Actual source code: ex3.c
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 10
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 PetscErrorCode femBg(PetscScalar[][3],PetscScalar*,PetscInt,PetscScalar*,PetscReal);
40: extern PetscErrorCode femA(AppCtx*,PetscInt,PetscScalar*);
41: extern PetscErrorCode rhs(AppCtx*,PetscScalar*, PetscInt, PetscScalar*,PetscReal);
42: extern PetscErrorCode RHSfunction(TS,PetscReal,Vec,Vec,void*);
44: int main(int argc,char **argv)
45: {
46: PetscInt i,m,nz,steps,max_steps,k,nphase=1;
47: PetscScalar zInitial,zFinal,val,*z;
48: PetscReal stepsz[4],T,ftime;
50: TS ts;
51: SNES snes;
52: Mat Jmat;
53: AppCtx appctx; /* user-defined application context */
54: Vec init_sol; /* ts solution vector */
55: PetscMPIInt size;
57: PetscInitialize(&argc,&argv,(char*)0,help);
58: MPI_Comm_size(PETSC_COMM_WORLD,&size);
61: /* initializations */
62: zInitial = 0.0;
63: zFinal = 1.0;
64: nz = num_z;
65: m = nz-2;
66: appctx.nz = nz;
67: max_steps = (PetscInt)10000;
69: appctx.m = m;
70: appctx.max_probsz = nz;
71: appctx.debug = PETSC_FALSE;
72: appctx.useAlhs = PETSC_FALSE;
74: PetscOptionsBegin(PETSC_COMM_WORLD,NULL,"","");
75: PetscOptionsName("-debug",NULL,NULL,&appctx.debug);
76: PetscOptionsName("-useAlhs",NULL,NULL,&appctx.useAlhs);
77: PetscOptionsRangeInt("-nphase",NULL,NULL,nphase,&nphase,NULL,1,3);
78: PetscOptionsEnd();
79: T = 0.014/nphase;
81: /* create vector to hold ts solution */
82: /*-----------------------------------*/
83: VecCreate(PETSC_COMM_WORLD, &init_sol);
84: VecSetSizes(init_sol, PETSC_DECIDE, m);
85: VecSetFromOptions(init_sol);
87: /* create vector to hold true ts soln for comparison */
88: VecDuplicate(init_sol, &appctx.solution);
90: /* create LHS matrix Amat */
91: /*------------------------*/
92: MatCreateSeqAIJ(PETSC_COMM_WORLD, m, m, 3, NULL, &appctx.Amat);
93: MatSetFromOptions(appctx.Amat);
94: MatSetUp(appctx.Amat);
95: /* set space grid points - interio points only! */
96: PetscMalloc1(nz+1,&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);
106: MatSetUp(Jmat);
108: /* create working vectors for formulating rhs=inv(Alhs)*(Arhs*U + g) */
109: VecDuplicate(init_sol,&appctx.ksp_rhs);
110: VecDuplicate(init_sol,&appctx.ksp_sol);
112: /* set initial guess */
113: /*-------------------*/
114: for (i=0; i<nz-2; i++) {
115: val = exact(z[i+1], 0.0);
116: VecSetValue(init_sol,i,(PetscScalar)val,INSERT_VALUES);
117: }
118: VecAssemblyBegin(init_sol);
119: VecAssemblyEnd(init_sol);
121: /*create a time-stepping context and set the problem type */
122: /*--------------------------------------------------------*/
123: TSCreate(PETSC_COMM_WORLD, &ts);
124: TSSetProblemType(ts,TS_NONLINEAR);
126: /* set time-step method */
127: TSSetType(ts,TSCN);
129: /* Set optional user-defined monitoring routine */
130: TSMonitorSet(ts,Monitor,&appctx,NULL);
131: /* set the right hand side of U_t = RHSfunction(U,t) */
132: TSSetRHSFunction(ts,NULL,(PetscErrorCode (*)(TS,PetscScalar,Vec,Vec,void*))RHSfunction,&appctx);
134: if (appctx.useAlhs) {
135: /* set the left hand side matrix of Amat*U_t = rhs(U,t) */
137: /* Note: this approach is incompatible with the finite differenced Jacobian set below because we can't restore the
138: * Alhs matrix without making a copy. Either finite difference the entire thing or use analytic Jacobians in both
139: * places.
140: */
141: TSSetIFunction(ts,NULL,TSComputeIFunctionLinear,&appctx);
142: TSSetIJacobian(ts,appctx.Amat,appctx.Amat,TSComputeIJacobianConstant,&appctx);
143: }
145: /* use petsc to compute the jacobian by finite differences */
146: TSGetSNES(ts,&snes);
147: SNESSetJacobian(snes,Jmat,Jmat,SNESComputeJacobianDefault,NULL);
149: /* get the command line options if there are any and set them */
150: TSSetFromOptions(ts);
152: #if defined(PETSC_HAVE_SUNDIALS2)
153: {
154: TSType type;
155: PetscBool sundialstype=PETSC_FALSE;
156: TSGetType(ts,&type);
157: PetscObjectTypeCompare((PetscObject)ts,TSSUNDIALS,&sundialstype);
159: }
160: #endif
161: /* Sets the initial solution */
162: TSSetSolution(ts,init_sol);
164: stepsz[0] = 1.0/(2.0*(nz-1)*(nz-1)); /* (mesh_size)^2/2.0 */
165: ftime = 0.0;
166: for (k=0; k<nphase; k++) {
167: if (nphase > 1) PetscPrintf(PETSC_COMM_WORLD,"Phase %D initial time %g, stepsz %g, duration: %g\n",k,(double)ftime,(double)stepsz[k],(double)((k+1)*T));
168: TSSetTime(ts,ftime);
169: TSSetTimeStep(ts,stepsz[k]);
170: TSSetMaxSteps(ts,max_steps);
171: TSSetMaxTime(ts,(k+1)*T);
172: TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);
174: /* loop over time steps */
175: /*----------------------*/
176: TSSolve(ts,init_sol);
177: TSGetSolveTime(ts,&ftime);
178: TSGetStepNumber(ts,&steps);
179: stepsz[k+1] = stepsz[k]*1.5; /* change step size for the next phase */
180: }
182: /* free space */
183: TSDestroy(&ts);
184: MatDestroy(&appctx.Amat);
185: MatDestroy(&Jmat);
186: VecDestroy(&appctx.ksp_rhs);
187: VecDestroy(&appctx.ksp_sol);
188: VecDestroy(&init_sol);
189: VecDestroy(&appctx.solution);
190: PetscFree(z);
192: PetscFinalize();
193: return 0;
194: }
196: /*------------------------------------------------------------------------
197: Set exact solution
198: u(z,t) = sin(6*PI*z)*exp(-36.*PI*PI*t) + 3.*sin(2*PI*z)*exp(-4.*PI*PI*t)
199: --------------------------------------------------------------------------*/
200: PetscScalar exact(PetscScalar z,PetscReal t)
201: {
202: PetscScalar val, ex1, ex2;
204: ex1 = PetscExpReal(-36.*PETSC_PI*PETSC_PI*t);
205: ex2 = PetscExpReal(-4.*PETSC_PI*PETSC_PI*t);
206: val = PetscSinScalar(6*PETSC_PI*z)*ex1 + 3.*PetscSinScalar(2*PETSC_PI*z)*ex2;
207: return val;
208: }
210: /*
211: Monitor - User-provided routine to monitor the solution computed at
212: each timestep. This example plots the solution and computes the
213: error in two different norms.
215: Input Parameters:
216: ts - the timestep context
217: step - the count of the current step (with 0 meaning the
218: initial condition)
219: time - the current time
220: u - the solution at this timestep
221: ctx - the user-provided context for this monitoring routine.
222: In this case we use the application context which contains
223: information about the problem size, workspace and the exact
224: solution.
225: */
226: PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal time,Vec u,void *ctx)
227: {
228: AppCtx *appctx = (AppCtx*)ctx;
229: PetscInt i,m=appctx->m;
230: PetscReal norm_2,norm_max,h=1.0/(m+1);
231: PetscScalar *u_exact;
233: /* Compute the exact solution */
234: VecGetArrayWrite(appctx->solution,&u_exact);
235: for (i=0; i<m; i++) u_exact[i] = exact(appctx->z[i+1],time);
236: VecRestoreArrayWrite(appctx->solution,&u_exact);
238: /* Print debugging information if desired */
239: if (appctx->debug) {
240: PetscPrintf(PETSC_COMM_SELF,"Computed solution vector at time %g\n",(double)time);
241: VecView(u,PETSC_VIEWER_STDOUT_SELF);
242: PetscPrintf(PETSC_COMM_SELF,"Exact solution vector\n");
243: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
244: }
246: /* Compute the 2-norm and max-norm of the error */
247: VecAXPY(appctx->solution,-1.0,u);
248: VecNorm(appctx->solution,NORM_2,&norm_2);
250: norm_2 = PetscSqrtReal(h)*norm_2;
251: VecNorm(appctx->solution,NORM_MAX,&norm_max);
252: PetscPrintf(PETSC_COMM_SELF,"Timestep %D: time = %g, 2-norm error = %6.4f, max norm error = %6.4f\n",step,(double)time,(double)norm_2,(double)norm_max);
254: /*
255: Print debugging information if desired
256: */
257: if (appctx->debug) {
258: PetscPrintf(PETSC_COMM_SELF,"Error vector\n");
259: VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
260: }
261: return 0;
262: }
264: /*%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
265: Function to solve a linear system using KSP
266: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%*/
268: PetscErrorCode Petsc_KSPSolve(AppCtx *obj)
269: {
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);
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 function.
294: ***********************************************************************
296: Arguments:
297: x = array of xpoints or nodal values
298: xx = point at which the basis function is to be
299: evaluated.
300: il = interval containing xx.
301: iq = indicates which of the two basis functions in
302: interval intrvl should be used
303: nll = array containing the endpoints of each interval.
304: id = If id ~= 2, the value of the basis function
305: is calculated; if id = 2, the value of the
306: derivative of the basis function is returned.
307: ***********************************************************************/
309: PetscScalar bspl(PetscScalar *x, PetscScalar xx,PetscInt il,PetscInt iq,PetscInt nll[][2],PetscInt id)
310: {
311: PetscScalar x1,x2,bfcn;
312: PetscInt i1,i2,iq1,iq2;
314: /* Determine which basis function in interval intrvl is to be used in */
315: iq1 = iq;
316: if (iq1==0) iq2 = 1;
317: else iq2 = 0;
319: /* Determine endpoint of the interval intrvl */
320: i1=nll[il][iq1];
321: i2=nll[il][iq2];
323: /* Determine nodal values at the endpoints of the interval intrvl */
324: x1=x[i1];
325: x2=x[i2];
327: /* Evaluate basis function */
328: if (id == 2) bfcn=(1.0)/(x1-x2);
329: else bfcn=(xx-x2)/(x1-x2);
330: return bfcn;
331: }
333: /*---------------------------------------------------------
334: Function called by rhs function to get B and g
335: ---------------------------------------------------------*/
336: PetscErrorCode femBg(PetscScalar btri[][3],PetscScalar *f,PetscInt nz,PetscScalar *z, PetscReal t)
337: {
338: PetscInt i,j,jj,il,ip,ipp,ipq,iq,iquad,iqq;
339: PetscInt nli[num_z][2],indx[num_z];
340: PetscScalar dd,dl,zip,zipq,zz,b_z,bb_z,bij;
341: PetscScalar zquad[num_z][3],dlen[num_z],qdwt[3];
343: /* initializing everything - btri and f are initialized in rhs.c */
344: for (i=0; i < nz; i++) {
345: nli[i][0] = 0;
346: nli[i][1] = 0;
347: indx[i] = 0;
348: zquad[i][0] = 0.0;
349: zquad[i][1] = 0.0;
350: zquad[i][2] = 0.0;
351: dlen[i] = 0.0;
352: } /*end for (i)*/
354: /* quadrature weights */
355: qdwt[0] = 1.0/6.0;
356: qdwt[1] = 4.0/6.0;
357: qdwt[2] = 1.0/6.0;
359: /* 1st and last nodes have Dirichlet boundary condition -
360: set indices there to -1 */
362: for (i=0; i < nz-1; i++) indx[i] = i-1;
363: indx[nz-1] = -1;
365: ipq = 0;
366: for (il=0; il < nz-1; il++) {
367: ip = ipq;
368: ipq = ip+1;
369: zip = z[ip];
370: zipq = z[ipq];
371: dl = zipq-zip;
372: zquad[il][0] = zip;
373: zquad[il][1] = (0.5)*(zip+zipq);
374: zquad[il][2] = zipq;
375: dlen[il] = PetscAbsScalar(dl);
376: nli[il][0] = ip;
377: nli[il][1] = ipq;
378: }
380: for (il=0; il < nz-1; il++) {
381: for (iquad=0; iquad < 3; iquad++) {
382: dd = (dlen[il])*(qdwt[iquad]);
383: zz = zquad[il][iquad];
385: for (iq=0; iq < 2; iq++) {
386: ip = nli[il][iq];
387: b_z = bspl(z,zz,il,iq,nli,2);
388: i = indx[ip];
390: if (i > -1) {
391: for (iqq=0; iqq < 2; iqq++) {
392: ipp = nli[il][iqq];
393: bb_z = bspl(z,zz,il,iqq,nli,2);
394: j = indx[ipp];
395: bij = -b_z*bb_z;
397: if (j > -1) {
398: jj = 1+j-i;
399: btri[i][jj] += bij*dd;
400: } else {
401: f[i] += bij*dd*exact(z[ipp], t);
402: /* f[i] += 0.0; */
403: /* if (il==0 && j==-1) { */
404: /* f[i] += bij*dd*exact(zz,t); */
405: /* }*/ /*end if*/
406: } /*end else*/
407: } /*end for (iqq)*/
408: } /*end if (i>0)*/
409: } /*end for (iq)*/
410: } /*end for (iquad)*/
411: } /*end for (il)*/
412: return 0;
413: }
415: PetscErrorCode femA(AppCtx *obj,PetscInt nz,PetscScalar *z)
416: {
417: PetscInt i,j,il,ip,ipp,ipq,iq,iquad,iqq;
418: PetscInt nli[num_z][2],indx[num_z];
419: PetscScalar dd,dl,zip,zipq,zz,bb,bbb,aij;
420: PetscScalar rquad[num_z][3],dlen[num_z],qdwt[3],add_term;
422: /* initializing everything */
423: for (i=0; i < nz; i++) {
424: nli[i][0] = 0;
425: nli[i][1] = 0;
426: indx[i] = 0;
427: rquad[i][0] = 0.0;
428: rquad[i][1] = 0.0;
429: rquad[i][2] = 0.0;
430: dlen[i] = 0.0;
431: } /*end for (i)*/
433: /* quadrature weights */
434: qdwt[0] = 1.0/6.0;
435: qdwt[1] = 4.0/6.0;
436: qdwt[2] = 1.0/6.0;
438: /* 1st and last nodes have Dirichlet boundary condition -
439: set indices there to -1 */
441: for (i=0; i < nz-1; i++) indx[i]=i-1;
442: indx[nz-1]=-1;
444: ipq = 0;
446: for (il=0; il < nz-1; il++) {
447: ip = ipq;
448: ipq = ip+1;
449: zip = z[ip];
450: zipq = z[ipq];
451: dl = zipq-zip;
452: rquad[il][0] = zip;
453: rquad[il][1] = (0.5)*(zip+zipq);
454: rquad[il][2] = zipq;
455: dlen[il] = PetscAbsScalar(dl);
456: nli[il][0] = ip;
457: nli[il][1] = ipq;
458: } /*end for (il)*/
460: for (il=0; il < nz-1; il++) {
461: for (iquad=0; iquad < 3; iquad++) {
462: dd = (dlen[il])*(qdwt[iquad]);
463: zz = rquad[il][iquad];
465: for (iq=0; iq < 2; iq++) {
466: ip = nli[il][iq];
467: bb = bspl(z,zz,il,iq,nli,1);
468: i = indx[ip];
469: if (i > -1) {
470: for (iqq=0; iqq < 2; iqq++) {
471: ipp = nli[il][iqq];
472: bbb = bspl(z,zz,il,iqq,nli,1);
473: j = indx[ipp];
474: aij = bb*bbb;
475: if (j > -1) {
476: add_term = aij*dd;
477: MatSetValue(obj->Amat,i,j,add_term,ADD_VALUES);
478: }/*endif*/
479: } /*end for (iqq)*/
480: } /*end if (i>0)*/
481: } /*end for (iq)*/
482: } /*end for (iquad)*/
483: } /*end for (il)*/
484: MatAssemblyBegin(obj->Amat,MAT_FINAL_ASSEMBLY);
485: MatAssemblyEnd(obj->Amat,MAT_FINAL_ASSEMBLY);
486: return 0;
487: }
489: /*---------------------------------------------------------
490: Function to fill the rhs vector with
491: By + g values ****
492: ---------------------------------------------------------*/
493: PetscErrorCode rhs(AppCtx *obj,PetscScalar *y, PetscInt nz, PetscScalar *z, PetscReal t)
494: {
495: PetscInt i,j,js,je,jj;
496: PetscScalar val,g[num_z],btri[num_z][3],add_term;
498: for (i=0; i < nz-2; i++) {
499: for (j=0; j <= 2; j++) btri[i][j]=0.0;
500: g[i] = 0.0;
501: }
503: /* call femBg to set the tri-diagonal b matrix and vector g */
504: femBg(btri,g,nz,z,t);
506: /* setting the entries of the right hand side vector */
507: for (i=0; i < nz-2; i++) {
508: val = 0.0;
509: js = 0;
510: if (i == 0) js = 1;
511: je = 2;
512: if (i == nz-2) je = 1;
514: for (jj=js; jj <= je; jj++) {
515: j = i+jj-1;
516: val += (btri[i][jj])*(y[j]);
517: }
518: add_term = val + g[i];
519: VecSetValue(obj->ksp_rhs,(PetscInt)i,(PetscScalar)add_term,INSERT_VALUES);
520: }
521: VecAssemblyBegin(obj->ksp_rhs);
522: VecAssemblyEnd(obj->ksp_rhs);
523: return 0;
524: }
526: /*%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
527: %% Function to form the right hand side of the time-stepping problem. %%
528: %% -------------------------------------------------------------------------------------------%%
529: if (useAlhs):
530: globalout = By+g
531: else if (!useAlhs):
532: globalout = f(y,t)=Ainv(By+g),
533: in which the ksp solver to transform the problem A*ydot=By+g
534: to the problem ydot=f(y,t)=inv(A)*(By+g)
535: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%*/
537: PetscErrorCode RHSfunction(TS ts,PetscReal t,Vec globalin,Vec globalout,void *ctx)
538: {
539: AppCtx *obj = (AppCtx*)ctx;
540: PetscScalar soln[num_z];
541: const PetscScalar *soln_ptr;
542: PetscInt i,nz=obj->nz;
543: PetscReal time;
545: /* get the previous solution to compute updated system */
546: VecGetArrayRead(globalin,&soln_ptr);
547: for (i=0; i < num_z-2; i++) soln[i] = soln_ptr[i];
548: VecRestoreArrayRead(globalin,&soln_ptr);
549: soln[num_z-1] = 0.0;
550: soln[num_z-2] = 0.0;
552: /* clear out the matrix and rhs for ksp to keep things straight */
553: VecSet(obj->ksp_rhs,(PetscScalar)0.0);
555: time = t;
556: /* get the updated system */
557: rhs(obj,soln,nz,obj->z,time); /* setup of the By+g rhs */
559: /* do a ksp solve to get the rhs for the ts problem */
560: if (obj->useAlhs) {
561: /* ksp_sol = ksp_rhs */
562: VecCopy(obj->ksp_rhs,globalout);
563: } else {
564: /* ksp_sol = inv(Amat)*ksp_rhs */
565: Petsc_KSPSolve(obj);
566: VecCopy(obj->ksp_sol,globalout);
567: }
568: return 0;
569: }
571: /*TEST
573: build:
574: requires: !complex
576: test:
577: suffix: euler
578: output_file: output/ex3.out
580: test:
581: suffix: 2
582: args: -useAlhs
583: output_file: output/ex3.out
584: TODO: Broken because SNESComputeJacobianDefault is incompatible with TSComputeIJacobianConstant
586: TEST*/