Actual source code: ex74.c

  1: static char help[] = "Solves the constant-coefficient 1D heat equation \n\
  2: with an Implicit Runge-Kutta method using MatKAIJ.                  \n\
  3:                                                                     \n\
  4:     du      d^2 u                                                   \n\
  5:     --  = a ----- ; 0 <= x <= 1;                                    \n\
  6:     dt      dx^2                                                    \n\
  7:                                                                     \n\
  8:   with periodic boundary conditions                                 \n\
  9:                                                                     \n\
 10: 2nd order central discretization in space:                          \n\
 11:                                                                     \n\
 12:    [ d^2 u ]     u_{i+1} - 2u_i + u_{i-1}                           \n\
 13:    [ ----- ]  =  ------------------------                           \n\
 14:    [ dx^2  ]i              h^2                                      \n\
 15:                                                                     \n\
 16:     i = grid index;    h = x_{i+1}-x_i (Uniform)                    \n\
 17:     0 <= i < n         h = 1.0/n                                    \n\
 18:                                                                     \n\
 19: Thus,                                                               \n\
 20:                                                                     \n\
 21:    du                                                               \n\
 22:    --  = Ju;  J = (a/h^2) tridiagonal(1,-2,1)_n                     \n\
 23:    dt                                                               \n\
 24:                                                                     \n\
 25: Implicit Runge-Kutta method:                                        \n\
 26:                                                                     \n\
 27:   U^(k)   = u^n + dt \\sum_i a_{ki} JU^{i}                          \n\
 28:   u^{n+1} = u^n + dt \\sum_i b_i JU^{i}                             \n\
 29:                                                                     \n\
 30:   i = 1,...,s (s -> number of stages)                               \n\
 31:                                                                     \n\
 32: At each time step, we solve                                         \n\
 33:                                                                     \n\
 34:  [  1                                  ]     1                      \n\
 35:  [ -- I \\otimes A^{-1} - J \\otimes I ] U = -- u^n \\otimes A^{-1} \n\
 36:  [ dt                                  ]     dt                     \n\
 37:                                                                     \n\
 38:   where A is the Butcher tableaux of the implicit                   \n\
 39:   Runge-Kutta method,                                               \n\
 40:                                                                     \n\
 41: with MATKAIJ and KSP.                                               \n\
 42:                                                                     \n\
 43: Available IRK Methods:                                              \n\
 44:   gauss       n-stage Gauss method                                  \n\
 45:                                                                     \n";

 47: /*
 48:   Include "petscksp.h" so that we can use KSP solvers.  Note that this file
 49:   automatically includes:
 50:   petscsys.h      - base PETSc routines
 51:   petscvec.h      - vectors
 52:   petscmat.h      - matrices
 53:   petscis.h       - index sets
 54:   petscviewer.h   - viewers
 55:   petscpc.h       - preconditioners
 56: */
 57: #include <petscksp.h>
 58: #include <petscdt.h>

 60: /* define the IRK methods available */
 61: #define IRKGAUSS      "gauss"

 63: typedef enum {
 64:   PHYSICS_DIFFUSION,
 65:   PHYSICS_ADVECTION
 66: } PhysicsType;
 67: const char *const PhysicsTypes[] = {"DIFFUSION","ADVECTION","PhysicsType","PHYSICS_",NULL};

 69: typedef struct __context__ {
 70:   PetscReal     a;              /* diffusion coefficient      */
 71:   PetscReal     xmin,xmax;      /* domain bounds              */
 72:   PetscInt      imax;           /* number of grid points      */
 73:   PetscInt      niter;          /* number of time iterations  */
 74:   PetscReal     dt;             /* time step size             */
 75:   PhysicsType   physics_type;
 76: } UserContext;

 78: static PetscErrorCode ExactSolution(Vec,void*,PetscReal);
 79: static PetscErrorCode RKCreate_Gauss(PetscInt,PetscScalar**,PetscScalar**,PetscReal**);
 80: static PetscErrorCode Assemble_AdvDiff(MPI_Comm,UserContext*,Mat*);

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

 84: int main(int argc, char **argv)
 85: {
 86:   PetscErrorCode    ierr;
 87:   Vec               u,uex,rhs,z;
 88:   UserContext       ctxt;
 89:   PetscInt          nstages,is,ie,matis,matie,*ix,*ix2;
 90:   PetscInt          n,i,s,t,total_its;
 91:   PetscScalar       *A,*B,*At,*b,*zvals,one = 1.0;
 92:   PetscReal         *c,err,time;
 93:   Mat               Identity,J,TA,SC,R;
 94:   KSP               ksp;
 95:   PetscFunctionList IRKList = NULL;
 96:   char              irktype[256] = IRKGAUSS;

 98:   PetscInitialize(&argc,&argv,(char*)0,help);
 99:   PetscFunctionListAdd(&IRKList,IRKGAUSS,RKCreate_Gauss);

101:   /* default value */
102:   ctxt.a       = 1.0;
103:   ctxt.xmin    = 0.0;
104:   ctxt.xmax    = 1.0;
105:   ctxt.imax    = 20;
106:   ctxt.niter   = 0;
107:   ctxt.dt      = 0.0;
108:   ctxt.physics_type = PHYSICS_DIFFUSION;

110:   PetscOptionsBegin(PETSC_COMM_WORLD,NULL,"IRK options","");
111:   PetscOptionsReal("-a","diffusion coefficient","<1.0>",ctxt.a,&ctxt.a,NULL);
112:   PetscOptionsInt ("-imax","grid size","<20>",ctxt.imax,&ctxt.imax,NULL);
113:   PetscOptionsReal("-xmin","xmin","<0.0>",ctxt.xmin,&ctxt.xmin,NULL);
114:   PetscOptionsReal("-xmax","xmax","<1.0>",ctxt.xmax,&ctxt.xmax,NULL);
115:   PetscOptionsInt ("-niter","number of time steps","<0>",ctxt.niter,&ctxt.niter,NULL);
116:   PetscOptionsReal("-dt","time step size","<0.0>",ctxt.dt,&ctxt.dt,NULL);
117:   PetscOptionsFList("-irk_type","IRK method family","",IRKList,irktype,irktype,sizeof(irktype),NULL);
118:   nstages = 2;
119:   PetscOptionsInt ("-irk_nstages","Number of stages in IRK method","",nstages,&nstages,NULL);
120:   PetscOptionsEnum("-physics_type","Type of process to discretize","",PhysicsTypes,(PetscEnum)ctxt.physics_type,(PetscEnum*)&ctxt.physics_type,NULL);
121:   PetscOptionsEnd();

123:   /* allocate and initialize solution vector and exact solution */
124:   VecCreate(PETSC_COMM_WORLD,&u);
125:   VecSetSizes(u,PETSC_DECIDE,ctxt.imax);
126:   VecSetFromOptions(u);
127:   VecDuplicate(u,&uex);
128:   /* initial solution */
129:   ExactSolution(u  ,&ctxt,0.0);
130:   /* exact   solution */
131:   ExactSolution(uex,&ctxt,ctxt.dt*ctxt.niter);

133:   {                             /* Create A,b,c */
134:     PetscErrorCode (*irkcreate)(PetscInt,PetscScalar**,PetscScalar**,PetscReal**);
135:     PetscFunctionListFind(IRKList,irktype,&irkcreate);
136:     (*irkcreate)(nstages,&A,&b,&c);
137:   }
138:   {                             /* Invert A */
139:     /* PETSc does not provide a routine to calculate the inverse of a general matrix.
140:      * To get the inverse of A, we form a sequential BAIJ matrix from it, consisting of a single block with block size
141:      * equal to the dimension of A, and then use MatInvertBlockDiagonal(). */
142:     Mat               A_baij;
143:     PetscInt          idxm[1]={0},idxn[1]={0};
144:     const PetscScalar *A_inv;
145:     MatCreateSeqBAIJ(PETSC_COMM_SELF,nstages,nstages,nstages,1,NULL,&A_baij);
146:     MatSetOption(A_baij,MAT_ROW_ORIENTED,PETSC_FALSE);
147:     MatSetValuesBlocked(A_baij,1,idxm,1,idxn,A,INSERT_VALUES);
148:     MatAssemblyBegin(A_baij,MAT_FINAL_ASSEMBLY);
149:     MatAssemblyEnd(A_baij,MAT_FINAL_ASSEMBLY);
150:     MatInvertBlockDiagonal(A_baij,&A_inv);
151:     PetscMemcpy(A,A_inv,nstages*nstages*sizeof(PetscScalar));
152:     MatDestroy(&A_baij);
153:   }
154:   /* Scale (1/dt)*A^{-1} and (1/dt)*b */
155:   for (s=0; s<nstages*nstages; s++) A[s] *= 1.0/ctxt.dt;
156:   for (s=0; s<nstages; s++) b[s] *= (-ctxt.dt);

158:   /* Compute row sums At and identity B */
159:   PetscMalloc2(nstages,&At,PetscSqr(nstages),&B);
160:   for (s=0; s<nstages; s++) {
161:     At[s] = 0;
162:     for (t=0; t<nstages; t++) {
163:       At[s] += A[s+nstages*t];      /* Row sums of  */
164:       B[s+nstages*t] = 1.*(s == t); /* identity */
165:     }
166:   }

168:   /* allocate and calculate the (-J) matrix */
169:   switch (ctxt.physics_type) {
170:   case PHYSICS_ADVECTION:
171:   case PHYSICS_DIFFUSION:
172:     Assemble_AdvDiff(PETSC_COMM_WORLD,&ctxt,&J);
173:   }
174:   MatCreate(PETSC_COMM_WORLD,&Identity);
175:   MatSetType(Identity,MATAIJ);
176:   MatGetOwnershipRange(J,&matis,&matie);
177:   MatSetSizes(Identity,matie-matis,matie-matis,ctxt.imax,ctxt.imax);
178:   MatSetUp(Identity);
179:   for (i=matis; i<matie; i++) {
180:     MatSetValues(Identity,1,&i,1,&i,&one,INSERT_VALUES);
181:   }
182:   MatAssemblyBegin(Identity,MAT_FINAL_ASSEMBLY);
183:   MatAssemblyEnd  (Identity,MAT_FINAL_ASSEMBLY);

185:   /* Create the KAIJ matrix for solving the stages */
186:   MatCreateKAIJ(J,nstages,nstages,A,B,&TA);

188:   /* Create the KAIJ matrix for step completion */
189:   MatCreateKAIJ(J,1,nstages,NULL,b,&SC);

191:   /* Create the KAIJ matrix to create the R for solving the stages */
192:   MatCreateKAIJ(Identity,nstages,1,NULL,At,&R);

194:   /* Create and set options for KSP */
195:   KSPCreate(PETSC_COMM_WORLD,&ksp);
196:   KSPSetOperators(ksp,TA,TA);
197:   KSPSetFromOptions(ksp);

199:   /* Allocate work and right-hand-side vectors */
200:   VecCreate(PETSC_COMM_WORLD,&z);
201:   VecSetFromOptions(z);
202:   VecSetSizes(z,PETSC_DECIDE,ctxt.imax*nstages);
203:   VecDuplicate(z,&rhs);

205:   VecGetOwnershipRange(u,&is,&ie);
206:   PetscMalloc3(nstages,&ix,nstages,&zvals,ie-is,&ix2);
207:   /* iterate in time */
208:   for (n=0,time=0.,total_its=0; n<ctxt.niter; n++) {
209:     PetscInt its;

211:     /* compute and set the right hand side */
212:     MatMult(R,u,rhs);

214:     /* Solve the system */
215:     KSPSolve(ksp,rhs,z);
216:     KSPGetIterationNumber(ksp,&its);
217:     total_its += its;

219:     /* Update the solution */
220:     MatMultAdd(SC,z,u,u);

222:     /* time step complete */
223:     time += ctxt.dt;
224:   }
225:   PetscFree3(ix,ix2,zvals);

227:   /* Deallocate work and right-hand-side vectors */
228:   VecDestroy(&z);
229:   VecDestroy(&rhs);

231:   /* Calculate error in final solution */
232:   VecAYPX(uex,-1.0,u);
233:   VecNorm(uex,NORM_2,&err);
234:   err  = PetscSqrtReal(err*err/((PetscReal)ctxt.imax));
235:   PetscPrintf(PETSC_COMM_WORLD,"L2 norm of the numerical error = %g (time=%g)\n",(double)err,(double)time);
236:   PetscPrintf(PETSC_COMM_WORLD,"Number of time steps: %D (%D Krylov iterations)\n",ctxt.niter,total_its);

238:   /* Free up memory */
239:   KSPDestroy(&ksp);
240:   MatDestroy(&TA);
241:   MatDestroy(&SC);
242:   MatDestroy(&R);
243:   MatDestroy(&J);
244:   MatDestroy(&Identity);
245:   PetscFree3(A,b,c);
246:   PetscFree2(At,B);
247:   VecDestroy(&uex);
248:   VecDestroy(&u);
249:   PetscFunctionListDestroy(&IRKList);

251:   PetscFinalize();
252:   return 0;
253: }

255: PetscErrorCode ExactSolution(Vec u,void *c,PetscReal t)
256: {
257:   UserContext     *ctxt = (UserContext*) c;
258:   PetscInt        i,is,ie;
259:   PetscScalar     *uarr;
260:   PetscReal       x,dx,a=ctxt->a,pi=PETSC_PI;

262:   dx = (ctxt->xmax - ctxt->xmin)/((PetscReal) ctxt->imax);
263:   VecGetOwnershipRange(u,&is,&ie);
264:   VecGetArray(u,&uarr);
265:   for (i=is; i<ie; i++) {
266:     x          = i * dx;
267:     switch (ctxt->physics_type) {
268:     case PHYSICS_DIFFUSION:
269:       uarr[i-is] = PetscExpScalar(-4.0*pi*pi*a*t)*PetscSinScalar(2*pi*x);
270:       break;
271:     case PHYSICS_ADVECTION:
272:       uarr[i-is] = PetscSinScalar(2*pi*(x - a*t));
273:       break;
274:     default: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"No support for physics type %s",PhysicsTypes[ctxt->physics_type]);
275:     }
276:   }
277:   VecRestoreArray(u,&uarr);
278:   return 0;
279: }

281: /* Arrays should be freed with PetscFree3(A,b,c) */
282: static PetscErrorCode RKCreate_Gauss(PetscInt nstages,PetscScalar **gauss_A,PetscScalar **gauss_b,PetscReal **gauss_c)
283: {
284:   PetscScalar       *A,*G0,*G1;
285:   PetscReal         *b,*c;
286:   PetscInt          i,j;
287:   Mat               G0mat,G1mat,Amat;

289:   PetscMalloc3(PetscSqr(nstages),&A,nstages,gauss_b,nstages,&c);
290:   PetscMalloc3(nstages,&b,PetscSqr(nstages),&G0,PetscSqr(nstages),&G1);
291:   PetscDTGaussQuadrature(nstages,0.,1.,c,b);
292:   for (i=0; i<nstages; i++) (*gauss_b)[i] = b[i]; /* copy to possibly-complex array */

294:   /* A^T = G0^{-1} G1 */
295:   for (i=0; i<nstages; i++) {
296:     for (j=0; j<nstages; j++) {
297:       G0[i*nstages+j] = PetscPowRealInt(c[i],j);
298:       G1[i*nstages+j] = PetscPowRealInt(c[i],j+1)/(j+1);
299:     }
300:   }
301:   /* The arrays above are row-aligned, but we create dense matrices as the transpose */
302:   MatCreateSeqDense(PETSC_COMM_SELF,nstages,nstages,G0,&G0mat);
303:   MatCreateSeqDense(PETSC_COMM_SELF,nstages,nstages,G1,&G1mat);
304:   MatCreateSeqDense(PETSC_COMM_SELF,nstages,nstages,A,&Amat);
305:   MatLUFactor(G0mat,NULL,NULL,NULL);
306:   MatMatSolve(G0mat,G1mat,Amat);
307:   MatTranspose(Amat,MAT_INPLACE_MATRIX,&Amat);

309:   MatDestroy(&G0mat);
310:   MatDestroy(&G1mat);
311:   MatDestroy(&Amat);
312:   PetscFree3(b,G0,G1);
313:   *gauss_A = A;
314:   *gauss_c = c;
315:   return 0;
316: }

318: static PetscErrorCode Assemble_AdvDiff(MPI_Comm comm,UserContext *user,Mat *J)
319: {
320:   PetscInt       matis,matie,i;
321:   PetscReal      dx,dx2;

323:   dx = (user->xmax - user->xmin)/((PetscReal)user->imax); dx2 = dx*dx;
324:   MatCreate(comm,J);
325:   MatSetType(*J,MATAIJ);
326:   MatSetSizes(*J,PETSC_DECIDE,PETSC_DECIDE,user->imax,user->imax);
327:   MatSetUp(*J);
328:   MatGetOwnershipRange(*J,&matis,&matie);
329:   for (i=matis; i<matie; i++) {
330:     PetscScalar values[3];
331:     PetscInt    col[3];
332:     switch (user->physics_type) {
333:     case PHYSICS_DIFFUSION:
334:       values[0] = -user->a*1.0/dx2;
335:       values[1] = user->a*2.0/dx2;
336:       values[2] = -user->a*1.0/dx2;
337:       break;
338:     case PHYSICS_ADVECTION:
339:       values[0] = -user->a*.5/dx;
340:       values[1] = 0.;
341:       values[2] = user->a*.5/dx;
342:       break;
343:     default: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"No support for physics type %s",PhysicsTypes[user->physics_type]);
344:     }
345:     /* periodic boundaries */
346:     if (i == 0) {
347:       col[0] = user->imax-1;
348:       col[1] = i;
349:       col[2] = i+1;
350:     } else if (i == user->imax-1) {
351:       col[0] = i-1;
352:       col[1] = i;
353:       col[2] = 0;
354:     } else {
355:       col[0] = i-1;
356:       col[1] = i;
357:       col[2] = i+1;
358:     }
359:     MatSetValues(*J,1,&i,3,col,values,INSERT_VALUES);
360:   }
361:   MatAssemblyBegin(*J,MAT_FINAL_ASSEMBLY);
362:   MatAssemblyEnd  (*J,MAT_FINAL_ASSEMBLY);
363:   return 0;
364: }

366: /*TEST
367:  testset:
368:    suffix: 1
369:    args: -a 0.1 -dt .125 -niter 5 -imax 40 -ksp_monitor_short -pc_type pbjacobi -irk_type gauss -irk_nstages 2
370:    test:
371:      args: -ksp_atol 1e-6
372:    test:
373:      requires: hpddm !single
374:      suffix: hpddm
375:      output_file: output/ex74_1.out
376:      args: -ksp_atol 1e-6 -ksp_type hpddm
377:    test:
378:      requires: hpddm
379:      suffix: hpddm_gcrodr
380:      output_file: output/ex74_1_hpddm.out
381:      args: -ksp_atol 1e-4 -ksp_view_final_residual -ksp_type hpddm -ksp_hpddm_type gcrodr -ksp_hpddm_recycle 2
382:  test:
383:    suffix: 2
384:    args: -a 0.1 -dt .125 -niter 5 -imax 40 -ksp_monitor_short -pc_type pbjacobi -ksp_atol 1e-6 -irk_type gauss -irk_nstages 4 -ksp_gmres_restart 100
385:  testset:
386:    suffix: 3
387:    requires: !single
388:    args: -a 1 -dt .33 -niter 3 -imax 40 -ksp_monitor_short -pc_type pbjacobi -ksp_atol 1e-6 -irk_type gauss -irk_nstages 4 -ksp_gmres_restart 100 -physics_type advection
389:    test:
390:      args:
391:    test:
392:      requires: hpddm
393:      suffix: hpddm
394:      output_file: output/ex74_3.out
395:      args: -ksp_type hpddm
396:    test:
397:      requires: hpddm
398:      suffix: hpddm_gcrodr
399:      output_file: output/ex74_3_hpddm.out
400:      args: -ksp_view_final_residual -ksp_type hpddm -ksp_hpddm_type gcrodr -ksp_hpddm_recycle 5

402: TEST*/