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*/