Date: 2022-03-04T17:39:41+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 */
     55:   PetscReal      dt;
     56:   DM             da;
     57:   PetscInt       M;

     59:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     60:      Initialize program and set problem parameters
     61:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

     71:   DMDACreate1d(PETSC_COMM_WORLD,DM_BOUNDARY_PERIODIC, 60, 1, 1,NULL,&da);
     72:   DMSetFromOptions(da);
     73:   DMSetUp(da);
     74:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     75:      Create vector data structures
     76:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

     83:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     84:      Create timestepping solver context
     85:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

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

     99:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    100:      Customize timestepping solver:
    101:        - Set timestepping duration info
    102:      Then set runtime options, which can override these defaults.
    103:      For example,
    104:           -ts_max_steps <maxsteps> -ts_max_time <maxtime>
    105:      to override the defaults set by TSSetMaxSteps()/TSSetMaxTime().
    106:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

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

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

    127:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    128:      Free work space.  All PETSc objects should be destroyed when they
    129:      are no longer needed.
    130:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

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

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

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

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

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

    186:   /*
    187:      Restore vector
    188:   */
    189:   DMDAVecRestoreArray(da,U,&u);
    190:   return 0;
    191: }
    192: /* --------------------------------------------------------------------- */
    193: /*
    194:    Solution - Computes the exact solution at a given time.

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

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

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

    221:   /*
    222:      Simply write the solution directly into the array locations.
    223:      Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
    224:   */
    225:   ex1 = PetscExpScalar(-36.*PETSC_PI*PETSC_PI*appctx->d*t);
    226:   ex2 = PetscExpScalar(-4.*PETSC_PI*PETSC_PI*appctx->d*t);
    227:   sc1 = PETSC_PI*6.*h;                 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:   DMDAVecRestoreArray(da,U,&u);
    234:   return 0;
    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 preconditioning matrix
    251:    str - flag indicating matrix structure

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

    267:   TSGetDM(ts,&da);
    268:   DMDAGetInfo(da,0,&M,0,0,0,0,0,0,0,0,0,0,0);
    269:   DMDAGetCorners(da,&mstart,0,0,&xm,0,0);
    270:   h    = 1.0/M;
    271:   mend = mstart + xm;
    272:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    273:      Compute entries for the locally owned part of the matrix
    274:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    275:   /*
    276:      Set matrix rows corresponding to boundary data
    277:   */

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

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

    295:   /*
    296:      Set matrix rows corresponding to interior data.  We construct the
    297:      matrix one row at a time.
    298:   */
    299:   for (i=mstart; i<mend; i++) {
    300:     idx[0] = i-1; idx[1] = i; idx[2] = i+1;
    301:     MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
    302:   }
    303:   MatAssemblyBegin(A,MAT_FLUSH_ASSEMBLY);
    304:   MatAssemblyEnd(A,MAT_FLUSH_ASSEMBLY);

    306:   DMDAGetCorners(da,&mstart,0,0,&xm,0,0);
    307:   mend = mstart + xm;
    308:   if (!appctx->upwind) {
    309:     /* advection -- centered differencing */
    310:     v[0] = -.5*appctx->a/(h);
    311:     v[1] = .5*appctx->a/(h);
    312:     if (!mstart) {
    313:       idx[0] = M-1; idx[1] = 1;
    314:       MatSetValues(A,1,&mstart,2,idx,v,ADD_VALUES);
    315:       mstart++;
    316:     }

    318:     if (mend == M) {
    319:       mend--;
    320:       idx[0] = M-2; idx[1] = 0;
    321:       MatSetValues(A,1,&mend,2,idx,v,ADD_VALUES);
    322:     }

    324:     for (i=mstart; i<mend; i++) {
    325:       idx[0] = i-1; idx[1] = i+1;
    326:       MatSetValues(A,1,&i,2,idx,v,ADD_VALUES);
    327:     }
    328:   } else {
    329:     /* advection -- upwinding */
    330:     v[0] = -appctx->a/(h);
    331:     v[1] = appctx->a/(h);
    332:     if (!mstart) {
    333:       idx[0] = 0; idx[1] = 1;
    334:       MatSetValues(A,1,&mstart,2,idx,v,ADD_VALUES);
    335:       mstart++;
    336:     }

    338:     if (mend == M) {
    339:       mend--;
    340:       idx[0] = M-1; idx[1] = 0;
    341:       MatSetValues(A,1,&mend,2,idx,v,ADD_VALUES);
    342:     }

    344:     for (i=mstart; i<mend; i++) {
    345:       idx[0] = i; idx[1] = i+1;
    346:       MatSetValues(A,1,&i,2,idx,v,ADD_VALUES);
    347:     }
    348:   }

    350:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    351:      Complete the matrix assembly process and set some options
    352:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    353:   /*
    354:      Assemble matrix, using the 2-step process:
    355:        MatAssemblyBegin(), MatAssemblyEnd()
    356:      Computations can be done while messages are in transition
    357:      by placing code between these two statements.
    358:   */
    359:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
    360:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);

    362:   /*
    363:      Set and option to indicate that we will never add a new nonzero location
    364:      to the matrix. If we do, it will generate an error.
    365:   */
    366:   MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);
    367:   return 0;
    368: }

    370: /*TEST

    372:    test:
    373:       args: -pc_type mg -da_refine 2  -ts_view  -ts_monitor -ts_max_time .3 -mg_levels_ksp_max_it 3
    374:       requires: double
    375:       filter: grep -v "total number of"

    377:    test:
    378:       suffix: 2
    379:       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
    380:       requires: x
    381:       output_file: output/ex3_1.out
    382:       requires: double
    383:       filter: grep -v "total number of"

    385: TEST*/