Date: 2026-03-31T05:46:36+00:00

Actual source code: ex3.c

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

      3: /*
      4:     Page 9, Section 1.2 Model Equations for Advection-Diffusion

      6:           u_t = a u_x + d u_xx

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

     10: */

     12: /*
     13:      Helpful runtime linear solver options:
     14:            -pc_type mg -da_refine 2 -snes_monitor -ksp_monitor -ts_view   (geometric multigrid with three levels)

     16: */

     18: /*
     19:    Include "petscts.h" so that we can use TS solvers.  Note that this file
     20:    automatically includes:
     21:      petscsys.h       - base PETSc routines   petscvec.h  - vectors
     22:      petscmat.h  - matrices
     23:      petscis.h     - index sets            petscksp.h  - Krylov subspace methods
     24:      petscviewer.h - viewers               petscpc.h   - preconditioners
     25:      petscksp.h   - linear solvers        petscsnes.h - nonlinear solvers
     26: */

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

     32: /*
     33:    User-defined application context - contains data needed by the
     34:    application-provided call-back routines.
     35: */
     36: typedef struct {
     37:   PetscScalar a, d; /* advection and diffusion strength */
     38:   PetscBool   upwind;
     39: } AppCtx;

     41: /*
     42:    User-defined routines
     43: */
     44: extern PetscErrorCode InitialConditions(TS, Vec, AppCtx *);
     45: extern PetscErrorCode RHSMatrixHeat(TS, PetscReal, Vec, Mat, Mat, void *);
     46: extern PetscErrorCode Solution(TS, PetscReal, Vec, AppCtx *);

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

     57:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     58:      Initialize program and set problem parameters
     59:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

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

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

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

     86:   PetscCall(TSCreate(PETSC_COMM_WORLD, &ts));
     87:   PetscCall(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:   PetscCall(TSSetRHSFunction(ts, NULL, TSComputeRHSFunctionLinear, &appctx));
     95:   PetscCall(TSSetRHSJacobian(ts, NULL, NULL, RHSMatrixHeat, &appctx));
     96:   PetscCall(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:   PetscCall(DMDAGetInfo(da, PETSC_IGNORE, &M, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0));
    108:   dt = .48 / (M * M);
    109:   PetscCall(TSSetTimeStep(ts, dt));
    110:   PetscCall(TSSetMaxSteps(ts, 1000));
    111:   PetscCall(TSSetMaxTime(ts, 100.0));
    112:   PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER));
    113:   PetscCall(TSSetType(ts, TSARKIMEX));
    114:   PetscCall(TSSetFromOptions(ts));

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

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

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

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

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

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

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

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

    209:   PetscFunctionBeginUser;
    210:   PetscCall(TSGetDM(ts, &da));
    211:   PetscCall(DMDAGetCorners(da, &mstart, 0, 0, &xm, 0, 0));
    212:   PetscCall(DMDAGetInfo(da, PETSC_IGNORE, &M, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0));
    213:   h    = 1.0 / M;
    214:   mend = mstart + xm;
    215:   /*
    216:      Get a pointer to vector data.
    217:   */
    218:   PetscCall(DMDAVecGetArray(da, U, &u));

    220:   /*
    221:      Simply write the solution directly into the array locations.
    222:      Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
    223:   */
    224:   ex1 = PetscExpScalar(-36. * PETSC_PI * PETSC_PI * appctx->d * t);
    225:   ex2 = PetscExpScalar(-4. * PETSC_PI * PETSC_PI * appctx->d * t);
    226:   sc1 = PETSC_PI * 6. * h;
    227:   sc2 = PETSC_PI * 2. * h;
    228:   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;

    230:   /*
    231:      Restore vector
    232:   */
    233:   PetscCall(DMDAVecRestoreArray(da, U, &u));
    234:   PetscFunctionReturn(PETSC_SUCCESS);
    235: }

    237: /* --------------------------------------------------------------------- */
    238: /*
    239:    RHSMatrixHeat - User-provided routine to compute the right-hand-side
    240:    matrix for the heat equation.

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

    248:    Output Parameters:
    249:    AA - Jacobian matrix
    250:    BB - optionally different matrix used to construct the preconditioner

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

    265:   PetscFunctionBeginUser;
    266:   PetscCall(TSGetDM(ts, &da));
    267:   PetscCall(DMDAGetInfo(da, 0, &M, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0));
    268:   PetscCall(DMDAGetCorners(da, &mstart, 0, 0, &xm, 0, 0));
    269:   h    = 1.0 / M;
    270:   mend = mstart + xm;
    271:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    272:      Compute entries for the locally owned part of the matrix
    273:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    274:   /*
    275:      Set matrix rows corresponding to boundary data
    276:   */

    278:   /* diffusion */
    279:   v[0] = appctx->d / (h * h);
    280:   v[1] = -2.0 * appctx->d / (h * h);
    281:   v[2] = appctx->d / (h * h);
    282:   if (!mstart) {
    283:     idx[0] = M - 1;
    284:     idx[1] = 0;
    285:     idx[2] = 1;
    286:     PetscCall(MatSetValues(A, 1, &mstart, 3, idx, v, INSERT_VALUES));
    287:     mstart++;
    288:   }

    290:   if (mend == M) {
    291:     mend--;
    292:     idx[0] = M - 2;
    293:     idx[1] = M - 1;
    294:     idx[2] = 0;
    295:     PetscCall(MatSetValues(A, 1, &mend, 3, idx, v, INSERT_VALUES));
    296:   }

    298:   /*
    299:      Set matrix rows corresponding to interior data.  We construct the
    300:      matrix one row at a time.
    301:   */
    302:   for (i = mstart; i < mend; i++) {
    303:     idx[0] = i - 1;
    304:     idx[1] = i;
    305:     idx[2] = i + 1;
    306:     PetscCall(MatSetValues(A, 1, &i, 3, idx, v, INSERT_VALUES));
    307:   }
    308:   PetscCall(MatAssemblyBegin(A, MAT_FLUSH_ASSEMBLY));
    309:   PetscCall(MatAssemblyEnd(A, MAT_FLUSH_ASSEMBLY));

    311:   PetscCall(DMDAGetCorners(da, &mstart, 0, 0, &xm, 0, 0));
    312:   mend = mstart + xm;
    313:   if (!appctx->upwind) {
    314:     /* advection -- centered differencing */
    315:     v[0] = -.5 * appctx->a / (h);
    316:     v[1] = .5 * appctx->a / (h);
    317:     if (!mstart) {
    318:       idx[0] = M - 1;
    319:       idx[1] = 1;
    320:       PetscCall(MatSetValues(A, 1, &mstart, 2, idx, v, ADD_VALUES));
    321:       mstart++;
    322:     }

    324:     if (mend == M) {
    325:       mend--;
    326:       idx[0] = M - 2;
    327:       idx[1] = 0;
    328:       PetscCall(MatSetValues(A, 1, &mend, 2, idx, v, ADD_VALUES));
    329:     }

    331:     for (i = mstart; i < mend; i++) {
    332:       idx[0] = i - 1;
    333:       idx[1] = i + 1;
    334:       PetscCall(MatSetValues(A, 1, &i, 2, idx, v, ADD_VALUES));
    335:     }
    336:   } else {
    337:     /* advection -- upwinding */
    338:     v[0] = -appctx->a / (h);
    339:     v[1] = appctx->a / (h);
    340:     if (!mstart) {
    341:       idx[0] = 0;
    342:       idx[1] = 1;
    343:       PetscCall(MatSetValues(A, 1, &mstart, 2, idx, v, ADD_VALUES));
    344:       mstart++;
    345:     }

    347:     if (mend == M) {
    348:       mend--;
    349:       idx[0] = M - 1;
    350:       idx[1] = 0;
    351:       PetscCall(MatSetValues(A, 1, &mend, 2, idx, v, ADD_VALUES));
    352:     }

    354:     for (i = mstart; i < mend; i++) {
    355:       idx[0] = i;
    356:       idx[1] = i + 1;
    357:       PetscCall(MatSetValues(A, 1, &i, 2, idx, v, ADD_VALUES));
    358:     }
    359:   }

    361:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    362:      Complete the matrix assembly process and set some options
    363:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    364:   /*
    365:      Assemble matrix, using the 2-step process:
    366:        MatAssemblyBegin(), MatAssemblyEnd()
    367:      Computations can be done while messages are in transition
    368:      by placing code between these two statements.
    369:   */
    370:   PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
    371:   PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));

    373:   /*
    374:      Set and option to indicate that we will never add a new nonzero location
    375:      to the matrix. If we do, it will generate an error.
    376:   */
    377:   PetscCall(MatSetOption(A, MAT_NEW_NONZERO_LOCATION_ERR, PETSC_TRUE));
    378:   PetscFunctionReturn(PETSC_SUCCESS);
    379: }

    381: /*TEST

    383:    test:
    384:       args: -pc_type mg -da_refine 2 -ts_view -ts_monitor -ts_max_time .3 -mg_levels_ksp_max_it 3
    385:       requires: double
    386:       filter: grep -v "total number of"

    388:    test:
    389:       suffix: 2
    390:       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
    391:       requires: x
    392:       output_file: output/ex3_1.out
    393:       requires: double
    394:       filter: grep -v "total number of"

    396: TEST*/