Date: 2022-08-02T03:15:06+00:00

Actual source code: ex3.c

      2: static char help[] ="Model Equations for Advection-Diffusion\n";

      4: /*
      5:     Page 9, Section 1.2 Model Equations for Advection-Diffusion

      7:           u_t = a u_x + d u_xx

      9:    The initial conditions used here different then in the book.

     11: */

     13: /*
     14:      Helpful runtime linear solver options:
     15:            -pc_type mg -da_refine 2 -snes_monitor -ksp_monitor -ts_view   (geometric multigrid with three levels)

     17: */

     19: /*
     20:    Include "petscts.h" so that we can use TS solvers.  Note that this file
     21:    automatically includes:
     22:      petscsys.h       - base PETSc routines   petscvec.h  - vectors
     23:      petscmat.h  - matrices
     24:      petscis.h     - index sets            petscksp.h  - Krylov subspace methods
     25:      petscviewer.h - viewers               petscpc.h   - preconditioners
     26:      petscksp.h   - linear solvers        petscsnes.h - nonlinear solvers
     27: */

     29: #include <petscts.h>
     30: #include <petscdm.h>
     31: #include <petscdmda.h>

     33: /*
     34:    User-defined application context - contains data needed by the
     35:    application-provided call-back routines.
     36: */
     37: typedef struct {
     38:   PetscScalar a,d;   /* advection and diffusion strength */
     39:   PetscBool   upwind;
     40: } AppCtx;

     42: /*
     43:    User-defined routines
     44: */
     45: extern PetscErrorCode InitialConditions(TS,Vec,AppCtx*);
     46: extern PetscErrorCode RHSMatrixHeat(TS,PetscReal,Vec,Mat,Mat,void*);
     47: extern PetscErrorCode Solution(TS,PetscReal,Vec,AppCtx*);

     49: int main(int argc,char **argv)
     50: {
     51:   AppCtx         appctx;                 /* user-defined application context */
     52:   TS             ts;                     /* timestepping context */
     53:   Vec            U;                      /* approximate solution vector */
     54:   PetscReal      dt;
     55:   DM             da;
     56:   PetscInt       M;

     58:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     59:      Initialize program and set problem parameters
     60:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

     62:   PetscInitialize(&argc,&argv,(char*)0,help);
     63:   appctx.a      = 1.0;
     64:   appctx.d      = 0.0;
     65:   PetscOptionsGetScalar(NULL,NULL,"-a",&appctx.a,NULL);
     66:   PetscOptionsGetScalar(NULL,NULL,"-d",&appctx.d,NULL);
     67:   appctx.upwind = PETSC_TRUE;
     68:   PetscOptionsGetBool(NULL,NULL,"-upwind",&appctx.upwind,NULL);

     70:   DMDACreate1d(PETSC_COMM_WORLD,DM_BOUNDARY_PERIODIC, 60, 1, 1,NULL,&da);
     71:   DMSetFromOptions(da);
     72:   DMSetUp(da);
     73:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     74:      Create vector data structures
     75:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

     77:   /*
     78:      Create vector data structures for approximate and exact solutions
     79:   */
     80:   DMCreateGlobalVector(da,&U);

     82:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     83:      Create timestepping solver context
     84:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

     86:   TSCreate(PETSC_COMM_WORLD,&ts);
     87:   TSSetDM(ts,da);

     89:   /*
     90:       For linear problems with a time-dependent f(U,t) in the equation
     91:      u_t = f(u,t), the user provides the discretized right-hand-side
     92:       as a time-dependent matrix.
     93:   */
     94:   TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
     95:   TSSetRHSJacobian(ts,NULL,NULL,RHSMatrixHeat,&appctx);
     96:   TSSetSolutionFunction(ts,(PetscErrorCode (*)(TS,PetscReal,Vec,void*))Solution,&appctx);

     98:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     99:      Customize timestepping solver:
    100:        - Set timestepping duration info
    101:      Then set runtime options, which can override these defaults.
    102:      For example,
    103:           -ts_max_steps <maxsteps> -ts_max_time <maxtime>
    104:      to override the defaults set by TSSetMaxSteps()/TSSetMaxTime().
    105:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    107:   DMDAGetInfo(da,PETSC_IGNORE,&M,0,0,0,0,0,0,0,0,0,0,0);
    108:   dt   = .48/(M*M);
    109:   TSSetTimeStep(ts,dt);
    110:   TSSetMaxSteps(ts,1000);
    111:   TSSetMaxTime(ts,100.0);
    112:   TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);
    113:   TSSetType(ts,TSARKIMEX);
    114:   TSSetFromOptions(ts);

    116:   /*
    117:      Evaluate initial conditions
    118:   */
    119:   InitialConditions(ts,U,&appctx);

    121:   /*
    122:      Run the timestepping solver
    123:   */
    124:   TSSolve(ts,U);

    126:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    127:      Free work space.  All PETSc objects should be destroyed when they
    128:      are no longer needed.
    129:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    131:   TSDestroy(&ts);
    132:   VecDestroy(&U);
    133:   DMDestroy(&da);

    135:   /*
    136:      Always call PetscFinalize() before exiting a program.  This routine
    137:        - finalizes the PETSc libraries as well as MPI
    138:        - provides summary and diagnostic information if certain runtime
    139:          options are chosen (e.g., -log_view).
    140:   */
    141:   PetscFinalize();
    142:   return 0;
    143: }
    144: /* --------------------------------------------------------------------- */
    145: /*
    146:    InitialConditions - Computes the solution at the initial time.

    148:    Input Parameter:
    149:    u - uninitialized solution vector (global)
    150:    appctx - user-defined application context

    152:    Output Parameter:
    153:    u - vector with solution at initial time (global)
    154: */
    155: PetscErrorCode InitialConditions(TS ts,Vec U,AppCtx *appctx)
    156: {
    157:   PetscScalar    *u,h;
    158:   PetscInt       i,mstart,mend,xm,M;
    159:   DM             da;

    161:   TSGetDM(ts,&da);
    162:   DMDAGetCorners(da,&mstart,0,0,&xm,0,0);
    163:   DMDAGetInfo(da,PETSC_IGNORE,&M,0,0,0,0,0,0,0,0,0,0,0);
    164:   h    = 1.0/M;
    165:   mend = mstart + xm;
    166:   /*
    167:     Get a pointer to vector data.
    168:     - For default PETSc vectors, VecGetArray() returns a pointer to
    169:       the data array.  Otherwise, the routine is implementation dependent.
    170:     - You MUST call VecRestoreArray() when you no longer need access to
    171:       the array.
    172:     - Note that the Fortran interface to VecGetArray() differs from the
    173:       C version.  See the users manual for details.
    174:   */
    175:   DMDAVecGetArray(da,U,&u);

    177:   /*
    178:      We initialize the solution array by simply writing the solution
    179:      directly into the array locations.  Alternatively, we could use
    180:      VecSetValues() or VecSetValuesLocal().
    181:   */
    182:   for (i=mstart; i<mend; i++) u[i] = PetscSinScalar(PETSC_PI*i*6.*h) + 3.*PetscSinScalar(PETSC_PI*i*2.*h);

    184:   /*
    185:      Restore vector
    186:   */
    187:   DMDAVecRestoreArray(da,U,&u);
    188:   return 0;
    189: }
    190: /* --------------------------------------------------------------------- */
    191: /*
    192:    Solution - Computes the exact solution at a given time.

    194:    Input Parameters:
    195:    t - current time
    196:    solution - vector in which exact solution will be computed
    197:    appctx - user-defined application context

    199:    Output Parameter:
    200:    solution - vector with the newly computed exact solution
    201: */
    202: PetscErrorCode Solution(TS ts,PetscReal t,Vec U,AppCtx *appctx)
    203: {
    204:   PetscScalar    *u,ex1,ex2,sc1,sc2,h;
    205:   PetscInt       i,mstart,mend,xm,M;
    206:   DM             da;

    208:   TSGetDM(ts,&da);
    209:   DMDAGetCorners(da,&mstart,0,0,&xm,0,0);
    210:   DMDAGetInfo(da,PETSC_IGNORE,&M,0,0,0,0,0,0,0,0,0,0,0);
    211:   h    = 1.0/M;
    212:   mend = mstart + xm;
    213:   /*
    214:      Get a pointer to vector data.
    215:   */
    216:   DMDAVecGetArray(da,U,&u);

    218:   /*
    219:      Simply write the solution directly into the array locations.
    220:      Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
    221:   */
    222:   ex1 = PetscExpScalar(-36.*PETSC_PI*PETSC_PI*appctx->d*t);
    223:   ex2 = PetscExpScalar(-4.*PETSC_PI*PETSC_PI*appctx->d*t);
    224:   sc1 = PETSC_PI*6.*h;                 sc2 = PETSC_PI*2.*h;
    225:   for (i=mstart; i<mend; i++) u[i] = PetscSinScalar(sc1*(PetscReal)i + appctx->a*PETSC_PI*6.*t)*ex1 + 3.*PetscSinScalar(sc2*(PetscReal)i + appctx->a*PETSC_PI*2.*t)*ex2;

    227:   /*
    228:      Restore vector
    229:   */
    230:   DMDAVecRestoreArray(da,U,&u);
    231:   return 0;
    232: }

    234: /* --------------------------------------------------------------------- */
    235: /*
    236:    RHSMatrixHeat - User-provided routine to compute the right-hand-side
    237:    matrix for the heat equation.

    239:    Input Parameters:
    240:    ts - the TS context
    241:    t - current time
    242:    global_in - global input vector
    243:    dummy - optional user-defined context, as set by TSetRHSJacobian()

    245:    Output Parameters:
    246:    AA - Jacobian matrix
    247:    BB - optionally different preconditioning matrix
    248:    str - flag indicating matrix structure

    250:    Notes:
    251:    Recall that MatSetValues() uses 0-based row and column numbers
    252:    in Fortran as well as in C.
    253: */
    254: PetscErrorCode RHSMatrixHeat(TS ts,PetscReal t,Vec U,Mat AA,Mat BB,void *ctx)
    255: {
    256:   Mat            A       = AA;                /* Jacobian matrix */
    257:   AppCtx         *appctx = (AppCtx*)ctx;     /* user-defined application context */
    258:   PetscInt       mstart, mend;
    259:   PetscInt       i,idx[3],M,xm;
    260:   PetscScalar    v[3],h;
    261:   DM             da;

    263:   TSGetDM(ts,&da);
    264:   DMDAGetInfo(da,0,&M,0,0,0,0,0,0,0,0,0,0,0);
    265:   DMDAGetCorners(da,&mstart,0,0,&xm,0,0);
    266:   h    = 1.0/M;
    267:   mend = mstart + xm;
    268:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    269:      Compute entries for the locally owned part of the matrix
    270:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    271:   /*
    272:      Set matrix rows corresponding to boundary data
    273:   */

    275:   /* diffusion */
    276:   v[0] = appctx->d/(h*h);
    277:   v[1] = -2.0*appctx->d/(h*h);
    278:   v[2] = appctx->d/(h*h);
    279:   if (!mstart) {
    280:     idx[0] = M-1; idx[1] = 0; idx[2] = 1;
    281:     MatSetValues(A,1,&mstart,3,idx,v,INSERT_VALUES);
    282:     mstart++;
    283:   }

    285:   if (mend == M) {
    286:     mend--;
    287:     idx[0] = M-2; idx[1] = M-1; idx[2] = 0;
    288:     MatSetValues(A,1,&mend,3,idx,v,INSERT_VALUES);
    289:   }

    291:   /*
    292:      Set matrix rows corresponding to interior data.  We construct the
    293:      matrix one row at a time.
    294:   */
    295:   for (i=mstart; i<mend; i++) {
    296:     idx[0] = i-1; idx[1] = i; idx[2] = i+1;
    297:     MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
    298:   }
    299:   MatAssemblyBegin(A,MAT_FLUSH_ASSEMBLY);
    300:   MatAssemblyEnd(A,MAT_FLUSH_ASSEMBLY);

    302:   DMDAGetCorners(da,&mstart,0,0,&xm,0,0);
    303:   mend = mstart + xm;
    304:   if (!appctx->upwind) {
    305:     /* advection -- centered differencing */
    306:     v[0] = -.5*appctx->a/(h);
    307:     v[1] = .5*appctx->a/(h);
    308:     if (!mstart) {
    309:       idx[0] = M-1; idx[1] = 1;
    310:       MatSetValues(A,1,&mstart,2,idx,v,ADD_VALUES);
    311:       mstart++;
    312:     }

    314:     if (mend == M) {
    315:       mend--;
    316:       idx[0] = M-2; idx[1] = 0;
    317:       MatSetValues(A,1,&mend,2,idx,v,ADD_VALUES);
    318:     }

    320:     for (i=mstart; i<mend; i++) {
    321:       idx[0] = i-1; idx[1] = i+1;
    322:       MatSetValues(A,1,&i,2,idx,v,ADD_VALUES);
    323:     }
    324:   } else {
    325:     /* advection -- upwinding */
    326:     v[0] = -appctx->a/(h);
    327:     v[1] = appctx->a/(h);
    328:     if (!mstart) {
    329:       idx[0] = 0; idx[1] = 1;
    330:       MatSetValues(A,1,&mstart,2,idx,v,ADD_VALUES);
    331:       mstart++;
    332:     }

    334:     if (mend == M) {
    335:       mend--;
    336:       idx[0] = M-1; idx[1] = 0;
    337:       MatSetValues(A,1,&mend,2,idx,v,ADD_VALUES);
    338:     }

    340:     for (i=mstart; i<mend; i++) {
    341:       idx[0] = i; idx[1] = i+1;
    342:       MatSetValues(A,1,&i,2,idx,v,ADD_VALUES);
    343:     }
    344:   }

    346:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    347:      Complete the matrix assembly process and set some options
    348:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    349:   /*
    350:      Assemble matrix, using the 2-step process:
    351:        MatAssemblyBegin(), MatAssemblyEnd()
    352:      Computations can be done while messages are in transition
    353:      by placing code between these two statements.
    354:   */
    355:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
    356:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);

    358:   /*
    359:      Set and option to indicate that we will never add a new nonzero location
    360:      to the matrix. If we do, it will generate an error.
    361:   */
    362:   MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);
    363:   return 0;
    364: }

    366: /*TEST

    368:    test:
    369:       args: -pc_type mg -da_refine 2  -ts_view  -ts_monitor -ts_max_time .3 -mg_levels_ksp_max_it 3
    370:       requires: double
    371:       filter: grep -v "total number of"

    373:    test:
    374:       suffix: 2
    375:       args:  -pc_type mg -da_refine 2  -ts_view  -ts_monitor_draw_solution -ts_monitor -ts_max_time .3 -mg_levels_ksp_max_it 3
    376:       requires: x
    377:       output_file: output/ex3_1.out
    378:       requires: double
    379:       filter: grep -v "total number of"

    381: TEST*/