Date: 2022-08-02T03:15:03+00:00

Actual source code: ex5.c

      2: static char help[] ="Solves a simple time-dependent linear PDE (the heat equation).\n\
      3: Input parameters include:\n\
      4:   -m <points>, where <points> = number of grid points\n\
      5:   -time_dependent_rhs : Treat the problem as having a time-dependent right-hand side\n\
      6:   -debug              : Activate debugging printouts\n\
      7:   -nox                : Deactivate x-window graphics\n\n";

      9: /* ------------------------------------------------------------------------

     11:    This program solves the one-dimensional heat equation (also called the
     12:    diffusion equation),
     13:        u_t = u_xx,
     14:    on the domain 0 <= x <= 1, with the boundary conditions
     15:        u(t,0) = 1, u(t,1) = 1,
     16:    and the initial condition
     17:        u(0,x) = cos(6*pi*x) + 3*cos(2*pi*x).
     18:    This is a linear, second-order, parabolic equation.

     20:    We discretize the right-hand side using finite differences with
     21:    uniform grid spacing h:
     22:        u_xx = (u_{i+1} - 2u_{i} + u_{i-1})/(h^2)
     23:    We then demonstrate time evolution using the various TS methods by
     24:    running the program via
     25:        ex3 -ts_type <timestepping solver>

     27:    We compare the approximate solution with the exact solution, given by
     28:        u_exact(x,t) = exp(-36*pi*pi*t) * cos(6*pi*x) +
     29:                       3*exp(-4*pi*pi*t) * cos(2*pi*x)

     31:    Notes:
     32:    This code demonstrates the TS solver interface to two variants of
     33:    linear problems, u_t = f(u,t), namely
     34:      - time-dependent f:   f(u,t) is a function of t
     35:      - time-independent f: f(u,t) is simply just f(u)

     37:     The parallel version of this code is ts/tutorials/ex4.c

     39:   ------------------------------------------------------------------------- */

     41: /*
     42:    Include "petscts.h" so that we can use TS solvers.  Note that this file
     43:    automatically includes:
     44:      petscsys.h       - base PETSc routines   petscvec.h  - vectors
     45:      petscmat.h  - matrices
     46:      petscis.h     - index sets            petscksp.h  - Krylov subspace methods
     47:      petscviewer.h - viewers               petscpc.h   - preconditioners
     48:      petscksp.h   - linear solvers        petscsnes.h - nonlinear solvers
     49: */
     50: #include <petscts.h>
     51: #include <petscdraw.h>

     53: /*
     54:    User-defined application context - contains data needed by the
     55:    application-provided call-back routines.
     56: */
     57: typedef struct {
     58:   Vec         solution;          /* global exact solution vector */
     59:   PetscInt    m;                      /* total number of grid points */
     60:   PetscReal   h;                 /* mesh width h = 1/(m-1) */
     61:   PetscBool   debug;             /* flag (1 indicates activation of debugging printouts) */
     62:   PetscViewer viewer1,viewer2;  /* viewers for the solution and error */
     63:   PetscReal   norm_2,norm_max;  /* error norms */
     64: } AppCtx;

     66: /*
     67:    User-defined routines
     68: */
     69: extern PetscErrorCode InitialConditions(Vec,AppCtx*);
     70: extern PetscErrorCode RHSMatrixHeat(TS,PetscReal,Vec,Mat,Mat,void*);
     71: extern PetscErrorCode Monitor(TS,PetscInt,PetscReal,Vec,void*);
     72: extern PetscErrorCode ExactSolution(PetscReal,Vec,AppCtx*);

     74: int main(int argc,char **argv)
     75: {
     76:   AppCtx         appctx;                 /* user-defined application context */
     77:   TS             ts;                     /* timestepping context */
     78:   Mat            A;                      /* matrix data structure */
     79:   Vec            u;                      /* approximate solution vector */
     80:   PetscReal      time_total_max = 100.0; /* default max total time */
     81:   PetscInt       time_steps_max = 100;   /* default max timesteps */
     82:   PetscDraw      draw;                   /* drawing context */
     83:   PetscInt       steps,m;
     84:   PetscMPIInt    size;
     85:   PetscBool      flg;
     86:   PetscReal      dt,ftime;

     88:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     89:      Initialize program and set problem parameters
     90:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

     92:   PetscInitialize(&argc,&argv,(char*)0,help);
     93:   MPI_Comm_size(PETSC_COMM_WORLD,&size);

     96:   m               = 60;
     97:   PetscOptionsGetInt(NULL,NULL,"-m",&m,NULL);
     98:   PetscOptionsHasName(NULL,NULL,"-debug",&appctx.debug);
     99:   appctx.m        = m;
    100:   appctx.h        = 1.0/(m-1.0);
    101:   appctx.norm_2   = 0.0;
    102:   appctx.norm_max = 0.0;

    104:   PetscPrintf(PETSC_COMM_SELF,"Solving a linear TS problem on 1 processor\n");

    106:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    107:      Create vector data structures
    108:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    110:   /*
    111:      Create vector data structures for approximate and exact solutions
    112:   */
    113:   VecCreateSeq(PETSC_COMM_SELF,m,&u);
    114:   VecDuplicate(u,&appctx.solution);

    116:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    117:      Set up displays to show graphs of the solution and error
    118:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    120:   PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,380,400,160,&appctx.viewer1);
    121:   PetscViewerDrawGetDraw(appctx.viewer1,0,&draw);
    122:   PetscDrawSetDoubleBuffer(draw);
    123:   PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,0,400,160,&appctx.viewer2);
    124:   PetscViewerDrawGetDraw(appctx.viewer2,0,&draw);
    125:   PetscDrawSetDoubleBuffer(draw);

    127:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    128:      Create timestepping solver context
    129:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    131:   TSCreate(PETSC_COMM_SELF,&ts);
    132:   TSSetProblemType(ts,TS_LINEAR);

    134:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    135:      Set optional user-defined monitoring routine
    136:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    138:   TSMonitorSet(ts,Monitor,&appctx,NULL);

    140:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

    142:      Create matrix data structure; set matrix evaluation routine.
    143:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    145:   MatCreate(PETSC_COMM_SELF,&A);
    146:   MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,m,m);
    147:   MatSetFromOptions(A);
    148:   MatSetUp(A);

    150:   PetscOptionsHasName(NULL,NULL,"-time_dependent_rhs",&flg);
    151:   if (flg) {
    152:     /*
    153:        For linear problems with a time-dependent f(u,t) in the equation
    154:        u_t = f(u,t), the user provides the discretized right-hand-side
    155:        as a time-dependent matrix.
    156:     */
    157:     TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
    158:     TSSetRHSJacobian(ts,A,A,RHSMatrixHeat,&appctx);
    159:   } else {
    160:     /*
    161:        For linear problems with a time-independent f(u) in the equation
    162:        u_t = f(u), the user provides the discretized right-hand-side
    163:        as a matrix only once, and then sets a null matrix evaluation
    164:        routine.
    165:     */
    166:     RHSMatrixHeat(ts,0.0,u,A,A,&appctx);
    167:     TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
    168:     TSSetRHSJacobian(ts,A,A,TSComputeRHSJacobianConstant,&appctx);
    169:   }

    171:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    172:      Set solution vector and initial timestep
    173:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    175:   dt   = appctx.h*appctx.h/2.0;
    176:   TSSetTimeStep(ts,dt);
    177:   TSSetSolution(ts,u);

    179:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    180:      Customize timestepping solver:
    181:        - Set the solution method to be the Backward Euler method.
    182:        - Set timestepping duration info
    183:      Then set runtime options, which can override these defaults.
    184:      For example,
    185:           -ts_max_steps <maxsteps> -ts_max_time <maxtime>
    186:      to override the defaults set by TSSetMaxSteps()/TSSetMaxTime().
    187:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    189:   TSSetMaxSteps(ts,time_steps_max);
    190:   TSSetMaxTime(ts,time_total_max);
    191:   TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);
    192:   TSSetFromOptions(ts);

    194:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    195:      Solve the problem
    196:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    198:   /*
    199:      Evaluate initial conditions
    200:   */
    201:   InitialConditions(u,&appctx);

    203:   /*
    204:      Run the timestepping solver
    205:   */
    206:   TSSolve(ts,u);
    207:   TSGetSolveTime(ts,&ftime);
    208:   TSGetStepNumber(ts,&steps);

    210:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    211:      View timestepping solver info
    212:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    214:   PetscPrintf(PETSC_COMM_SELF,"avg. error (2 norm) = %g, avg. error (max norm) = %g\n",(double)(appctx.norm_2/steps),(double)(appctx.norm_max/steps));
    215:   TSView(ts,PETSC_VIEWER_STDOUT_SELF);

    217:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    218:      Free work space.  All PETSc objects should be destroyed when they
    219:      are no longer needed.
    220:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    222:   TSDestroy(&ts);
    223:   MatDestroy(&A);
    224:   VecDestroy(&u);
    225:   PetscViewerDestroy(&appctx.viewer1);
    226:   PetscViewerDestroy(&appctx.viewer2);
    227:   VecDestroy(&appctx.solution);

    229:   /*
    230:      Always call PetscFinalize() before exiting a program.  This routine
    231:        - finalizes the PETSc libraries as well as MPI
    232:        - provides summary and diagnostic information if certain runtime
    233:          options are chosen (e.g., -log_view).
    234:   */
    235:   PetscFinalize();
    236:   return 0;
    237: }
    238: /* --------------------------------------------------------------------- */
    239: /*
    240:    InitialConditions - Computes the solution at the initial time.

    242:    Input Parameter:
    243:    u - uninitialized solution vector (global)
    244:    appctx - user-defined application context

    246:    Output Parameter:
    247:    u - vector with solution at initial time (global)
    248: */
    249: PetscErrorCode InitialConditions(Vec u,AppCtx *appctx)
    250: {
    251:   PetscScalar    *u_localptr,h = appctx->h;
    252:   PetscInt       i;

    254:   /*
    255:     Get a pointer to vector data.
    256:     - For default PETSc vectors, VecGetArray() returns a pointer to
    257:       the data array.  Otherwise, the routine is implementation dependent.
    258:     - You MUST call VecRestoreArray() when you no longer need access to
    259:       the array.
    260:     - Note that the Fortran interface to VecGetArray() differs from the
    261:       C version.  See the users manual for details.
    262:   */
    263:   VecGetArray(u,&u_localptr);

    265:   /*
    266:      We initialize the solution array by simply writing the solution
    267:      directly into the array locations.  Alternatively, we could use
    268:      VecSetValues() or VecSetValuesLocal().
    269:   */
    270:   for (i=0; i<appctx->m; i++) u_localptr[i] = PetscCosScalar(PETSC_PI*i*6.*h) + 3.*PetscCosScalar(PETSC_PI*i*2.*h);

    272:   /*
    273:      Restore vector
    274:   */
    275:   VecRestoreArray(u,&u_localptr);

    277:   /*
    278:      Print debugging information if desired
    279:   */
    280:   if (appctx->debug) {
    281:     printf("initial guess vector\n");
    282:     VecView(u,PETSC_VIEWER_STDOUT_SELF);
    283:   }

    285:   return 0;
    286: }
    287: /* --------------------------------------------------------------------- */
    288: /*
    289:    ExactSolution - Computes the exact solution at a given time.

    291:    Input Parameters:
    292:    t - current time
    293:    solution - vector in which exact solution will be computed
    294:    appctx - user-defined application context

    296:    Output Parameter:
    297:    solution - vector with the newly computed exact solution
    298: */
    299: PetscErrorCode ExactSolution(PetscReal t,Vec solution,AppCtx *appctx)
    300: {
    301:   PetscScalar    *s_localptr,h = appctx->h,ex1,ex2,sc1,sc2,tc = t;
    302:   PetscInt       i;

    304:   /*
    305:      Get a pointer to vector data.
    306:   */
    307:   VecGetArray(solution,&s_localptr);

    309:   /*
    310:      Simply write the solution directly into the array locations.
    311:      Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
    312:   */
    313:   ex1 = PetscExpScalar(-36.*PETSC_PI*PETSC_PI*tc); ex2 = PetscExpScalar(-4.*PETSC_PI*PETSC_PI*tc);
    314:   sc1 = PETSC_PI*6.*h;                 sc2 = PETSC_PI*2.*h;
    315:   for (i=0; i<appctx->m; i++) s_localptr[i] = PetscCosScalar(sc1*(PetscReal)i)*ex1 + 3.*PetscCosScalar(sc2*(PetscReal)i)*ex2;

    317:   /*
    318:      Restore vector
    319:   */
    320:   VecRestoreArray(solution,&s_localptr);
    321:   return 0;
    322: }
    323: /* --------------------------------------------------------------------- */
    324: /*
    325:    Monitor - User-provided routine to monitor the solution computed at
    326:    each timestep.  This example plots the solution and computes the
    327:    error in two different norms.

    329:    Input Parameters:
    330:    ts     - the timestep context
    331:    step   - the count of the current step (with 0 meaning the
    332:              initial condition)
    333:    time   - the current time
    334:    u      - the solution at this timestep
    335:    ctx    - the user-provided context for this monitoring routine.
    336:             In this case we use the application context which contains
    337:             information about the problem size, workspace and the exact
    338:             solution.
    339: */
    340: PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal time,Vec u,void *ctx)
    341: {
    342:   AppCtx         *appctx = (AppCtx*) ctx;   /* user-defined application context */
    343:   PetscReal      norm_2,norm_max;

    345:   /*
    346:      View a graph of the current iterate
    347:   */
    348:   VecView(u,appctx->viewer2);

    350:   /*
    351:      Compute the exact solution
    352:   */
    353:   ExactSolution(time,appctx->solution,appctx);

    355:   /*
    356:      Print debugging information if desired
    357:   */
    358:   if (appctx->debug) {
    359:     printf("Computed solution vector\n");
    360:     VecView(u,PETSC_VIEWER_STDOUT_SELF);
    361:     printf("Exact solution vector\n");
    362:     VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
    363:   }

    365:   /*
    366:      Compute the 2-norm and max-norm of the error
    367:   */
    368:   VecAXPY(appctx->solution,-1.0,u);
    369:   VecNorm(appctx->solution,NORM_2,&norm_2);
    370:   norm_2 = PetscSqrtReal(appctx->h)*norm_2;
    371:   VecNorm(appctx->solution,NORM_MAX,&norm_max);
    372:   if (norm_2   < 1e-14) norm_2   = 0;
    373:   if (norm_max < 1e-14) norm_max = 0;

    375:   PetscPrintf(PETSC_COMM_WORLD,"Timestep %D: time = %g, 2-norm error = %g, max norm error = %g\n",step,(double)time,(double)norm_2,(double)norm_max);
    376:   appctx->norm_2   += norm_2;
    377:   appctx->norm_max += norm_max;

    379:   /*
    380:      View a graph of the error
    381:   */
    382:   VecView(appctx->solution,appctx->viewer1);

    384:   /*
    385:      Print debugging information if desired
    386:   */
    387:   if (appctx->debug) {
    388:     printf("Error vector\n");
    389:     VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
    390:   }

    392:   return 0;
    393: }
    394: /* --------------------------------------------------------------------- */
    395: /*
    396:    RHSMatrixHeat - User-provided routine to compute the right-hand-side
    397:    matrix for the heat equation.

    399:    Input Parameters:
    400:    ts - the TS context
    401:    t - current time
    402:    global_in - global input vector
    403:    dummy - optional user-defined context, as set by TSetRHSJacobian()

    405:    Output Parameters:
    406:    AA - Jacobian matrix
    407:    BB - optionally different preconditioning matrix
    408:    str - flag indicating matrix structure

    410:   Notes:
    411:   Recall that MatSetValues() uses 0-based row and column numbers
    412:   in Fortran as well as in C.
    413: */
    414: PetscErrorCode RHSMatrixHeat(TS ts,PetscReal t,Vec X,Mat AA,Mat BB,void *ctx)
    415: {
    416:   Mat            A       = AA;                /* Jacobian matrix */
    417:   AppCtx         *appctx = (AppCtx*)ctx;     /* user-defined application context */
    418:   PetscInt       mstart  = 0;
    419:   PetscInt       mend    = appctx->m;
    420:   PetscInt       i,idx[3];
    421:   PetscScalar    v[3],stwo = -2./(appctx->h*appctx->h),sone = -.5*stwo;

    423:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    424:      Compute entries for the locally owned part of the matrix
    425:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    426:   /*
    427:      Set matrix rows corresponding to boundary data
    428:   */

    430:   mstart = 0;
    431:   v[0]   = 1.0;
    432:   MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
    433:   mstart++;

    435:   mend--;
    436:   v[0] = 1.0;
    437:   MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);

    439:   /*
    440:      Set matrix rows corresponding to interior data.  We construct the
    441:      matrix one row at a time.
    442:   */
    443:   v[0] = sone; v[1] = stwo; v[2] = sone;
    444:   for (i=mstart; i<mend; i++) {
    445:     idx[0] = i-1; idx[1] = i; idx[2] = i+1;
    446:     MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
    447:   }

    449:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    450:      Complete the matrix assembly process and set some options
    451:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    452:   /*
    453:      Assemble matrix, using the 2-step process:
    454:        MatAssemblyBegin(), MatAssemblyEnd()
    455:      Computations can be done while messages are in transition
    456:      by placing code between these two statements.
    457:   */
    458:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
    459:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);

    461:   /*
    462:      Set and option to indicate that we will never add a new nonzero location
    463:      to the matrix. If we do, it will generate an error.
    464:   */
    465:   MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);

    467:   return 0;
    468: }

    470: /*TEST

    472:     test:
    473:       requires: x

    475:     test:
    476:       suffix: nox
    477:       args: -nox

    479: TEST*/