Actual source code: ex3.c
petsc-3.3-p7 2013-05-11
2: static char help[] = "Solves 1D heat equation with FEM formulation.\n\
3: Input arguments are\n\
4: -useAlhs: solve Alhs*U' = (Arhs*U + g) \n\
5: otherwise, solve U' = inv(Alhs)*(Arhs*U + g) \n\n";
7: /*--------------------------------------------------------------------------
8: Solves 1D heat equation U_t = U_xx with FEM formulation:
9: Alhs*U' = rhs (= Arhs*U + g)
10: We thank Chris Cox <clcox@clemson.edu> for contributing the original code
11: ----------------------------------------------------------------------------*/
13: #include <petscksp.h>
14: #include <petscts.h>
16: /* special variable - max size of all arrays */
17: #define num_z 60
19: /*
20: User-defined application context - contains data needed by the
21: application-provided call-back routines.
22: */
23: typedef struct{
24: Mat Amat; /* left hand side matrix */
25: Vec ksp_rhs,ksp_sol; /* working vectors for formulating inv(Alhs)*(Arhs*U+g) */
26: int max_probsz; /* max size of the problem */
27: PetscBool useAlhs; /* flag (1 indicates solving Alhs*U' = Arhs*U+g */
28: int nz; /* total number of grid points */
29: PetscInt m; /* total number of interio grid points */
30: Vec solution; /* global exact ts solution vector */
31: PetscScalar *z; /* array of grid points */
32: PetscBool debug; /* flag (1 indicates activation of debugging printouts) */
33: } AppCtx;
35: extern PetscScalar exact(PetscScalar,PetscReal);
36: extern PetscErrorCode Monitor(TS,PetscInt,PetscReal,Vec,void*);
37: extern PetscErrorCode Petsc_KSPSolve(AppCtx*);
38: extern PetscScalar bspl(PetscScalar*,PetscScalar,PetscInt,PetscInt,PetscInt[][2],PetscInt);
39: extern void femBg(PetscScalar[][3],PetscScalar*,PetscInt,PetscScalar*,PetscReal);
40: extern void femA(AppCtx*,PetscInt,PetscScalar*);
41: extern void rhs(AppCtx*,PetscScalar*, PetscInt, PetscScalar*,PetscReal);
42: extern PetscErrorCode RHSfunction(TS,PetscReal,Vec,Vec,void*);
46: int main(int argc,char **argv)
47: {
48: PetscInt i,m,nz,steps,max_steps,k,nphase=1;
49: PetscScalar zInitial,zFinal,val,*z;
50: PetscReal stepsz[4],T,ftime;
52: TS ts;
53: SNES snes;
54: Mat Jmat;
55: AppCtx appctx; /* user-defined application context */
56: Vec init_sol; /* ts solution vector */
57: PetscMPIInt size;
59: PetscInitialize(&argc,&argv,(char*)0,help);
60: MPI_Comm_size(PETSC_COMM_WORLD,&size);
61: if (size != 1) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"This is a uniprocessor example only");
63: PetscOptionsHasName(PETSC_NULL,"-debug",&appctx.debug);
64: PetscOptionsHasName(PETSC_NULL,"-useAlhs",&appctx.useAlhs);
65: PetscOptionsGetInt(PETSC_NULL,"-nphase",&nphase,PETSC_NULL);
66: if (nphase > 3) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"nphase must be an integer between 1 and 3");
68: /* initializations */
69: zInitial = 0.0;
70: zFinal = 1.0;
71: T = 0.014/nphase;
72: nz = num_z;
73: m = nz-2;
74: appctx.nz = nz;
75: max_steps = (PetscInt)10000;
77: appctx.m = m;
78: appctx.max_probsz = nz;
79: appctx.debug = PETSC_FALSE;
80: appctx.useAlhs = PETSC_FALSE;
82: /* create vector to hold ts solution */
83: /*-----------------------------------*/
84: VecCreate(PETSC_COMM_WORLD, &init_sol);
85: VecSetSizes(init_sol, PETSC_DECIDE, m);
86: VecSetFromOptions(init_sol);
88: /* create vector to hold true ts soln for comparison */
89: VecDuplicate(init_sol, &appctx.solution);
91: /* create LHS matrix Amat */
92: /*------------------------*/
93: MatCreateSeqAIJ(PETSC_COMM_WORLD, m, m, 3, PETSC_NULL, &appctx.Amat);
94: MatSetFromOptions(appctx.Amat);
95: /* set space grid points - interio points only! */
96: PetscMalloc((nz+1)*sizeof(PetscScalar),&z);
97: for (i=0; i<nz; i++) z[i]=(i)*((zFinal-zInitial)/(nz-1));
98: appctx.z = z;
99: femA(&appctx,nz,z);
101: /* create the jacobian matrix */
102: /*----------------------------*/
103: MatCreate(PETSC_COMM_WORLD, &Jmat);
104: MatSetSizes(Jmat,PETSC_DECIDE,PETSC_DECIDE,m,m);
105: MatSetFromOptions(Jmat);
107: /* create working vectors for formulating rhs=inv(Alhs)*(Arhs*U + g) */
108: VecDuplicate(init_sol,&appctx.ksp_rhs);
109: VecDuplicate(init_sol,&appctx.ksp_sol);
111: /* set intial guess */
112: /*------------------*/
113: for(i=0; i<nz-2; i++){
114: val = exact(z[i+1], 0.0);
115: VecSetValue(init_sol,i,(PetscScalar)val,INSERT_VALUES);
116: }
117: VecAssemblyBegin(init_sol);
118: VecAssemblyEnd(init_sol);
120: /*create a time-stepping context and set the problem type */
121: /*--------------------------------------------------------*/
122: TSCreate(PETSC_COMM_WORLD, &ts);
123: TSSetProblemType(ts,TS_NONLINEAR);
125: /* set time-step method */
126: TSSetType(ts,TSCN);
128: /* Set optional user-defined monitoring routine */
129: TSMonitorSet(ts,Monitor,&appctx,PETSC_NULL);
130: /* set the right hand side of U_t = RHSfunction(U,t) */
131: TSSetRHSFunction(ts,PETSC_NULL,(PetscErrorCode (*)(TS,PetscScalar,Vec,Vec,void*))RHSfunction,&appctx);
133: if (appctx.useAlhs){
134: /* set the left hand side matrix of Amat*U_t = rhs(U,t) */
135: TSSetIFunction(ts,PETSC_NULL,TSComputeIFunctionLinear,&appctx);
136: TSSetIJacobian(ts,appctx.Amat,appctx.Amat,TSComputeIJacobianConstant,&appctx);
137: }
139: /* use petsc to compute the jacobian by finite differences */
140: TSGetSNES(ts,&snes);
141: SNESSetJacobian(snes,Jmat,Jmat,SNESDefaultComputeJacobian,PETSC_NULL);
143: /* get the command line options if there are any and set them */
144: TSSetFromOptions(ts);
146: #ifdef PETSC_HAVE_SUNDIALS
147: {
148: const TSType type;
149: PetscBool sundialstype=PETSC_FALSE;
150: TSGetType(ts,&type);
151: PetscObjectTypeCompare((PetscObject)ts,TSSUNDIALS,&sundialstype);
152: if (sundialstype && appctx.useAlhs) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Cannot use Alhs formulation for TSSUNDIALS type");
153: }
154: #endif
155: /* Sets the initial solution */
156: TSSetSolution(ts,init_sol);
158: stepsz[0] = 1.0/(2.0*(nz-1)*(nz-1)); /* (mesh_size)^2/2.0 */
159: ftime = 0.0;
160: for (k=0; k<nphase; k++){
161: if (nphase > 1) {
162: printf("Phase %d: initial time %g, stepsz %g, duration: %g\n",k,ftime,stepsz[k],(k+1)*T);
163: }
164: TSSetInitialTimeStep(ts,ftime,stepsz[k]);
165: TSSetDuration(ts,max_steps,(k+1)*T);
167: /* loop over time steps */
168: /*----------------------*/
169: TSSolve(ts,init_sol,&ftime);
170: TSGetTimeStepNumber(ts,&steps);
171: stepsz[k+1] = stepsz[k]*1.5; /* change step size for the next phase */
172: }
174: /* free space */
175: TSDestroy(&ts);
176: MatDestroy(&appctx.Amat);
177: MatDestroy(&Jmat);
178: VecDestroy(&appctx.ksp_rhs);
179: VecDestroy(&appctx.ksp_sol);
180: VecDestroy(&init_sol);
181: VecDestroy(&appctx.solution);
182: PetscFree(z);
184: PetscFinalize();
185: return 0;
186: }
188: /*------------------------------------------------------------------------
189: Set exact solution
190: u(z,t) = sin(6*PI*z)*exp(-36.*PI*PI*t) + 3.*sin(2*PI*z)*exp(-4.*PI*PI*t)
191: --------------------------------------------------------------------------*/
192: PetscScalar exact(PetscScalar z,PetscReal t)
193: {
194: PetscScalar val, ex1, ex2;
196: ex1 = exp(-36.*PETSC_PI*PETSC_PI*t);
197: ex2 = exp(-4.*PETSC_PI*PETSC_PI*t);
198: val = sin(6*PETSC_PI*z)*ex1 + 3.*sin(2*PETSC_PI*z)*ex2;
199: return val;
200: }
204: /*
205: Monitor - User-provided routine to monitor the solution computed at
206: each timestep. This example plots the solution and computes the
207: error in two different norms.
209: Input Parameters:
210: ts - the timestep context
211: step - the count of the current step (with 0 meaning the
212: initial condition)
213: time - the current time
214: u - the solution at this timestep
215: ctx - the user-provided context for this monitoring routine.
216: In this case we use the application context which contains
217: information about the problem size, workspace and the exact
218: solution.
219: */
220: PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal time,Vec u,void *ctx)
221: {
222: AppCtx *appctx = (AppCtx*)ctx;
224: PetscInt i,m=appctx->m;
225: PetscReal norm_2,norm_max,h=1.0/(m+1);
226: PetscScalar *u_exact;
228: /* Compute the exact solution */
229: VecGetArray(appctx->solution,&u_exact);
230: for (i=0; i<m; i++){
231: u_exact[i] = exact(appctx->z[i+1],time);
232: }
233: VecRestoreArray(appctx->solution,&u_exact);
235: /* Print debugging information if desired */
236: if (appctx->debug) {
237: PetscPrintf(PETSC_COMM_SELF,"Computed solution vector at time %g\n",time);
238: VecView(u,PETSC_VIEWER_STDOUT_SELF);
239: PetscPrintf(PETSC_COMM_SELF,"Exact solution vector\n");
240: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
241: }
243: /* Compute the 2-norm and max-norm of the error */
244: VecAXPY(appctx->solution,-1.0,u);
245: VecNorm(appctx->solution,NORM_2,&norm_2);
247: norm_2 = PetscSqrtReal(h)*norm_2;
248: VecNorm(appctx->solution,NORM_MAX,&norm_max);
250: PetscPrintf(PETSC_COMM_SELF,"Timestep %D: time = %G, 2-norm error = %6.4f, max norm error = %6.4f\n",
251: step,time,norm_2,norm_max);
253: /*
254: Print debugging information if desired
255: */
256: if (appctx->debug) {
257: PetscPrintf(PETSC_COMM_SELF,"Error vector\n");
258: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
259: }
260: return 0;
261: }
263: /*%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
264: %% Function to solve a linear system using KSP %%
265: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%*/
267: PetscErrorCode Petsc_KSPSolve(AppCtx *obj)
268: {
269: PetscErrorCode ierr;
270: KSP ksp;
271: PC pc;
273: /*create the ksp context and set the operators,that is, associate the system matrix with it*/
274: KSPCreate(PETSC_COMM_WORLD,&ksp);
275: KSPSetOperators(ksp,obj->Amat,obj->Amat,DIFFERENT_NONZERO_PATTERN);
277: /*get the preconditioner context, set its type and the tolerances*/
278: KSPGetPC(ksp,&pc);
279: PCSetType(pc,PCLU);
280: KSPSetTolerances(ksp,1.e-7,PETSC_DEFAULT,PETSC_DEFAULT,PETSC_DEFAULT);
282: /*get the command line options if there are any and set them*/
283: KSPSetFromOptions(ksp);
285: /*get the linear system (ksp) solve*/
286: KSPSolve(ksp,obj->ksp_rhs,obj->ksp_sol);
288: KSPDestroy(&ksp);
289: return 0;
290: }
292: /***********************************************************************
293: * Function to return value of basis function or derivative of basis *
294: * function. *
295: ***********************************************************************
296: * *
297: * Arguments: *
298: * x = array of xpoints or nodal values *
299: * xx = point at which the basis function is to be *
300: * evaluated. *
301: * il = interval containing xx. *
302: * iq = indicates which of the two basis functions in *
303: * interval intrvl should be used *
304: * nll = array containing the endpoints of each interval. *
305: * id = If id ~= 2, the value of the basis function *
306: * is calculated; if id = 2, the value of the *
307: * derivative of the basis function is returned. *
308: ***********************************************************************/
310: PetscScalar bspl(PetscScalar *x, PetscScalar xx,PetscInt il,PetscInt iq,PetscInt nll[][2],PetscInt id)
311: {
312: PetscScalar x1,x2,bfcn;
313: PetscInt i1,i2,iq1,iq2;
315: /*** Determine which basis function in interval intrvl is to be used in ***/
316: iq1 = iq;
317: if(iq1==0) iq2 = 1;
318: else iq2 = 0;
320: /*** Determine endpoint of the interval intrvl ***/
321: i1=nll[il][iq1];
322: i2=nll[il][iq2];
324: /*** Determine nodal values at the endpoints of the interval intrvl ***/
325: x1=x[i1];
326: x2=x[i2];
327: //printf("x1=%g\tx2=%g\txx=%g\n",x1,x2,xx);
328: /*** Evaluate basis function ***/
329: if(id == 2) bfcn=(1.0)/(x1-x2);
330: else bfcn=(xx-x2)/(x1-x2);
331: //printf("bfcn=%g\n",bfcn);
332: return bfcn;
333: }
335: /*---------------------------------------------------------
336: Function called by rhs function to get B and g
337: ---------------------------------------------------------*/
338: void femBg(PetscScalar btri[][3],PetscScalar *f,PetscInt nz,PetscScalar *z, PetscReal t)
339: {
340: PetscInt i,j,jj,il,ip,ipp,ipq,iq,iquad,iqq;
341: PetscInt nli[num_z][2],indx[num_z];
342: PetscScalar dd,dl,zip,zipq,zz,bb,b_z,bbb,bb_z,bij;
343: PetscScalar zquad[num_z][3],dlen[num_z],qdwt[3];
345: /* initializing everything - btri and f are initialized in rhs.c */
346: for(i=0; i < nz; i++){
347: nli[i][0] = 0;
348: nli[i][1] = 0;
349: indx[i] = 0;
350: zquad[i][0] = 0.0;
351: zquad[i][1] = 0.0;
352: zquad[i][2] = 0.0;
353: dlen[i] = 0.0;
354: }/*end for(i)*/
356: /* quadrature weights */
357: qdwt[0] = 1.0/6.0;
358: qdwt[1] = 4.0/6.0;
359: qdwt[2] = 1.0/6.0;
361: /* 1st and last nodes have Dirichlet boundary condition -
362: set indices there to -1 */
364: for(i=0; i < nz-1; i++){
365: indx[i]=i-1;
366: }
367: indx[nz-1]=-1;
369: ipq = 0;
370: for (il=0; il < nz-1; il++){
371: ip = ipq;
372: ipq = ip+1;
373: zip = z[ip];
374: zipq = z[ipq];
375: dl = zipq-zip;
376: zquad[il][0] = zip;
377: zquad[il][1] = (0.5)*(zip+zipq);
378: zquad[il][2] = zipq;
379: dlen[il] = fabs(dl);
380: nli[il][0] = ip;
381: nli[il][1] = ipq;
382: }
384: for (il=0; il < nz-1; il++){
385: for (iquad=0; iquad < 3; iquad++){
386: dd = (dlen[il])*(qdwt[iquad]);
387: zz = zquad[il][iquad];
389: for (iq=0; iq < 2; iq++){
390: ip = nli[il][iq];
391: bb = bspl(z,zz,il,iq,nli,1);
392: b_z = bspl(z,zz,il,iq,nli,2);
393: i = indx[ip];
395: if(i > -1){
396: for(iqq=0; iqq < 2; iqq++){
397: ipp = nli[il][iqq];
398: bbb = bspl(z,zz,il,iqq,nli,1);
399: bb_z = bspl(z,zz,il,iqq,nli,2);
400: j = indx[ipp];
401: bij = -b_z*bb_z;
403: if (j > -1){
404: jj = 1+j-i;
405: btri[i][jj] += bij*dd;
406: } else {
407: f[i] += bij*dd*exact(z[ipp], t);
408: // f[i] += 0.0;
409: // if(il==0 && j==-1){
410: // f[i] += bij*dd*exact(zz,t);
411: // }/*end if*/
412: } /*end else*/
413: }/*end for(iqq)*/
414: }/*end if(i>0)*/
415: }/*end for(iq)*/
416: }/*end for(iquad)*/
417: }/*end for(il)*/
418: return;
419: }
421: void femA(AppCtx *obj,PetscInt nz,PetscScalar *z)
422: {
423: PetscInt i,j,il,ip,ipp,ipq,iq,iquad,iqq;
424: PetscInt nli[num_z][2],indx[num_z];
425: PetscScalar dd,dl,zip,zipq,zz,bb,bbb,aij;
426: PetscScalar rquad[num_z][3],dlen[num_z],qdwt[3],add_term;
427: PetscErrorCode ierr;
429: /* initializing everything */
431: for(i=0; i < nz; i++)
432: {
433: nli[i][0] = 0;
434: nli[i][1] = 0;
435: indx[i] = 0;
436: rquad[i][0] = 0.0;
437: rquad[i][1] = 0.0;
438: rquad[i][2] = 0.0;
439: dlen[i] = 0.0;
440: }/*end for(i)*/
442: /* quadrature weights */
443: qdwt[0] = 1.0/6.0;
444: qdwt[1] = 4.0/6.0;
445: qdwt[2] = 1.0/6.0;
447: /* 1st and last nodes have Dirichlet boundary condition -
448: set indices there to -1 */
450: for(i=0; i < nz-1; i++)
451: {
452: indx[i]=i-1;
454: }/*end for(i)*/
455: indx[nz-1]=-1;
457: ipq = 0;
459: for(il=0; il < nz-1; il++)
460: {
461: ip = ipq;
462: ipq = ip+1;
463: zip = z[ip];
464: zipq = z[ipq];
465: dl = zipq-zip;
466: rquad[il][0] = zip;
467: rquad[il][1] = (0.5)*(zip+zipq);
468: rquad[il][2] = zipq;
469: dlen[il] = fabs(dl);
470: nli[il][0] = ip;
471: nli[il][1] = ipq;
473: }/*end for(il)*/
475: for(il=0; il < nz-1; il++){
476: for(iquad=0; iquad < 3; iquad++){
477: dd = (dlen[il])*(qdwt[iquad]);
478: zz = rquad[il][iquad];
480: for(iq=0; iq < 2; iq++){
481: ip = nli[il][iq];
482: bb = bspl(z,zz,il,iq,nli,1);
483: i = indx[ip];
484: if(i > -1){
485: for(iqq=0; iqq < 2; iqq++){
486: ipp = nli[il][iqq];
487: bbb = bspl(z,zz,il,iqq,nli,1);
488: j = indx[ipp];
489: aij = bb*bbb;
490: if(j > -1) {
491: add_term = aij*dd;
492: MatSetValue(obj->Amat,i,j,add_term,ADD_VALUES);
493: }/*endif*/
494: }/*end for(iqq)*/
495: }/*end if(i>0)*/
496: }/*end for(iq)*/
497: }/*end for(iquad)*/
498: }/*end for(il)*/
499: MatAssemblyBegin(obj->Amat,MAT_FINAL_ASSEMBLY);
500: MatAssemblyEnd(obj->Amat,MAT_FINAL_ASSEMBLY);
501: return;
502: }
504: /*---------------------------------------------------------
505: Function to fill the rhs vector with
506: By + g values ****
507: ---------------------------------------------------------*/
508: void rhs(AppCtx *obj,PetscScalar *y, PetscInt nz, PetscScalar *z, PetscReal t)
509: {
510: PetscInt i,j,js,je,jj;
511: PetscScalar val,g[num_z],btri[num_z][3],add_term;
512: PetscErrorCode ierr;
514: for(i=0; i < nz-2; i++){
515: for(j=0; j <= 2; j++){
516: btri[i][j]=0.0;
517: }
518: g[i] = 0.0;
519: }
521: /* call femBg to set the tri-diagonal b matrix and vector g */
522: femBg(btri,g,nz,z,t);
524: /* setting the entries of the right hand side vector */
525: for(i=0; i < nz-2; i++){
526: val = 0.0;
527: js = 0;
528: if(i == 0) js = 1;
529: je = 2;
530: if(i == nz-2) je = 1;
532: for(jj=js; jj <= je; jj++){
533: j = i+jj-1;
534: val += (btri[i][jj])*(y[j]);
535: }
536: add_term = val + g[i];
537: VecSetValue(obj->ksp_rhs,(PetscInt)i,(PetscScalar)add_term,INSERT_VALUES);
538: }
539: VecAssemblyBegin(obj->ksp_rhs);
540: VecAssemblyEnd(obj->ksp_rhs);
542: /* return to main driver function */
543: return;
544: }
546: /*%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
547: %% Function to form the right hand side of the time-stepping problem. %%
548: %% -------------------------------------------------------------------------------------------%%
549: if (useAlhs):
550: globalout = By+g
551: else if (!useAlhs):
552: globalout = f(y,t)=Ainv(By+g),
553: in which the ksp solver to transform the problem A*ydot=By+g
554: to the problem ydot=f(y,t)=inv(A)*(By+g)
555: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%*/
557: PetscErrorCode RHSfunction(TS ts,PetscReal t,Vec globalin,Vec globalout,void *ctx)
558: {
560: AppCtx *obj = (AppCtx*)ctx;
561: PetscScalar *soln_ptr,soln[num_z-2];
562: PetscInt i,nz=obj->nz;
563: PetscReal time;
565: /* get the previous solution to compute updated system */
566: VecGetArray(globalin,&soln_ptr);
567: for(i=0;i < num_z-2;i++){
568: soln[i] = soln_ptr[i];
569: }
570: VecRestoreArray(globalin,&soln_ptr);
572: /* clear out the matrix and rhs for ksp to keep things straight */
573: VecSet(obj->ksp_rhs,(PetscScalar)0.0);
575: time = t;
576: /* get the updated system */
577: rhs(obj,soln,nz,obj->z,time); /* setup of the By+g rhs */
579: /* do a ksp solve to get the rhs for the ts problem */
580: if (obj->useAlhs){
581: /* ksp_sol = ksp_rhs */
582: VecCopy(obj->ksp_rhs,globalout);
583: } else {
584: /* ksp_sol = inv(Amat)*ksp_rhs */
585: Petsc_KSPSolve(obj);
586: VecCopy(obj->ksp_sol,globalout);
587: }
588: return 0;
589: }