Date: 2022-08-02T03:14:46+00:00

Actual source code: ex13.c

      2: static char help[] = "Solves a variable Poisson problem with KSP.\n\n";

      4: /*
      5:   Include "petscksp.h" so that we can use KSP solvers.  Note that this file
      6:   automatically includes:
      7:      petscsys.h       - base PETSc routines   petscvec.h - vectors
      8:      petscmat.h - matrices
      9:      petscis.h     - index sets            petscksp.h - Krylov subspace methods
     10:      petscviewer.h - viewers               petscpc.h  - preconditioners
     11: */
     12: #include <petscksp.h>

     14: /*
     15:     User-defined context that contains all the data structures used
     16:     in the linear solution process.
     17: */
     18: typedef struct {
     19:   Vec         x,b;       /* solution vector, right-hand-side vector */
     20:   Mat         A;          /* sparse matrix */
     21:   KSP         ksp;       /* linear solver context */
     22:   PetscInt    m,n;       /* grid dimensions */
     23:   PetscScalar hx2,hy2;   /* 1/(m+1)*(m+1) and 1/(n+1)*(n+1) */
     24: } UserCtx;

     26: extern PetscErrorCode UserInitializeLinearSolver(PetscInt,PetscInt,UserCtx*);
     27: extern PetscErrorCode UserFinalizeLinearSolver(UserCtx*);
     28: extern PetscErrorCode UserDoLinearSolver(PetscScalar*,UserCtx *userctx,PetscScalar *b,PetscScalar *x);

     30: int main(int argc,char **args)
     31: {
     32:   UserCtx        userctx;
     33:   PetscInt       m = 6,n = 7,t,tmax = 2,i,Ii,j,N;
     34:   PetscScalar    *userx,*rho,*solution,*userb,hx,hy,x,y;
     35:   PetscReal      enorm;

     37:   /*
     38:      Initialize the PETSc libraries
     39:   */
     40:   PetscInitialize(&argc,&args,(char*)0,help);
     41:   /*
     42:      The next two lines are for testing only; these allow the user to
     43:      decide the grid size at runtime.
     44:   */
     45:   PetscOptionsGetInt(NULL,NULL,"-m",&m,NULL);
     46:   PetscOptionsGetInt(NULL,NULL,"-n",&n,NULL);

     48:   /*
     49:      Create the empty sparse matrix and linear solver data structures
     50:   */
     51:   UserInitializeLinearSolver(m,n,&userctx);
     52:   N    = m*n;

     54:   /*
     55:      Allocate arrays to hold the solution to the linear system.
     56:      This is not normally done in PETSc programs, but in this case,
     57:      since we are calling these routines from a non-PETSc program, we
     58:      would like to reuse the data structures from another code. So in
     59:      the context of a larger application these would be provided by
     60:      other (non-PETSc) parts of the application code.
     61:   */
     62:   PetscMalloc1(N,&userx);
     63:   PetscMalloc1(N,&userb);
     64:   PetscMalloc1(N,&solution);

     66:   /*
     67:       Allocate an array to hold the coefficients in the elliptic operator
     68:   */
     69:   PetscMalloc1(N,&rho);

     71:   /*
     72:      Fill up the array rho[] with the function rho(x,y) = x; fill the
     73:      right-hand-side b[] and the solution with a known problem for testing.
     74:   */
     75:   hx = 1.0/(m+1);
     76:   hy = 1.0/(n+1);
     77:   y  = hy;
     78:   Ii = 0;
     79:   for (j=0; j<n; j++) {
     80:     x = hx;
     81:     for (i=0; i<m; i++) {
     82:       rho[Ii]      = x;
     83:       solution[Ii] = PetscSinScalar(2.*PETSC_PI*x)*PetscSinScalar(2.*PETSC_PI*y);
     84:       userb[Ii]    = -2*PETSC_PI*PetscCosScalar(2*PETSC_PI *x)*PetscSinScalar(2*PETSC_PI*y) +
     85:                      8*PETSC_PI*PETSC_PI*x*PetscSinScalar(2*PETSC_PI *x)*PetscSinScalar(2*PETSC_PI*y);
     86:       x += hx;
     87:       Ii++;
     88:     }
     89:     y += hy;
     90:   }

     92:   /*
     93:      Loop over a bunch of timesteps, setting up and solver the linear
     94:      system for each time-step.

     96:      Note this is somewhat artificial. It is intended to demonstrate how
     97:      one may reuse the linear solver stuff in each time-step.
     98:   */
     99:   for (t=0; t<tmax; t++) {
    100:     UserDoLinearSolver(rho,&userctx,userb,userx);

    102:     /*
    103:         Compute error: Note that this could (and usually should) all be done
    104:         using the PETSc vector operations. Here we demonstrate using more
    105:         standard programming practices to show how they may be mixed with
    106:         PETSc.
    107:     */
    108:     enorm = 0.0;
    109:     for (i=0; i<N; i++) enorm += PetscRealPart(PetscConj(solution[i]-userx[i])*(solution[i]-userx[i]));
    110:     enorm *= PetscRealPart(hx*hy);
    111:     PetscPrintf(PETSC_COMM_WORLD,"m %D n %D error norm %g\n",m,n,(double)enorm);
    112:   }

    114:   /*
    115:      We are all finished solving linear systems, so we clean up the
    116:      data structures.
    117:   */
    118:   PetscFree(rho);
    119:   PetscFree(solution);
    120:   PetscFree(userx);
    121:   PetscFree(userb);
    122:   UserFinalizeLinearSolver(&userctx);
    123:   PetscFinalize();
    124:   return 0;
    125: }

    127: /* ------------------------------------------------------------------------*/
    128: PetscErrorCode UserInitializeLinearSolver(PetscInt m,PetscInt n,UserCtx *userctx)
    129: {
    130:   PetscInt       N;

    132:   /*
    133:      Here we assume use of a grid of size m x n, with all points on the
    134:      interior of the domain, i.e., we do not include the points corresponding
    135:      to homogeneous Dirichlet boundary conditions.  We assume that the domain
    136:      is [0,1]x[0,1].
    137:   */
    138:   userctx->m   = m;
    139:   userctx->n   = n;
    140:   userctx->hx2 = (m+1)*(m+1);
    141:   userctx->hy2 = (n+1)*(n+1);
    142:   N            = m*n;

    144:   /*
    145:      Create the sparse matrix. Preallocate 5 nonzeros per row.
    146:   */
    147:   MatCreateSeqAIJ(PETSC_COMM_SELF,N,N,5,0,&userctx->A);

    149:   /*
    150:      Create vectors. Here we create vectors with no memory allocated.
    151:      This way, we can use the data structures already in the program
    152:      by using VecPlaceArray() subroutine at a later stage.
    153:   */
    154:   VecCreateSeqWithArray(PETSC_COMM_SELF,1,N,NULL,&userctx->b);
    155:   VecDuplicate(userctx->b,&userctx->x);

    157:   /*
    158:      Create linear solver context. This will be used repeatedly for all
    159:      the linear solves needed.
    160:   */
    161:   KSPCreate(PETSC_COMM_SELF,&userctx->ksp);

    163:   return 0;
    164: }

    166: /*
    167:    Solves -div (rho grad psi) = F using finite differences.
    168:    rho is a 2-dimensional array of size m by n, stored in Fortran
    169:    style by columns. userb is a standard one-dimensional array.
    170: */
    171: /* ------------------------------------------------------------------------*/
    172: PetscErrorCode UserDoLinearSolver(PetscScalar *rho,UserCtx *userctx,PetscScalar *userb,PetscScalar *userx)
    173: {
    174:   PetscInt       i,j,Ii,J,m = userctx->m,n = userctx->n;
    175:   Mat            A = userctx->A;
    176:   PC             pc;
    177:   PetscScalar    v,hx2 = userctx->hx2,hy2 = userctx->hy2;

    179:   /*
    180:      This is not the most efficient way of generating the matrix
    181:      but let's not worry about it. We should have separate code for
    182:      the four corners, each edge and then the interior. Then we won't
    183:      have the slow if-tests inside the loop.

    185:      Computes the operator
    186:              -div rho grad
    187:      on an m by n grid with zero Dirichlet boundary conditions. The rho
    188:      is assumed to be given on the same grid as the finite difference
    189:      stencil is applied.  For a staggered grid, one would have to change
    190:      things slightly.
    191:   */
    192:   Ii = 0;
    193:   for (j=0; j<n; j++) {
    194:     for (i=0; i<m; i++) {
    195:       if (j>0) {
    196:         J    = Ii - m;
    197:         v    = -.5*(rho[Ii] + rho[J])*hy2;
    198:         MatSetValues(A,1,&Ii,1,&J,&v,INSERT_VALUES);
    199:       }
    200:       if (j<n-1) {
    201:         J    = Ii + m;
    202:         v    = -.5*(rho[Ii] + rho[J])*hy2;
    203:         MatSetValues(A,1,&Ii,1,&J,&v,INSERT_VALUES);
    204:       }
    205:       if (i>0) {
    206:         J    = Ii - 1;
    207:         v    = -.5*(rho[Ii] + rho[J])*hx2;
    208:         MatSetValues(A,1,&Ii,1,&J,&v,INSERT_VALUES);
    209:       }
    210:       if (i<m-1) {
    211:         J    = Ii + 1;
    212:         v    = -.5*(rho[Ii] + rho[J])*hx2;
    213:         MatSetValues(A,1,&Ii,1,&J,&v,INSERT_VALUES);
    214:       }
    215:       v    = 2.0*rho[Ii]*(hx2+hy2);
    216:       MatSetValues(A,1,&Ii,1,&Ii,&v,INSERT_VALUES);
    217:       Ii++;
    218:     }
    219:   }

    221:   /*
    222:      Assemble matrix
    223:   */
    224:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
    225:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);

    227:   /*
    228:      Set operators. Here the matrix that defines the linear system
    229:      also serves as the preconditioning matrix. Since all the matrices
    230:      will have the same nonzero pattern here, we indicate this so the
    231:      linear solvers can take advantage of this.
    232:   */
    233:   KSPSetOperators(userctx->ksp,A,A);

    235:   /*
    236:      Set linear solver defaults for this problem (optional).
    237:      - Here we set it to use direct LU factorization for the solution
    238:   */
    239:   KSPGetPC(userctx->ksp,&pc);
    240:   PCSetType(pc,PCLU);

    242:   /*
    243:      Set runtime options, e.g.,
    244:         -ksp_type <type> -pc_type <type> -ksp_monitor -ksp_rtol <rtol>
    245:      These options will override those specified above as long as
    246:      KSPSetFromOptions() is called _after_ any other customization
    247:      routines.

    249:      Run the program with the option -help to see all the possible
    250:      linear solver options.
    251:   */
    252:   KSPSetFromOptions(userctx->ksp);

    254:   /*
    255:      This allows the PETSc linear solvers to compute the solution
    256:      directly in the user's array rather than in the PETSc vector.

    258:      This is essentially a hack and not highly recommend unless you
    259:      are quite comfortable with using PETSc. In general, users should
    260:      write their entire application using PETSc vectors rather than
    261:      arrays.
    262:   */
    263:   VecPlaceArray(userctx->x,userx);
    264:   VecPlaceArray(userctx->b,userb);

    266:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    267:                       Solve the linear system
    268:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    270:   KSPSolve(userctx->ksp,userctx->b,userctx->x);

    272:   /*
    273:     Put back the PETSc array that belongs in the vector xuserctx->x
    274:   */
    275:   VecResetArray(userctx->x);
    276:   VecResetArray(userctx->b);

    278:   return 0;
    279: }

    281: /* ------------------------------------------------------------------------*/
    282: PetscErrorCode UserFinalizeLinearSolver(UserCtx *userctx)
    283: {
    284:   /*
    285:      We are all done and don't need to solve any more linear systems, so
    286:      we free the work space.  All PETSc objects should be destroyed when
    287:      they are no longer needed.
    288:   */
    289:   KSPDestroy(&userctx->ksp);
    290:   VecDestroy(&userctx->x);
    291:   VecDestroy(&userctx->b);
    292:   MatDestroy(&userctx->A);
    293:   return 0;
    294: }

    296: /*TEST

    298:    test:
    299:       args: -m 19 -n 20 -ksp_gmres_cgs_refinement_type refine_always

    301: TEST*/