = 0.5*PetscSqr(uR[0]);
446: flux[0] = 0.5*(fL + fR) - 0.5*c*(uR[0] - uL[0]);
447: *maxspeed = c;
448: return(0);
449: }
453: static PetscErrorCode PhysicsRiemann_Burgers_Rusanov(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
454: {
455: PetscReal c;
456: PetscScalar fL,fR;
459: c = PetscMax(PetscAbs(uL[0]),PetscAbs(uR[0]));
460: fL = 0.5*PetscSqr(uL[0]);
461: fR = 0.5*PetscSqr(uR[0]);
462: flux[0] = 0.5*(fL + fR) - 0.5*c*(uR[0] - uL[0]);
463: *maxspeed = c;
464: return(0);
465: }
469: static PetscErrorCode PhysicsCreate_Burgers(FVCtx *ctx)
470: {
471: BurgersCtx *user;
473: RiemannFunction r;
474: PetscFList rlist = 0;
475: char rname[256] = "exact";
478: PetscNew(*user,&user);
479: ctx->physics.sample = PhysicsSample_Burgers;
480: ctx->physics.characteristic = PhysicsCharacteristic_Conservative;
481: ctx->physics.destroy = PhysicsDestroy_SimpleFree;
482: ctx->physics.user = user;
483: ctx->physics.dof = 1;
484: PetscStrallocpy("u",&ctx->physics.fieldname[0]);
485: RiemannListAdd(&rlist,"exact", PhysicsRiemann_Burgers_Exact);
486: RiemannListAdd(&rlist,"roe", PhysicsRiemann_Burgers_Roe);
487: RiemannListAdd(&rlist,"lxf", PhysicsRiemann_Burgers_LxF);
488: RiemannListAdd(&rlist,"rusanov",PhysicsRiemann_Burgers_Rusanov);
489: PetscOptionsBegin(ctx->comm,ctx->prefix,"Options for advection","");
490: {
491: PetscOptionsList("-physics_burgers_riemann","Riemann solver","",rlist,rname,rname,sizeof rname,PETSC_NULL);
492: }
493: PetscOptionsEnd();
494: RiemannListFind(rlist,rname,&r);
495: PetscFListDestroy(&rlist);
496: ctx->physics.riemann = r;
498: /* *
499: * Hack to deal with LxF in semi-discrete form
500: * max speed is 1 for the basic initial conditions (where |u| <= 1)
501: * */
502: if (r == PhysicsRiemann_Burgers_LxF) user->lxf_speed = 1;
503: return(0);
504: }
508: /* --------------------------------- Traffic ----------------------------------- */
510: typedef struct {
511: PetscReal lxf_speed;
512: PetscReal a;
513: } TrafficCtx;
515: static inline PetscScalar TrafficFlux(PetscScalar a,PetscScalar u) { return a*u*(1-u); }
519: static PetscErrorCode PhysicsSample_Traffic(void *vctx,PetscInt initial,FVBCType bctype,PetscReal xmin,PetscReal xmax,PetscReal t,PetscReal x,PetscReal *u)
520: {
521: PetscReal a = ((TrafficCtx*)vctx)->a;
524: if (bctype == FVBC_PERIODIC && t > 0) SETERRQ(PETSC_COMM_SELF,1,"Exact solution not implemented for periodic");
525: switch (initial) {
526: case 0:
527: u[0] = (-a*t < x) ? 2 : 0; break;
528: case 1:
529: if (x < PetscMin(2*a*t,0.5+a*t)) u[0] = -1;
530: else if (x < 1) u[0] = 0;
531: else u[0] = 1;
532: break;
533: case 2:
534: if (t > 0) SETERRQ(PETSC_COMM_SELF,1,"Only initial condition available");
535: u[0] = 0.7 + 0.3*sin(2*M_PI*((x-xmin)/(xmax-xmin)));
536: break;
537: default: SETERRQ(PETSC_COMM_SELF,1,"unknown initial condition");
538: }
539: return(0);
540: }
544: static PetscErrorCode PhysicsRiemann_Traffic_Exact(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
545: {
546: PetscReal a = ((TrafficCtx*)vctx)->a;
549: if (uL[0] < uR[0]) {
550: flux[0] = PetscMin(TrafficFlux(a,uL[0]),
551: TrafficFlux(a,uR[0]));
552: } else {
553: flux[0] = (uR[0] < 0.5 && 0.5 < uL[0])
554: ? TrafficFlux(a,0.5)
555: : PetscMax(TrafficFlux(a,uL[0]),
556: TrafficFlux(a,uR[0]));
557: }
558: *maxspeed = a*MaxAbs(1-2*uL[0],1-2*uR[0]);
559: return(0);
560: }
564: static PetscErrorCode PhysicsRiemann_Traffic_Roe(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
565: {
566: PetscReal a = ((TrafficCtx*)vctx)->a;
567: PetscReal speed;
570: speed = a*(1 - (uL[0] + uR[0]));
571: flux[0] = 0.5*(TrafficFlux(a,uL[0]) + TrafficFlux(a,uR[0])) - 0.5*PetscAbs(speed)*(uR[0]-uL[0]);
572: *maxspeed = speed;
573: return(0);
574: }
578: static PetscErrorCode PhysicsRiemann_Traffic_LxF(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
579: {
580: TrafficCtx *phys = (TrafficCtx*)vctx;
581: PetscReal a = phys->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*phys->lxf_speed*(uR[0]-uL[0]);
587: *maxspeed = speed;
588: return(0);
589: }
593: static PetscErrorCode PhysicsRiemann_Traffic_Rusanov(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
594: {
595: PetscReal a = ((TrafficCtx*)vctx)->a;
596: PetscReal speed;
599: speed = a*PetscMax(PetscAbs(1-2*uL[0]),PetscAbs(1-2*uR[0]));
600: flux[0] = 0.5*(TrafficFlux(a,uL[0]) + TrafficFlux(a,uR[0])) - 0.5*speed*(uR[0]-uL[0]);
601: *maxspeed = speed;
602: return(0);
603: }
607: static PetscErrorCode PhysicsCreate_Traffic(FVCtx *ctx)
608: {
610: TrafficCtx *user;
611: RiemannFunction r;
612: PetscFList rlist = 0;
613: char rname[256] = "exact";
616: PetscNew(*user,&user);
617: ctx->physics.sample = PhysicsSample_Traffic;
618: ctx->physics.characteristic = PhysicsCharacteristic_Conservative;
619: ctx->physics.destroy = PhysicsDestroy_SimpleFree;
620: ctx->physics.user = user;
621: ctx->physics.dof = 1;
622: PetscStrallocpy("density",&ctx->physics.fieldname[0]);
623: user->a = 0.5;
624: RiemannListAdd(&rlist,"exact", PhysicsRiemann_Traffic_Exact);
625: RiemannListAdd(&rlist,"roe", PhysicsRiemann_Traffic_Roe);
626: RiemannListAdd(&rlist,"lxf", PhysicsRiemann_Traffic_LxF);
627: RiemannListAdd(&rlist,"rusanov",PhysicsRiemann_Traffic_Rusanov);
628: PetscOptionsBegin(ctx->comm,ctx->prefix,"Options for Traffic","");
629: {
630: PetscOptionsReal("-physics_traffic_a","Flux = a*u*(1-u)","",user->a,&user->a,PETSC_NULL);
631: PetscOptionsList("-physics_traffic_riemann","Riemann solver","",rlist,rname,rname,sizeof rname,PETSC_NULL);
632: }
633: PetscOptionsEnd();
635: RiemannListFind(rlist,rname,&r);
636: PetscFListDestroy(&rlist);
637: ctx->physics.riemann = r;
639: /* *
640: * Hack to deal with LxF in semi-discrete form
641: * max speed is 3*a for the basic initial conditions (-1 <= u <= 2)
642: * */
643: if (r == PhysicsRiemann_Traffic_LxF) user->lxf_speed = 3*user->a;
645: return(0);
646: }
651: /* --------------------------------- Isothermal Gas Dynamics ----------------------------------- */
653: typedef struct {
654: PetscReal acoustic_speed;
655: } IsoGasCtx;
657: static inline void IsoGasFlux(PetscReal c,const PetscScalar *u,PetscScalar *f)
658: {
659: f[0] = u[1];
660: f[1] = PetscSqr(u[1])/u[0] + c*c*u[0];
661: }
665: static PetscErrorCode PhysicsSample_IsoGas(void *vctx,PetscInt initial,FVBCType bctype,PetscReal xmin,PetscReal xmax,PetscReal t,PetscReal x,PetscReal *u)
666: {
669: if (t > 0) SETERRQ(PETSC_COMM_SELF,1,"Exact solutions not implemented for t > 0");
670: switch (initial) {
671: case 0:
672: u[0] = (x < 0) ? 1 : 0.5;
673: u[1] = (x < 0) ? 1 : 0.7;
674: break;
675: case 1:
676: u[0] = 1+0.5*sin(2*M_PI*x);
677: u[1] = 1*u[0];
678: break;
679: default: SETERRQ(PETSC_COMM_SELF,1,"unknown initial condition");
680: }
681: return(0);
682: }
686: static PetscErrorCode PhysicsRiemann_IsoGas_Roe(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
687: {
688: IsoGasCtx *phys = (IsoGasCtx*)vctx;
689: PetscReal c = phys->acoustic_speed;
690: PetscScalar ubar,du[2],a[2],fL[2],fR[2],lam[2],ustar[2],R[2][2];
691: PetscInt i;
694: ubar = (uL[1]/PetscSqrtScalar(uL[0]) + uR[1]/PetscSqrtScalar(uR[0])) / (PetscSqrtScalar(uL[0]) + PetscSqrtScalar(uR[0]));
695: /* write fluxuations in characteristic basis */
696: du[0] = uR[0] - uL[0];
697: du[1] = uR[1] - uL[1];
698: a[0] = (1/(2*c)) * ((ubar + c)*du[0] - du[1]);
699: a[1] = (1/(2*c)) * ((-ubar + c)*du[0] + du[1]);
700: /* wave speeds */
701: lam[0] = ubar - c;
702: lam[1] = ubar + c;
703: /* Right eigenvectors */
704: R[0][0] = 1; R[0][1] = ubar-c;
705: R[1][0] = 1; R[1][1] = ubar+c;
706: /* Compute state in star region (between the 1-wave and 2-wave) */
707: for (i=0; i<2; i++) ustar[i] = uL[i] + a[0]*R[0][i];
708: if (uL[1]/uL[0] < c && c < ustar[1]/ustar[0]) { /* 1-wave is sonic rarefaction */
709: PetscScalar ufan[2];
710: ufan[0] = uL[0]*PetscExpScalar(uL[1]/(uL[0]*c) - 1);
711: ufan[1] = c*ufan[0];
712: IsoGasFlux(c,ufan,flux);
713: } else if (ustar[1]/ustar[0] < -c && -c < uR[1]/uR[0]) { /* 2-wave is sonic rarefaction */
714: PetscScalar ufan[2];
715: ufan[0] = uR[0]*PetscExpScalar(-uR[1]/(uR[0]*c) - 1);
716: ufan[1] = -c*ufan[0];
717: IsoGasFlux(c,ufan,flux);
718: } else { /* Centered form */
719: IsoGasFlux(c,uL,fL);
720: IsoGasFlux(c,uR,fR);
721: for (i=0; i<2; i++) {
722: PetscScalar absdu = PetscAbsScalar(lam[0])*a[0]*R[0][i] + PetscAbsScalar(lam[1])*a[1]*R[1][i];
723: flux[i] = 0.5*(fL[i]+fR[i]) - 0.5*absdu;
724: }
725: }
726: *maxspeed = MaxAbs(lam[0],lam[1]);
727: return(0);
728: }
732: static PetscErrorCode PhysicsRiemann_IsoGas_Exact(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
733: {
734: IsoGasCtx *phys = (IsoGasCtx*)vctx;
735: PetscReal c = phys->acoustic_speed;
736: PetscScalar ustar[2];
737: struct {PetscScalar rho,u;} L = {uL[0],uL[1]/uL[0]},R = {uR[0],uR[1]/uR[0]},star;
738: PetscInt i;
741: if (!(L.rho > 0 && R.rho > 0)) SETERRQ(PETSC_COMM_SELF,1,"Reconstructed density is negative");
742: {
743: /* Solve for star state */
744: PetscScalar res,tmp,rho = 0.5*(L.rho + R.rho); /* initial guess */
745: for (i=0; i<20; i++) {
746: PetscScalar fr,fl,dfr,dfl;
747: fl = (L.rho < rho)
748: ? (rho-L.rho)/PetscSqrtScalar(L.rho*rho) /* shock */
749: : PetscLogScalar(rho) - PetscLogScalar(L.rho); /* rarefaction */
750: fr = (R.rho < rho)
751: ? (rho-R.rho)/PetscSqrtScalar(R.rho*rho) /* shock */
752: : PetscLogScalar(rho) - PetscLogScalar(R.rho); /* rarefaction */
753: res = R.u-L.u + c*(fr+fl);
754: if (!isfinite(res)) SETERRQ1(PETSC_COMM_SELF,1,"non-finite residual=%g",res);
755: if (PetscAbsScalar(res) < 1e-10) {
756: star.rho = rho;
757: star.u = L.u - c*fl;
758: goto converged;
759: }
760: dfl = (L.rho < rho)
761: ? 1/PetscSqrtScalar(L.rho*rho)*(1 - 0.5*(rho-L.rho)/rho)
762: : 1/rho;
763: dfr = (R.rho < rho)
764: ? 1/PetscSqrtScalar(R.rho*rho)*(1 - 0.5*(rho-R.rho)/rho)
765: : 1/rho;
766: tmp = rho - res/(c*(dfr+dfl));
767: if (tmp <= 0) rho /= 2; /* Guard against Newton shooting off to a negative density */
768: else rho = tmp;
769: if (!((rho > 0) && isnormal(rho))) SETERRQ1(PETSC_COMM_SELF,1,"non-normal iterate rho=%g",rho);
770: }
771: SETERRQ1(PETSC_COMM_SELF,1,"Newton iteration for star.rho diverged after %d iterations",i);
772: }
773: converged:
774: if (L.u-c < 0 && 0 < star.u-c) { /* 1-wave is sonic rarefaction */
775: PetscScalar ufan[2];
776: ufan[0] = L.rho*PetscExpScalar(L.u/c - 1);
777: ufan[1] = c*ufan[0];
778: IsoGasFlux(c,ufan,flux);
779: } else if (star.u+c < 0 && 0 < R.u+c) { /* 2-wave is sonic rarefaction */
780: PetscScalar ufan[2];
781: ufan[0] = R.rho*PetscExpScalar(-R.u/c - 1);
782: ufan[1] = -c*ufan[0];
783: IsoGasFlux(c,ufan,flux);
784: } else if ((L.rho >= star.rho && L.u-c >= 0)
785: || (L.rho < star.rho && (star.rho*star.u-L.rho*L.u)/(star.rho-L.rho) > 0)) {
786: /* 1-wave is supersonic rarefaction, or supersonic shock */
787: IsoGasFlux(c,uL,flux);
788: } else if ((star.rho <= R.rho && R.u+c <= 0)
789: || (star.rho > R.rho && (R.rho*R.u-star.rho*star.u)/(R.rho-star.rho) < 0)) {
790: /* 2-wave is supersonic rarefaction or supersonic shock */
791: IsoGasFlux(c,uR,flux);
792: } else {
793: ustar[0] = star.rho;
794: ustar[1] = star.rho*star.u;
795: IsoGasFlux(c,ustar,flux);
796: }
797: *maxspeed = MaxAbs(MaxAbs(star.u-c,star.u+c),MaxAbs(L.u-c,R.u+c));
798: return(0);
799: }
803: static PetscErrorCode PhysicsRiemann_IsoGas_Rusanov(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
804: {
805: IsoGasCtx *phys = (IsoGasCtx*)vctx;
806: PetscScalar c = phys->acoustic_speed,fL[2],fR[2],s;
807: struct {PetscScalar rho,u;} L = {uL[0],uL[1]/uL[0]},R = {uR[0],uR[1]/uR[0]};
810: if (!(L.rho > 0 && R.rho > 0)) SETERRQ(PETSC_COMM_SELF,1,"Reconstructed density is negative");
811: IsoGasFlux(c,uL,fL);
812: IsoGasFlux(c,uR,fR);
813: s = PetscMax(PetscAbs(L.u),PetscAbs(R.u))+c;
814: flux[0] = 0.5*(fL[0] + fR[0]) + 0.5*s*(uL[0] - uR[0]);
815: flux[1] = 0.5*(fL[1] + fR[1]) + 0.5*s*(uL[1] - uR[1]);
816: *maxspeed = s;
817: return(0);
818: }
822: static PetscErrorCode PhysicsCharacteristic_IsoGas(void *vctx,PetscInt m,const PetscScalar *u,PetscScalar *X,PetscScalar *Xi)
823: {
824: IsoGasCtx *phys = (IsoGasCtx*)vctx;
825: PetscReal c = phys->acoustic_speed;
829: X[0*2+0] = 1;
830: X[0*2+1] = u[1]/u[0] - c;
831: X[1*2+0] = 1;
832: X[1*2+1] = u[1]/u[0] + c;
833: PetscMemcpy(Xi,X,4*sizeof(X[0]));
834: Kernel_A_gets_inverse_A_2(Xi,0);
835: return(0);
836: }
840: static PetscErrorCode PhysicsCreate_IsoGas(FVCtx *ctx)
841: {
843: IsoGasCtx *user;
844: PetscFList rlist = 0,rclist = 0;
845: char rname[256] = "exact",rcname[256] = "characteristic";
848: PetscNew(*user,&user);
849: ctx->physics.sample = PhysicsSample_IsoGas;
850: ctx->physics.destroy = PhysicsDestroy_SimpleFree;
851: ctx->physics.user = user;
852: ctx->physics.dof = 2;
853: PetscStrallocpy("density",&ctx->physics.fieldname[0]);
854: PetscStrallocpy("momentum",&ctx->physics.fieldname[1]);
855: user->acoustic_speed = 1;
856: RiemannListAdd(&rlist,"exact", PhysicsRiemann_IsoGas_Exact);
857: RiemannListAdd(&rlist,"roe", PhysicsRiemann_IsoGas_Roe);
858: RiemannListAdd(&rlist,"rusanov",PhysicsRiemann_IsoGas_Rusanov);
859: ReconstructListAdd(&rclist,"characteristic",PhysicsCharacteristic_IsoGas);
860: ReconstructListAdd(&rclist,"conservative",PhysicsCharacteristic_Conservative);
861: PetscOptionsBegin(ctx->comm,ctx->prefix,"Options for IsoGas","");
862: {
863: PetscOptionsReal("-physics_isogas_acoustic_speed","Acoustic speed","",user->acoustic_speed,&user->acoustic_speed,PETSC_NULL);
864: PetscOptionsList("-physics_isogas_riemann","Riemann solver","",rlist,rname,rname,sizeof rname,PETSC_NULL);
865: PetscOptionsList("-physics_isogas_reconstruct","Reconstruction","",rclist,rcname,rcname,sizeof rcname,PETSC_NULL);
866: }
867: PetscOptionsEnd();
868: RiemannListFind(rlist,rname,&ctx->physics.riemann);
869: ReconstructListFind(rclist,rcname,&ctx->physics.characteristic);
870: PetscFListDestroy(&rlist);
871: PetscFListDestroy(&rclist);
872: return(0);
873: }
877: /* --------------------------------- Shallow Water ----------------------------------- */
879: typedef struct {
880: PetscReal gravity;
881: } ShallowCtx;
883: static inline void ShallowFlux(ShallowCtx *phys,const PetscScalar *u,PetscScalar *f)
884: {
885: f[0] = u[1];
886: f[1] = PetscSqr(u[1])/u[0] + 0.5*phys->gravity*PetscSqr(u[0]);
887: }
891: static PetscErrorCode PhysicsRiemann_Shallow_Exact(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
892: {
893: ShallowCtx *phys = (ShallowCtx*)vctx;
894: PetscScalar g = phys->gravity,ustar[2],cL,cR,c,cstar;
895: struct {PetscScalar h,u;} L = {uL[0],uL[1]/uL[0]},R = {uR[0],uR[1]/uR[0]},star;
896: PetscInt i;
899: if (!(L.h > 0 && R.h > 0)) SETERRQ(PETSC_COMM_SELF,1,"Reconstructed thickness is negative");
900: cL = PetscSqrtScalar(g*L.h);
901: cR = PetscSqrtScalar(g*R.h);
902: c = PetscMax(cL,cR);
903: {
904: /* Solve for star state */
905: const PetscInt maxits = 50;
906: PetscScalar tmp,res,res0=0,h0,h = 0.5*(L.h + R.h); /* initial guess */
907: h0 = h;
908: for (i=0; i<maxits; i++) {
909: PetscScalar fr,fl,dfr,dfl;
910: fl = (L.h < h)
911: ? PetscSqrtScalar(0.5*g*(h*h - L.h*L.h)*(1/L.h - 1/h)) /* shock */
912: : 2*PetscSqrtScalar(g*h) - 2*PetscSqrtScalar(g*L.h); /* rarefaction */
913: fr = (R.h < h)
914: ? PetscSqrtScalar(0.5*g*(h*h - R.h*R.h)*(1/R.h - 1/h)) /* shock */
915: : 2*PetscSqrtScalar(g*h) - 2*PetscSqrtScalar(g*R.h); /* rarefaction */
916: res = R.u - L.u + fr + fl;
917: if (!isfinite(res)) SETERRQ1(PETSC_COMM_SELF,1,"non-finite residual=%g",res);
918: //PetscPrintf(PETSC_COMM_WORLD,"h=%g, res[%d] = %g\n",h,i,res);
919: if (PetscAbsScalar(res) < 1e-8 || (i > 0 && PetscAbsScalar(h-h0) < 1e-8)) {
920: star.h = h;
921: star.u = L.u - fl;
922: goto converged;
923: } else if (i > 0 && PetscAbsScalar(res) >= PetscAbsScalar(res0)) { /* Line search */
924: h = 0.8*h0 + 0.2*h;
925: continue;
926: }
927: /* Accept the last step and take another */
928: res0 = res;
929: h0 = h;
930: dfl = (L.h < h)
931: ? 0.5/fl*0.5*g*(-L.h*L.h/(h*h) - 1 + 2*h/L.h)
932: : PetscSqrtScalar(g/h);
933: dfr = (R.h < h)
934: ? 0.5/fr*0.5*g*(-R.h*R.h/(h*h) - 1 + 2*h/R.h)
935: : PetscSqrtScalar(g/h);
936: tmp = h - res/(dfr+dfl);
937: if (tmp <= 0) h /= 2; /* Guard against Newton shooting off to a negative thickness */
938: else h = tmp;
939: if (!((h > 0) && isnormal(h))) SETERRQ1(PETSC_COMM_SELF,1,"non-normal iterate h=%g",h);
940: }
941: SETERRQ1(PETSC_COMM_SELF,1,"Newton iteration for star.h diverged after %d iterations",i);
942: }
943: converged:
944: cstar = PetscSqrtScalar(g*star.h);
945: if (L.u-cL < 0 && 0 < star.u-cstar) { /* 1-wave is sonic rarefaction */
946: PetscScalar ufan[2];
947: ufan[0] = 1/g*PetscSqr(L.u/3 + 2./3*cL);
948: ufan[1] = PetscSqrtScalar(g*ufan[0])*ufan[0];
949: ShallowFlux(phys,ufan,flux);
950: } else if (star.u+cstar < 0 && 0 < R.u+cR) { /* 2-wave is sonic rarefaction */
951: PetscScalar ufan[2];
952: ufan[0] = 1/g*PetscSqr(R.u/3 - 2./3*cR);
953: ufan[1] = -PetscSqrtScalar(g*ufan[0])*ufan[0];
954: ShallowFlux(phys,ufan,flux);
955: } else if ((L.h >= star.h && L.u-c >= 0)
956: || (L.h<star.h && (star.h*star.u-L.h*L.u)/(star.h-L.h) > 0)) {
957: /* 1-wave is right-travelling shock (supersonic) */
958: ShallowFlux(phys,uL,flux);
959: } else if ((star.h <= R.h && R.u+c <= 0)
960: || (star.h>R.h && (R.h*R.u-star.h*star.h)/(R.h-star.h) < 0)) {
961: /* 2-wave is left-travelling shock (supersonic) */
962: ShallowFlux(phys,uR,flux);
963: } else {
964: ustar[0] = star.h;
965: ustar[1] = star.h*star.u;
966: ShallowFlux(phys,ustar,flux);
967: }
968: *maxspeed = MaxAbs(MaxAbs(star.u-cstar,star.u+cstar),MaxAbs(L.u-cL,R.u+cR));
969: return(0);
970: }
974: static PetscErrorCode PhysicsRiemann_Shallow_Rusanov(void *vctx,PetscInt m,const PetscScalar *uL,const PetscScalar *uR,PetscScalar *flux,PetscReal *maxspeed)
975: {
976: ShallowCtx *phys = (ShallowCtx*)vctx;
977: PetscScalar g = phys->gravity,fL[2],fR[2],s;
978: struct {PetscScalar h,u;} L = {uL[0],uL[1]/uL[0]},R = {uR[0],uR[1]/uR[0]};
981: if (!(L.h > 0 && R.h > 0)) SETERRQ(PETSC_COMM_SELF,1,"Reconstructed thickness is negative");
982: ShallowFlux(phys,uL,fL);
983: ShallowFlux(phys,uR,fR);
984: s = PetscMax(PetscAbs(L.u)+PetscSqrtScalar(g*L.h),PetscAbs(R.u)+PetscSqrtScalar(g*R.h));
985: flux[0] = 0.5*(fL[0] + fR[0]) + 0.5*s*(uL[0] - uR[0]);
986: flux[1] = 0.5*(fL[1] + fR[1]) + 0.5*s*(uL[1] - uR[1]);
987: *maxspeed = s;
988: return(0);
989: }
993: static PetscErrorCode PhysicsCharacteristic_Shallow(void *vctx,PetscInt m,const PetscScalar *u,PetscScalar *X,PetscScalar *Xi)
994: {
995: ShallowCtx *phys = (ShallowCtx*)vctx;
996: PetscReal c;
1000: c = PetscSqrtScalar(u[0]*phys->gravity);
1001: X[0*2+0] = 1;
1002: X[0*2+1] = u[1]/u[0] - c;
1003: X[1*2+0] = 1;
1004: X[1*2+1] = u[1]/u[0] + c;
1005: PetscMemcpy(Xi,X,4*sizeof(X[0]));
1006: Kernel_A_gets_inverse_A_2(Xi,0);
1007: return(0);
1008: }
1012: static PetscErrorCode PhysicsCreate_Shallow(FVCtx *ctx)
1013: {
1015: ShallowCtx *user;
1016: PetscFList rlist = 0,rclist = 0;
1017: char rname[256] = "exact",rcname[256] = "characteristic";
1020: PetscNew(*user,&user);
1021: /* Shallow water and Isothermal Gas dynamics are similar so we reuse initial conditions for now */
1022: ctx->physics.sample = PhysicsSample_IsoGas;
1023: ctx->physics.destroy = PhysicsDestroy_SimpleFree;
1024: ctx->physics.user = user;
1025: ctx->physics.dof = 2;
1026: PetscStrallocpy("density",&ctx->physics.fieldname[0]);
1027: PetscStrallocpy("momentum",&ctx->physics.fieldname[1]);
1028: user->gravity = 1;
1029: RiemannListAdd(&rlist,"exact", PhysicsRiemann_Shallow_Exact);
1030: RiemannListAdd(&rlist,"rusanov",PhysicsRiemann_Shallow_Rusanov);
1031: ReconstructListAdd(&rclist,"characteristic",PhysicsCharacteristic_Shallow);
1032: ReconstructListAdd(&rclist,"conservative",PhysicsCharacteristic_Conservative);
1033: PetscOptionsBegin(ctx->comm,ctx->prefix,"Options for Shallow","");
1034: {
1035: PetscOptionsReal("-physics_shallow_gravity","Gravity","",user->gravity,&user->gravity,PETSC_NULL);
1036: PetscOptionsList("-physics_shallow_riemann","Riemann solver","",rlist,rname,rname,sizeof rname,PETSC_NULL);
1037: PetscOptionsList("-physics_shallow_reconstruct","Reconstruction","",rclist,rcname,rcname,sizeof rcname,PETSC_NULL);
1038: }
1039: PetscOptionsEnd();
1040: RiemannListFind(rlist,rname,&ctx->physics.riemann);
1041: ReconstructListFind(rclist,rcname,&ctx->physics.characteristic);
1042: PetscFListDestroy(&rlist);
1043: PetscFListDestroy(&rclist);
1044: return(0);
1045: }
1049: /* --------------------------------- Finite Volume Solver ----------------------------------- */
1053: static PetscErrorCode FVRHSFunction(TS ts,PetscReal time,Vec X,Vec F,void *vctx)
1054: {
1055: FVCtx *ctx = (FVCtx*)vctx;
1056: PetscErrorCode ierr;
1057: PetscInt i,j,k,Mx,dof,xs,xm;
1058: PetscReal hx,cfl_idt = 0;
1059: PetscScalar *x,*f,*slope;
1060: Vec Xloc;
1063: DMGetLocalVector(ctx->da,&Xloc);
1064: DMDAGetInfo(ctx->da,0, &Mx,0,0, 0,0,0, &dof,0,0,0,0,0);
1065: hx = (ctx->xmax - ctx->xmin)/Mx;
1066: DMGlobalToLocalBegin(ctx->da,X,INSERT_VALUES,Xloc);
1067: DMGlobalToLocalEnd (ctx->da,X,INSERT_VALUES,Xloc);
1069: VecZeroEntries(F);
1071: DMDAVecGetArray(ctx->da,Xloc,&x);
1072: DMDAVecGetArray(ctx->da,F,&f);
1073: DMDAGetArray(ctx->da,PETSC_TRUE,&slope);
1075: DMDAGetCorners(ctx->da,&xs,0,0,&xm,0,0);
1077: if (ctx->bctype == FVBC_OUTFLOW) {
1078: for (i=xs-2; i<0; i++) {
1079: for (j=0; j<dof; j++) x[i*dof+j] = x[j];
1080: }
1081: for (i=Mx; i<xs+xm+2; i++) {
1082: for (j=0; j<dof; j++) x[i*dof+j] = x[(xs+xm-1)*dof+j];
1083: }
1084: }
1085: for (i=xs-1; i<xs+xm+1; i++) {
1086: struct _LimitInfo info;
1087: PetscScalar *cjmpL,*cjmpR;
1088: /* Determine the right eigenvectors R, where A = R \Lambda R^{-1} */
1089: (*ctx->physics.characteristic)(ctx->physics.user,dof,&x[i*dof],ctx->R,ctx->Rinv);
1090: /* Evaluate jumps across interfaces (i-1, i) and (i, i+1), put in characteristic basis */
1091: PetscMemzero(ctx->cjmpLR,2*dof*sizeof(ctx->cjmpLR[0]));
1092: cjmpL = &ctx->cjmpLR[0];
1093: cjmpR = &ctx->cjmpLR[dof];
1094: for (j=0; j<dof; j++) {
1095: PetscScalar jmpL,jmpR;
1096: jmpL = x[(i+0)*dof+j] - x[(i-1)*dof+j];
1097: jmpR = x[(i+1)*dof+j] - x[(i+0)*dof+j];
1098: for (k=0; k<dof; k++) {
1099: cjmpL[k] += ctx->Rinv[k+j*dof] * jmpL;
1100: cjmpR[k] += ctx->Rinv[k+j*dof] * jmpR;
1101: }
1102: }
1103: /* Apply limiter to the left and right characteristic jumps */
1104: info.m = dof;
1105: info.hx = hx;
1106: (*ctx->limit)(&info,cjmpL,cjmpR,ctx->cslope);
1107: for (j=0; j<dof; j++) ctx->cslope[j] /= hx; /* rescale to a slope */
1108: for (j=0; j<dof; j++) {
1109: PetscScalar tmp = 0;
1110: for (k=0; k<dof; k++) tmp += ctx->R[j+k*dof] * ctx->cslope[k];
1111: slope[i*dof+j] = tmp;
1112: }
1113: }
1115: for (i=xs; i<xs+xm+1; i++) {
1116: PetscReal maxspeed;
1117: PetscScalar *uL,*uR;
1118: uL = &ctx->uLR[0];
1119: uR = &ctx->uLR[dof];
1120: for (j=0; j<dof; j++) {
1121: uL[j] = x[(i-1)*dof+j] + slope[(i-1)*dof+j]*hx/2;
1122: uR[j] = x[(i-0)*dof+j] - slope[(i-0)*dof+j]*hx/2;
1123: }
1124: (*ctx->physics.riemann)(ctx->physics.user,dof,uL,uR,ctx->flux,&maxspeed);
1125: cfl_idt = PetscMax(cfl_idt,PetscAbsScalar(maxspeed/hx)); /* Max allowable value of 1/Delta t */
1127: if (i > xs) {
1128: for (j=0; j<dof; j++) f[(i-1)*dof+j] -= ctx->flux[j]/hx;
1129: }
1130: if (i < xs+xm) {
1131: for (j=0; j<dof; j++) f[i*dof+j] += ctx->flux[j]/hx;
1132: }
1133: }
1135: DMDAVecRestoreArray(ctx->da,Xloc,&x);
1136: DMDAVecRestoreArray(ctx->da,F,&f);
1137: DMDARestoreArray(ctx->da,PETSC_TRUE,&slope);
1138: DMRestoreLocalVector(ctx->da,&Xloc);
1140: MPI_Allreduce(&cfl_idt,&ctx->cfl_idt,1,MPIU_REAL,MPIU_MAX,((PetscObject)ctx->da)->comm);
1141: if (0) {
1142: /* We need to a way to inform the TS of a CFL constraint, this is a debugging fragment */
1143: PetscReal dt,tnow;
1144: TSGetTimeStep(ts,&dt);
1145: TSGetTime(ts,&tnow);
1146: if (dt > 0.5/ctx->cfl_idt) {
1147: if (1) {
1148: PetscPrintf(ctx->comm,"Stability constraint exceeded at t=%g, dt %g > %g\n",tnow,dt,0.5/ctx->cfl_idt);
1149: } else {
1150: SETERRQ2(PETSC_COMM_SELF,1,"Stability constraint exceeded, %g > %g",dt,ctx->cfl/ctx->cfl_idt);
1151: }
1152: }
1153: }
1154: return(0);
1155: }
1159: static PetscErrorCode FVSample(FVCtx *ctx,PetscReal time,Vec U)
1160: {
1162: PetscScalar *u,*uj;
1163: PetscInt i,j,k,dof,xs,xm,Mx;
1166: if (!ctx->physics.sample) SETERRQ(PETSC_COMM_SELF,1,"Physics has not provided a sampling function");
1167: DMDAGetInfo(ctx->da,0, &Mx,0,0, 0,0,0, &dof,0,0,0,0,0);
1168: DMDAGetCorners(ctx->da,&xs,0,0,&xm,0,0);
1169: DMDAVecGetArray(ctx->da,U,&u);
1170: PetscMalloc(dof*sizeof uj[0],&uj);
1171: for (i=xs; i<xs+xm; i++) {
1172: const PetscReal h = (ctx->xmax-ctx->xmin)/Mx,xi = ctx->xmin+h/2+i*h;
1173: const PetscInt N = 200;
1174: /* Integrate over cell i using trapezoid rule with N points. */
1175: for (k=0; k<dof; k++) u[i*dof+k] = 0;
1176: for (j=0; j<N+1; j++) {
1177: PetscScalar xj = xi+h*(j-N/2)/(PetscReal)N;
1178: (*ctx->physics.sample)(ctx->physics.user,ctx->initial,ctx->bctype,ctx->xmin,ctx->xmax,time,xj,uj);
1179: for (k=0; k<dof; k++) u[i*dof+k] += ((j==0 || j==N)?0.5:1.0)*uj[k]/N;
1180: }
1181: }
1182: DMDAVecRestoreArray(ctx->da,U,&u);
1183: PetscFree(uj);
1184: return(0);
1185: }
1189: static PetscErrorCode SolutionStatsView(DM da,Vec X,PetscViewer viewer)
1190: {
1192: PetscReal xmin,xmax;
1193: PetscScalar sum,*x,tvsum,tvgsum;
1194: PetscInt imin,imax,Mx,i,j,xs,xm,dof;
1195: Vec Xloc;
1196: PetscBool iascii;
1199: PetscTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&iascii);
1200: if (iascii) {
1201: /* PETSc lacks a function to compute total variation norm (difficult in multiple dimensions), we do it here */
1202: DMGetLocalVector(da,&Xloc);
1203: DMGlobalToLocalBegin(da,X,INSERT_VALUES,Xloc);
1204: DMGlobalToLocalEnd (da,X,INSERT_VALUES,Xloc);
1205: DMDAVecGetArray(da,Xloc,&x);
1206: DMDAGetCorners(da,&xs,0,0,&xm,0,0);
1207: DMDAGetInfo(da,0, &Mx,0,0, 0,0,0, &dof,0,0,0,0,0);
1208: tvsum = 0;
1209: for (i=xs; i<xs+xm; i++) {
1210: for (j=0; j<dof; j++) tvsum += PetscAbsScalar(x[i*dof+j] - x[(i-1)*dof+j]);
1211: }
1212: MPI_Allreduce(&tvsum,&tvgsum,1,MPIU_REAL,MPIU_MAX,((PetscObject)da)->comm);
1213: DMDAVecRestoreArray(da,Xloc,&x);
1214: DMRestoreLocalVector(da,&Xloc);
1216: VecMin(X,&imin,&xmin);
1217: VecMax(X,&imax,&xmax);
1218: VecSum(X,&sum);
1219: PetscViewerASCIIPrintf(viewer,"Solution range [%8.5f,%8.5f] with extrema at %d and %d, mean %8.5f, ||x||_TV %8.5f\n",xmin,xmax,imin,imax,sum/Mx,tvgsum/Mx);
1220: } else {
1221: SETERRQ(PETSC_COMM_SELF,1,"Viewer type not supported");
1222: }
1223: return(0);
1224: }
1228: static PetscErrorCode SolutionErrorNorms(FVCtx *ctx,PetscReal t,Vec X,PetscReal *nrm1,PetscReal *nrmsup)
1229: {
1231: Vec Y;
1232: PetscInt Mx;
1235: VecGetSize(X,&Mx);
1236: VecDuplicate(X,&Y);
1237: FVSample(ctx,t,Y);
1238: VecAYPX(Y,-1,X);
1239: VecNorm(Y,NORM_1,nrm1);
1240: VecNorm(Y,NORM_INFINITY,nrmsup);
1241: *nrm1 /= Mx;
1242: VecDestroy(&Y);
1243: return(0);
1244: }
1249: int main(int argc,char *argv[])
1250: {
1251: char lname[256] = "mc",physname[256] = "advect",final_fname[256] = "solution.m";
1252: PetscFList limiters = 0,physics = 0;
1253: MPI_Comm comm;
1254: TS ts;
1255: Vec X,X0,R;
1256: FVCtx ctx;
1257: PetscInt i,dof,xs,xm,Mx,draw = 0;
1258: PetscBool view_final = PETSC_FALSE;
1259: PetscReal ptime;
1262: PetscInitialize(&argc,&argv,0,help);
1263: comm = PETSC_COMM_WORLD;
1264: PetscMemzero(&ctx,sizeof(ctx));
1266: /* Register limiters to be available on the command line */
1267: PetscFListAdd(&limiters,"upwind" ,"",(void(*)(void))Limit_Upwind);
1268: PetscFListAdd(&limiters,"lax-wendroff" ,"",(void(*)(void))Limit_LaxWendroff);
1269: PetscFListAdd(&limiters,"beam-warming" ,"",(void(*)(void))Limit_BeamWarming);
1270: PetscFListAdd(&limiters,"fromm" ,"",(void(*)(void))Limit_Fromm);
1271: PetscFListAdd(&limiters,"minmod" ,"",(void(*)(void))Limit_Minmod);
1272: PetscFListAdd(&limiters,"superbee" ,"",(void(*)(void))Limit_Superbee);
1273: PetscFListAdd(&limiters,"mc" ,"",(void(*)(void))Limit_MC);
1274: PetscFListAdd(&limiters,"vanleer" ,"",(void(*)(void))Limit_VanLeer);
1275: PetscFListAdd(&limiters,"vanalbada" ,"",(void(*)(void))Limit_VanAlbada);
1276: PetscFListAdd(&limiters,"vanalbadatvd" ,"",(void(*)(void))Limit_VanAlbadaTVD);
1277: PetscFListAdd(&limiters,"koren" ,"",(void(*)(void))Limit_Koren);
1278: PetscFListAdd(&limiters,"korensym" ,"",(void(*)(void))Limit_KorenSym);
1279: PetscFListAdd(&limiters,"koren3" ,"",(void(*)(void))Limit_Koren3);
1280: PetscFListAdd(&limiters,"cada-torrilhon2" ,"",(void(*)(void))Limit_CadaTorrilhon2);
1281: PetscFListAdd(&limiters,"cada-torrilhon3-r0p1","",(void(*)(void))Limit_CadaTorrilhon3R0p1);
1282: PetscFListAdd(&limiters,"cada-torrilhon3-r1" ,"",(void(*)(void))Limit_CadaTorrilhon3R1);
1283: PetscFListAdd(&limiters,"cada-torrilhon3-r10" ,"",(void(*)(void))Limit_CadaTorrilhon3R10);
1284: PetscFListAdd(&limiters,"cada-torrilhon3-r100","",(void(*)(void))Limit_CadaTorrilhon3R100);
1286: /* Register physical models to be available on the command line */
1287: PetscFListAdd(&physics,"advect" ,"",(void(*)(void))PhysicsCreate_Advect);
1288: PetscFListAdd(&physics,"burgers" ,"",(void(*)(void))PhysicsCreate_Burgers);
1289: PetscFListAdd(&physics,"traffic" ,"",(void(*)(void))PhysicsCreate_Traffic);
1290: PetscFListAdd(&physics,"isogas" ,"",(void(*)(void))PhysicsCreate_IsoGas);
1291: PetscFListAdd(&physics,"shallow" ,"",(void(*)(void))PhysicsCreate_Shallow);
1293: ctx.cfl = 0.9; ctx.bctype = FVBC_PERIODIC;
1294: ctx.xmin = 0; ctx.xmax = 1;
1295: PetscOptionsBegin(comm,PETSC_NULL,"Finite Volume solver options","");
1296: {
1297: PetscOptionsReal("-xmin","X min","",ctx.xmin,&ctx.xmin,PETSC_NULL);
1298: PetscOptionsReal("-xmax","X max","",ctx.xmax,&ctx.xmax,PETSC_NULL);
1299: PetscOptionsList("-limit","Name of flux limiter to use","",limiters,lname,lname,sizeof(lname),PETSC_NULL);
1300: PetscOptionsList("-physics","Name of physics (Riemann solver and characteristics) to use","",physics,physname,physname,sizeof(physname),PETSC_NULL);
1301: PetscOptionsInt("-draw","Draw solution vector, bitwise OR of (1=initial,2=final,4=final error)","",draw,&draw,PETSC_NULL);
1302: PetscOptionsString("-view_final","Write final solution in ASCII MATLAB format to given file name","",final_fname,final_fname,sizeof final_fname,&view_final);
1303: PetscOptionsInt("-initial","Initial condition (depends on the physics)","",ctx.initial,&ctx.initial,PETSC_NULL);
1304: PetscOptionsBool("-exact","Compare errors with exact solution","",ctx.exact,&ctx.exact,PETSC_NULL);
1305: PetscOptionsReal("-cfl","CFL number to time step at","",ctx.cfl,&ctx.cfl,PETSC_NULL);
1306: PetscOptionsEnum("-bc_type","Boundary condition","",FVBCTypes,(PetscEnum)ctx.bctype,(PetscEnum*)&ctx.bctype,PETSC_NULL);
1307: }
1308: PetscOptionsEnd();
1310: /* Choose the limiter from the list of registered limiters */
1311: PetscFListFind(limiters,comm,lname,PETSC_FALSE,(void(**)(void))&ctx.limit);
1312: if (!ctx.limit) SETERRQ1(PETSC_COMM_SELF,1,"Limiter '%s' not found",lname);
1314: /* Choose the physics from the list of registered models */
1315: {
1316: PetscErrorCode (*r)(FVCtx*);
1317: PetscFListFind(physics,comm,physname,PETSC_FALSE,(void(**)(void))&r);
1318: if (!r) SETERRQ1(PETSC_COMM_SELF,1,"Physics '%s' not found",physname);
1319: /* Create the physics, will set the number of fields and their names */
1320: (*r)(&ctx);
1321: }
1323: /* Create a DMDA to manage the parallel grid */
1324: DMDACreate1d(comm,DMDA_BOUNDARY_PERIODIC,-50,ctx.physics.dof,2,PETSC_NULL,&ctx.da);
1325: /* Inform the DMDA of the field names provided by the physics. */
1326: /* The names will be shown in the title bars when run with -ts_monitor_solution */
1327: for (i=0; i<ctx.physics.dof; i++) {
1328: DMDASetFieldName(ctx.da,i,ctx.physics.fieldname[i]);
1329: }
1330: DMDAGetInfo(ctx.da,0, &Mx,0,0, 0,0,0, &dof,0,0,0,0,0);
1331: DMDAGetCorners(ctx.da,&xs,0,0,&xm,0,0);
1333: /* Set coordinates of cell centers */
1334: DMDASetUniformCoordinates(ctx.da,ctx.xmin+0.5*(ctx.xmax-ctx.xmin)/Mx,ctx.xmax+0.5*(ctx.xmax-ctx.xmin)/Mx,0,0,0,0);
1336: /* Allocate work space for the Finite Volume solver (so it doesn't have to be reallocated on each function evaluation) */
1337: PetscMalloc4(dof*dof,PetscScalar,&ctx.R,dof*dof,PetscScalar,&ctx.Rinv,2*dof,PetscScalar,&ctx.cjmpLR,1*dof,PetscScalar,&ctx.cslope);
1338: PetscMalloc2(2*dof,PetscScalar,&ctx.uLR,dof,PetscScalar,&ctx.flux);
1340: /* Create a vector to store the solution and to save the initial state */
1341: DMCreateGlobalVector(ctx.da,&X);
1342: VecDuplicate(X,&X0);
1343: VecDuplicate(X,&R);
1345: /* Create a time-stepping object */
1346: TSCreate(comm,&ts);
1347: TSSetRHSFunction(ts,R,FVRHSFunction,&ctx);
1348: TSSetType(ts,TSSSP);
1349: TSSetDuration(ts,1000,10);
1351: /* Compute initial conditions and starting time step */
1352: FVSample(&ctx,0,X0);
1353: FVRHSFunction(ts,0,X0,X,(void*)&ctx); /* Initial function evaluation, only used to determine max speed */
1354: VecCopy(X0,X); /* The function value was not used so we set X=X0 again */
1355: TSSetInitialTimeStep(ts,0,ctx.cfl/ctx.cfl_idt);
1357: TSSetFromOptions(ts); /* Take runtime options */
1359: SolutionStatsView(ctx.da,X,PETSC_VIEWER_STDOUT_WORLD);
1361: {
1362: PetscReal nrm1,nrmsup;
1363: PetscInt steps;
1365: TSSolve(ts,X,&ptime);
1366: TSGetTimeStepNumber(ts,&steps);
1368: PetscPrintf(comm,"Final time %8.5f, steps %d\n",ptime,steps);
1369: if (ctx.exact) {
1370: SolutionErrorNorms(&ctx,ptime,X,&nrm1,&nrmsup);
1371: PetscPrintf(comm,"Error ||x-x_e||_1 %8.4e ||x-x_e||_sup %8.4e\n",nrm1,nrmsup);
1372: }
1373: }
1375: SolutionStatsView(ctx.da,X,PETSC_VIEWER_STDOUT_WORLD);
1377: if (draw & 0x1) {VecView(X0,PETSC_VIEWER_DRAW_WORLD);}
1378: if (draw & 0x2) {VecView(X,PETSC_VIEWER_DRAW_WORLD);}
1379: if (draw & 0x4) {
1380: Vec Y;
1381: VecDuplicate(X,&Y);
1382: FVSample(&ctx,ptime,Y);
1383: VecAYPX(Y,-1,X);
1384: VecView(Y,PETSC_VIEWER_DRAW_WORLD);
1385: VecDestroy(&Y);
1386: }
1388: if (view_final) {
1389: PetscViewer viewer;
1390: PetscViewerASCIIOpen(PETSC_COMM_WORLD,final_fname,&viewer);
1391: PetscViewerSetFormat(viewer,PETSC_VIEWER_ASCII_MATLAB);
1392: VecView(X,viewer);
1393: PetscViewerDestroy(&viewer);
1394: }
1396: /* Clean up */
1397: (*ctx.physics.destroy)(ctx.physics.user);
1398: for (i=0; i<ctx.physics.dof; i++) {PetscFree(ctx.physics.fieldname[i]);}
1399: PetscFree4(ctx.R,ctx.Rinv,ctx.cjmpLR,ctx.cslope);
1400: PetscFree2(ctx.uLR,ctx.flux);
1401: VecDestroy(&X);
1402: VecDestroy(&X0);
1403: VecDestroy(&R);
1404: DMDestroy(&ctx.da);
1405: TSDestroy(&ts);
1406: PetscFinalize();
1407: return 0;
1408: }