Actual source code: ex4.c
1: /*
2: The Problem:
3: Solve the convection-diffusion equation:
4:
5: u_t+a*(u_x+u_y)=epsilon*(u_xx+u_yy)
6: u=0 at x=0, y=0
7: u_x=0 at x=1
8: u_y=0 at y=1
9:
10: This program tests the routine of computing the Jacobian by the
11: finite difference method as well as PETSc with PVODE.
13: */
15: static char help[] = "Solve the convection-diffusion equation. \n\n";
17: #include petscsys.h
18: #include petscts.h
23: typedef struct
24: {
25: PetscInt m; /* the number of mesh points in x-direction */
26: PetscInt n; /* the number of mesh points in y-direction */
27: PetscReal dx; /* the grid space in x-direction */
28: PetscReal dy; /* the grid space in y-direction */
29: PetscReal a; /* the convection coefficient */
30: PetscReal epsilon; /* the diffusion coefficient */
31: } Data;
33: /* two temporal functions */
39: /* the initial function */
40: PetscReal f_ini(PetscReal x,PetscReal y)
41: {
42: PetscReal f;
43: f=exp(-20.0*(pow(x-0.5,2.0)+pow(y-0.5,2.0)));
44: return f;
45: }
48: #define linear_no_matrix 0
49: #define linear_no_time 1
50: #define linear 2
51: #define nonlinear_no_jacobian 3
52: #define nonlinear 4
56: int main(int argc,char **argv)
57: {
59: PetscInt time_steps = 100,steps;
60: PetscMPIInt size;
61: Vec global;
62: PetscReal dt,ftime;
63: TS ts;
64: PetscViewer viewfile;
65: MatStructure A_structure;
66: Mat A = 0;
67: TSProblemType tsproblem = TS_NONLINEAR; /* Need to be TS_NONLINEAR */
68: SNES snes;
69: Vec x;
70: Data data;
71: PetscInt mn;
72: #if defined(PETSC_HAVE_PVODE)
73: PC pc;
74: PetscViewer viewer;
75: char pcinfo[120],tsinfo[120];
76: #endif
78: PetscInitialize(&argc,&argv,(char*)0,help);
79: MPI_Comm_size(PETSC_COMM_WORLD,&size);
80:
81: /* set Data */
82: data.m = 9;
83: data.n = 9;
84: data.a = 1.0;
85: data.epsilon = 0.1;
86: data.dx = 1.0/(data.m+1.0);
87: data.dy = 1.0/(data.n+1.0);
88: mn = (data.m)*(data.n);
90: PetscOptionsGetInt(PETSC_NULL,"-time",&time_steps,PETSC_NULL);
91:
92: /* set initial conditions */
93: VecCreate(PETSC_COMM_WORLD,&global);
94: VecSetSizes(global,PETSC_DECIDE,mn);
95: VecSetFromOptions(global);
96: Initial(global,&data);
97: VecDuplicate(global,&x);
98:
99: /* make timestep context */
100: TSCreate(PETSC_COMM_WORLD,&ts);
101: TSSetProblemType(ts,tsproblem);
102: TSSetMonitor(ts,Monitor,PETSC_NULL,PETSC_NULL);
104: dt = 0.1;
106: /*
107: The user provides the RHS and Jacobian
108: */
109: MatCreate(PETSC_COMM_WORLD,&A);
110: MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,mn,mn);
111: MatSetFromOptions(A);
113: /* Create SNES context */
114: SNESCreate(PETSC_COMM_WORLD,&snes);
115:
117: /* setting the RHS function and the Jacobian's non-zero structutre */
118: SNESSetFunction(snes,global,FormFunction,&data);
119: SNESSetJacobian(snes,A,A,FormJacobian,&data);
121: /* set TSPVodeRHSFunction and TSPVodeRHSJacobian, so PETSc will pick up the
122: RHS function from SNES and compute the Jacobian by FD */
123: /*
124: TSSetRHSFunction(ts,TSPVodeSetRHSFunction,snes);
125: TSPVodeSetRHSJacobian(ts,0.0,global,&A,&A,&A_structure,snes);
126: TSSetRHSJacobian(ts,A,A,TSPVodeSetRHSJacobian,snes);
127: */
128:
129: TSSetRHSFunction(ts,RHSFunction,&data);
130: RHSJacobian(ts,0.0,global,&A,&A,&A_structure,&data);
131: TSSetRHSJacobian(ts,A,A,RHSJacobian,&data);
133: /* Use PVODE */
134: TSSetType(ts,TS_PVODE);
136: TSSetFromOptions(ts);
138: TSSetInitialTimeStep(ts,0.0,dt);
139: TSSetDuration(ts,time_steps,1);
140: TSSetSolution(ts,global);
143: /* Pick up a Petsc preconditioner */
144: /* one can always set method or preconditioner during the run time */
145: #if defined(PETSC_HAVE_PVODE)
146: TSPVodeGetPC(ts,&pc);
147: PCSetType(pc,PCJACOBI);
148: TSPVodeSetType(ts,PVODE_BDF);
149: /* TSPVodeSetMethodFromOptions(ts); */
150: #endif
152: TSSetUp(ts);
153: TSStep(ts,&steps,&ftime);
155: TSGetSolution(ts,&global);
156: PetscViewerASCIIOpen(PETSC_COMM_SELF,"out.m",&viewfile);
157: PetscViewerSetFormat(viewfile,PETSC_VIEWER_ASCII_MATLAB);
158: VecView(global,viewfile);
160: #if defined(PETSC_HAVE_PVODE)
161: /* extracts the PC from ts */
162: TSPVodeGetPC(ts,&pc);
163: PetscViewerStringOpen(PETSC_COMM_WORLD,tsinfo,120,&viewer);
164: TSView(ts,viewer);
165: PetscViewerStringOpen(PETSC_COMM_WORLD,pcinfo,120,&viewer);
166: PCView(pc,viewer);
167: PetscPrintf(PETSC_COMM_WORLD,"%d Procs,%s Preconditioner,%s\n",
168: size,tsinfo,pcinfo);
169: PCDestroy(pc);
170: #endif
172: /* free the memories */
173: TSDestroy(ts);
174: VecDestroy(global);
175: if (A) {ierr= MatDestroy(A);}
176: PetscFinalize();
177: return 0;
178: }
180: /* -------------------------------------------------------------------*/
183: PetscErrorCode Initial(Vec global,void *ctx)
184: {
185: Data *data = (Data*)ctx;
186: PetscInt m,row,col;
187: PetscReal x,y,dx,dy;
188: PetscScalar *localptr;
189: PetscInt i,mybase,myend,locsize;
192: /* make the local copies of parameters */
193: m = data->m;
194: dx = data->dx;
195: dy = data->dy;
197: /* determine starting point of each processor */
198: VecGetOwnershipRange(global,&mybase,&myend);
199: VecGetLocalSize(global,&locsize);
201: /* Initialize the array */
202: VecGetArray(global,&localptr);
204: for (i=0; i<locsize; i++) {
205: row = 1+(mybase+i)-((mybase+i)/m)*m;
206: col = (mybase+i)/m+1;
207: x = dx*row;
208: y = dy*col;
209: localptr[i] = f_ini(x,y);
210: }
211:
212: VecRestoreArray(global,&localptr);
213: return 0;
214: }
218: PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal time,Vec global,void *ctx)
219: {
220: VecScatter scatter;
221: IS from,to;
222: PetscInt i,n,*idx;
223: Vec tmp_vec;
225: PetscScalar *tmp;
227: /* Get the size of the vector */
228: VecGetSize(global,&n);
230: /* Set the index sets */
231: PetscMalloc(n*sizeof(PetscInt),&idx);
232: for(i=0; i<n; i++) idx[i]=i;
233:
234: /* Create local sequential vectors */
235: VecCreateSeq(PETSC_COMM_SELF,n,&tmp_vec);
237: /* Create scatter context */
238: ISCreateGeneral(PETSC_COMM_SELF,n,idx,&from);
239: ISCreateGeneral(PETSC_COMM_SELF,n,idx,&to);
240: VecScatterCreate(global,from,tmp_vec,to,&scatter);
241: VecScatterBegin(global,tmp_vec,INSERT_VALUES,SCATTER_FORWARD,scatter);
242: VecScatterEnd(global,tmp_vec,INSERT_VALUES,SCATTER_FORWARD,scatter);
244: VecGetArray(tmp_vec,&tmp);
245: PetscPrintf(PETSC_COMM_WORLD,"At t =%14.6e u= %14.6e at the center \n",time,PetscRealPart(tmp[n/2]));
246: VecRestoreArray(tmp_vec,&tmp);
248: PetscFree(idx);
249: return 0;
250: }
255: PetscErrorCode FormFunction(SNES snes,Vec globalin,Vec globalout,void *ptr)
256: {
257: Data *data = (Data*)ptr;
258: PetscInt m,n,mn;
259: PetscReal dx,dy;
260: PetscReal xc,xl,xr,yl,yr;
261: PetscReal a,epsilon;
262: PetscScalar *inptr,*outptr;
263: PetscInt i,j,len;
265: IS from,to;
266: PetscInt *idx;
267: VecScatter scatter;
268: Vec tmp_in,tmp_out;
270: m = data->m;
271: n = data->n;
272: mn = m*n;
273: dx = data->dx;
274: dy = data->dy;
275: a = data->a;
276: epsilon = data->epsilon;
278: xc = -2.0*epsilon*(1.0/(dx*dx)+1.0/(dy*dy));
279: xl = 0.5*a/dx+epsilon/(dx*dx);
280: xr = -0.5*a/dx+epsilon/(dx*dx);
281: yl = 0.5*a/dy+epsilon/(dy*dy);
282: yr = -0.5*a/dy+epsilon/(dy*dy);
283:
284: /* Get the length of parallel vector */
285: VecGetSize(globalin,&len);
287: /* Set the index sets */
288: PetscMalloc(len*sizeof(PetscInt),&idx);
289: for(i=0; i<len; i++) idx[i]=i;
290:
291: /* Create local sequential vectors */
292: VecCreateSeq(PETSC_COMM_SELF,len,&tmp_in);
293: VecDuplicate(tmp_in,&tmp_out);
295: /* Create scatter context */
296: ISCreateGeneral(PETSC_COMM_SELF,len,idx,&from);
297: ISCreateGeneral(PETSC_COMM_SELF,len,idx,&to);
298: VecScatterCreate(globalin,from,tmp_in,to,&scatter);
299: VecScatterBegin(globalin,tmp_in,INSERT_VALUES,SCATTER_FORWARD,scatter);
300:
302:
303: VecScatterEnd(globalin,tmp_in,INSERT_VALUES,SCATTER_FORWARD,scatter);
304:
306: /*Extract income array */
307: VecGetArray(tmp_in,&inptr);
309: /* Extract outcome array*/
310: VecGetArray(tmp_out,&outptr);
312: outptr[0] = xc*inptr[0]+xr*inptr[1]+yr*inptr[m];
313: outptr[m-1] = 2.0*xl*inptr[m-2]+xc*inptr[m-1]+yr*inptr[m-1+m];
314: for (i=1; i<m-1; i++) {
315: outptr[i] = xc*inptr[i]+xl*inptr[i-1]+xr*inptr[i+1]
316: +yr*inptr[i+m];
317: }
319: for (j=1; j<n-1; j++) {
320: outptr[j*m] = xc*inptr[j*m]+xr*inptr[j*m+1]+
321: yl*inptr[j*m-m]+yr*inptr[j*m+m];
322: outptr[j*m+m-1] = xc*inptr[j*m+m-1]+2.0*xl*inptr[j*m+m-1-1]+
323: yl*inptr[j*m+m-1-m]+yr*inptr[j*m+m-1+m];
324: for(i=1; i<m-1; i++) {
325: outptr[j*m+i] = xc*inptr[j*m+i]+xl*inptr[j*m+i-1]+xr*inptr[j*m+i+1]
326: +yl*inptr[j*m+i-m]+yr*inptr[j*m+i+m];
327: }
328: }
330: outptr[mn-m] = xc*inptr[mn-m]+xr*inptr[mn-m+1]+2.0*yl*inptr[mn-m-m];
331: outptr[mn-1] = 2.0*xl*inptr[mn-2]+xc*inptr[mn-1]+2.0*yl*inptr[mn-1-m];
332: for (i=1; i<m-1; i++) {
333: outptr[mn-m+i] = xc*inptr[mn-m+i]+xl*inptr[mn-m+i-1]+xr*inptr[mn-m+i+1]
334: +2*yl*inptr[mn-m+i-m];
335: }
337: VecRestoreArray(tmp_in,&inptr);
338: VecRestoreArray(tmp_out,&outptr);
340: VecScatterCreate(tmp_out,from,globalout,to,&scatter);
341: VecScatterBegin(tmp_out,globalout,INSERT_VALUES,SCATTER_FORWARD,scatter
342: );
343:
344: VecScatterEnd(tmp_out,globalout,INSERT_VALUES,SCATTER_FORWARD,scatter);
345:
346:
347: /* Destroy idx aand scatter */
348: ISDestroy(from);
349: ISDestroy(to);
350: VecScatterDestroy(scatter);
352: PetscFree(idx);
353: return 0;
354: }
358: PetscErrorCode FormJacobian(SNES snes,Vec x,Mat *AA,Mat *BB,MatStructure *flag,void *ptr)
359: {
360: Data *data = (Data*)ptr;
361: Mat A = *AA;
362: PetscScalar v[1],one = 1.0;
363: PetscInt idx[1],i,j,row;
365: PetscInt m,n,mn;
367: m = data->m;
368: n = data->n;
369: mn = (data->m)*(data->n);
370:
371: for(i=0; i<mn; i++) {
372: idx[0] = i;
373: v[0] = one;
374: MatSetValues(A,1,&i,1,idx,v,INSERT_VALUES);
375: }
377: for(i=0; i<mn-m; i++) {
378: idx[0] = i+m;
379: v[0]= one;
380: MatSetValues(A,1,&i,1,idx,v,INSERT_VALUES);
381: }
383: for(i=m; i<mn; i++) {
384: idx[0] = i-m;
385: v[0] = one;
386: MatSetValues(A,1,&i,1,idx,v,INSERT_VALUES);
387: }
389: for(j=0; j<n; j++) {
390: for(i=0; i<m-1; i++) {
391: row = j*m+i;
392: idx[0] = j*m+i+1;
393: v[0] = one;
394: MatSetValues(A,1,&row,1,idx,v,INSERT_VALUES);
395: }
396: }
398: for(j=0; j<n; j++) {
399: for(i=1; i<m; i++) {
400: row = j*m+i;
401: idx[0] = j*m+i-1;
402: v[0] = one;
403: MatSetValues(A,1,&row,1,idx,v,INSERT_VALUES);
404: }
405: }
406:
407: MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
408: MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
411: *flag = SAME_NONZERO_PATTERN;
412: return 0;
413: }
417: PetscErrorCode RHSJacobian(TS ts,PetscReal t,Vec x,Mat *AA,Mat *BB,MatStructure *flag,void *ptr)
418: {
419: Data *data = (Data*)ptr;
420: Mat A = *AA;
421: PetscScalar v[5];
422: PetscInt idx[5],i,j,row;
424: PetscInt m,n,mn;
425: PetscReal dx,dy,a,epsilon,xc,xl,xr,yl,yr;
427: m = data->m;
428: n = data->n;
429: mn = m*n;
430: dx = data->dx;
431: dy = data->dy;
432: a = data->a;
433: epsilon = data->epsilon;
435: xc = -2.0*epsilon*(1.0/(dx*dx)+1.0/(dy*dy));
436: xl = 0.5*a/dx+epsilon/(dx*dx);
437: xr = -0.5*a/dx+epsilon/(dx*dx);
438: yl = 0.5*a/dy+epsilon/(dy*dy);
439: yr = -0.5*a/dy+epsilon/(dy*dy);
441: row=0;
442: v[0] = xc; v[1]=xr; v[2]=yr;
443: idx[0]=0; idx[1]=2; idx[2]=m;
444: MatSetValues(A,1,&row,3,idx,v,INSERT_VALUES);
446: row=m-1;
447: v[0]=2.0*xl; v[1]=xc; v[2]=yr;
448: idx[0]=m-2; idx[1]=m-1; idx[2]=m-1+m;
449: MatSetValues(A,1,&row,3,idx,v,INSERT_VALUES);
451: for (i=1; i<m-1; i++) {
452: row=i;
453: v[0]=xl; v[1]=xc; v[2]=xr; v[3]=yr;
454: idx[0]=i-1; idx[1]=i; idx[2]=i+1; idx[3]=i+m;
455: MatSetValues(A,1,&row,4,idx,v,INSERT_VALUES);
456: }
458: for (j=1; j<n-1; j++) {
459: row=j*m;
460: v[0]=xc; v[1]=xr; v[2]=yl; v[3]=yr;
461: idx[0]=j*m; idx[1]=j*m; idx[2]=j*m-m; idx[3]=j*m+m;
462: MatSetValues(A,1,&row,4,idx,v,INSERT_VALUES);
463:
464: row=j*m+m-1;
465: v[0]=xc; v[1]=2.0*xl; v[2]=yl; v[3]=yr;
466: idx[0]=j*m+m-1; idx[1]=j*m+m-1-1; idx[2]=j*m+m-1-m; idx[3]=j*m+m-1+m;
467: MatSetValues(A,1,&row,4,idx,v,INSERT_VALUES);
469: for(i=1; i<m-1; i++) {
470: row=j*m+i;
471: v[0]=xc; v[1]=xl; v[2]=xr; v[3]=yl; v[4]=yr;
472: idx[0]=j*m+i; idx[1]=j*m+i-1; idx[2]=j*m+i+1; idx[3]=j*m+i-m;
473: idx[4]=j*m+i+m;
474: MatSetValues(A,1,&row,5,idx,v,INSERT_VALUES);
475: }
476: }
478: row=mn-m;
479: v[0] = xc; v[1]=xr; v[2]=2.0*yl;
480: idx[0]=mn-m; idx[1]=mn-m+1; idx[2]=mn-m-m;
481: MatSetValues(A,1,&row,3,idx,v,INSERT_VALUES);
482:
483: row=mn-1;
484: v[0] = xc; v[1]=2.0*xl; v[2]=2.0*yl;
485: idx[0]=mn-1; idx[1]=mn-2; idx[2]=mn-1-m;
486: MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
488: for (i=1; i<m-1; i++) {
489: row=mn-m+i;
490: v[0]=xl; v[1]=xc; v[2]=xr; v[3]=2.0*yl;
491: idx[0]=mn-m+i-1; idx[1]=mn-m+i; idx[2]=mn-m+i+1; idx[3]=mn-m+i-m;
492: MatSetValues(A,1,&row,4,idx,v,INSERT_VALUES);
493: }
496: MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
497: MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
500: *flag = SAME_NONZERO_PATTERN;
501: return 0;
502: }
506: PetscErrorCode RHSFunction(TS ts,PetscReal t,Vec globalin,Vec globalout,void *ctx)
507: {
509: SNES snes = PETSC_NULL;
511: FormFunction(snes,globalin,globalout,ctx);
514: return 0;
515: }