Date: 2022-03-31T04:48:45+00:00

Actual source code: ex6.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: /*
     10:    Concepts: TS^time-dependent linear problems
     11:    Concepts: TS^heat equation
     12:    Concepts: TS^diffusion equation
     13:    Routines: TSCreate(); TSSetSolution(); TSSetRHSJacobian(), TSSetIJacobian();
     14:    Routines: TSSetTimeStep(); TSSetMaxTime(); TSMonitorSet();
     15:    Routines: TSSetFromOptions(); TSStep(); TSDestroy();
     16:    Routines: TSSetTimeStep(); TSGetTimeStep();
     17:    Processors: 1
     18: */

     20: /* ------------------------------------------------------------------------

     22:    This program solves the one-dimensional heat equation (also called the
     23:    diffusion equation),
     24:        u_t = u_xx,
     25:    on the domain 0 <= x <= 1, with the boundary conditions
     26:        u(t,0) = 0, u(t,1) = 0,
     27:    and the initial condition
     28:        u(0,x) = sin(6*pi*x) + 3*sin(2*pi*x).
     29:    This is a linear, second-order, parabolic equation.

     31:    We discretize the right-hand side using finite differences with
     32:    uniform grid spacing h:
     33:        u_xx = (u_{i+1} - 2u_{i} + u_{i-1})/(h^2)
     34:    We then demonstrate time evolution using the various TS methods by
     35:    running the program via
     36:        ex3 -ts_type <timestepping solver>

     38:    We compare the approximate solution with the exact solution, given by
     39:        u_exact(x,t) = exp(-36*pi*pi*t) * sin(6*pi*x) +
     40:                       3*exp(-4*pi*pi*t) * sin(2*pi*x)

     42:    Notes:
     43:    This code demonstrates the TS solver interface to two variants of
     44:    linear problems, u_t = f(u,t), namely
     45:      - time-dependent f:   f(u,t) is a function of t
     46:      - time-independent f: f(u,t) is simply f(u)

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

     50:   ------------------------------------------------------------------------- */

     52: /*
     53:    Include "ts.h" so that we can use TS solvers.  Note that this file
     54:    automatically includes:
     55:      petscsys.h  - base PETSc routines   vec.h  - vectors
     56:      sys.h    - system routines       mat.h  - matrices
     57:      is.h     - index sets            ksp.h  - Krylov subspace methods
     58:      viewer.h - viewers               pc.h   - preconditioners
     59:      snes.h - nonlinear solvers
     60: */

     62: #include <petscts.h>
     63: #include <petscdraw.h>

     65: /*
     66:    User-defined application context - contains data needed by the
     67:    application-provided call-back routines.
     68: */
     69: typedef struct {
     70:   Vec         solution;          /* global exact solution vector */
     71:   PetscInt    m;                 /* total number of grid points */
     72:   PetscReal   h;                 /* mesh width h = 1/(m-1) */
     73:   PetscBool   debug;             /* flag (1 indicates activation of debugging printouts) */
     74:   PetscViewer viewer1, viewer2;  /* viewers for the solution and error */
     75:   PetscReal   norm_2, norm_max;  /* error norms */
     76: } AppCtx;

     78: /*
     79:    User-defined routines
     80: */
     81: extern PetscErrorCode InitialConditions(Vec,AppCtx*);
     82: extern PetscErrorCode RHSMatrixHeat(TS,PetscReal,Vec,Mat,Mat,void*);
     83: extern PetscErrorCode Monitor(TS,PetscInt,PetscReal,Vec,void*);
     84: extern PetscErrorCode ExactSolution(PetscReal,Vec,AppCtx*);
     85: extern PetscErrorCode MyBCRoutine(TS,PetscReal,Vec,void*);

     87: int main(int argc,char **argv)
     88: {
     89:   AppCtx         appctx;                 /* user-defined application context */
     90:   TS             ts;                     /* timestepping context */
     91:   Mat            A;                      /* matrix data structure */
     92:   Vec            u;                      /* approximate solution vector */
     93:   PetscReal      time_total_max = 100.0; /* default max total time */
     94:   PetscInt       time_steps_max = 100;   /* default max timesteps */
     95:   PetscDraw      draw;                   /* drawing context */
     96:   PetscInt       steps, m;
     97:   PetscMPIInt    size;
     98:   PetscReal      dt;
     99:   PetscReal      ftime;
    100:   PetscBool      flg;
    101:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    102:      Initialize program and set problem parameters
    103:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

    109:   m    = 60;
    110:   PetscOptionsGetInt(NULL,NULL,"-m",&m,NULL);
    111:   PetscOptionsHasName(NULL,NULL,"-debug",&appctx.debug);

    113:   appctx.m        = m;
    114:   appctx.h        = 1.0/(m-1.0);
    115:   appctx.norm_2   = 0.0;
    116:   appctx.norm_max = 0.0;

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

    120:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    121:      Create vector data structures
    122:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    124:   /*
    125:      Create vector data structures for approximate and exact solutions
    126:   */
    127:   VecCreateSeq(PETSC_COMM_SELF,m,&u);
    128:   VecDuplicate(u,&appctx.solution);

    130:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    131:      Set up displays to show graphs of the solution and error
    132:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    134:   PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,380,400,160,&appctx.viewer1);
    135:   PetscViewerDrawGetDraw(appctx.viewer1,0,&draw);
    136:   PetscDrawSetDoubleBuffer(draw);
    137:   PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,0,400,160,&appctx.viewer2);
    138:   PetscViewerDrawGetDraw(appctx.viewer2,0,&draw);
    139:   PetscDrawSetDoubleBuffer(draw);

    141:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    142:      Create timestepping solver context
    143:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    145:   TSCreate(PETSC_COMM_SELF,&ts);
    146:   TSSetProblemType(ts,TS_LINEAR);

    148:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    149:      Set optional user-defined monitoring routine
    150:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

    154:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

    156:      Create matrix data structure; set matrix evaluation routine.
    157:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    159:   MatCreate(PETSC_COMM_SELF,&A);
    160:   MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,m,m);
    161:   MatSetFromOptions(A);
    162:   MatSetUp(A);

    164:   PetscOptionsHasName(NULL,NULL,"-time_dependent_rhs",&flg);
    165:   if (flg) {
    166:     /*
    167:        For linear problems with a time-dependent f(u,t) in the equation
    168:        u_t = f(u,t), the user provides the discretized right-hand-side
    169:        as a time-dependent matrix.
    170:     */
    171:     TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
    172:     TSSetRHSJacobian(ts,A,A,RHSMatrixHeat,&appctx);
    173:   } else {
    174:     /*
    175:        For linear problems with a time-independent f(u) in the equation
    176:        u_t = f(u), the user provides the discretized right-hand-side
    177:        as a matrix only once, and then sets a null matrix evaluation
    178:        routine.
    179:     */
    180:     RHSMatrixHeat(ts,0.0,u,A,A,&appctx);
    181:     TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
    182:     TSSetRHSJacobian(ts,A,A,TSComputeRHSJacobianConstant,&appctx);
    183:   }

    185:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    186:      Set solution vector and initial timestep
    187:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    189:   dt   = appctx.h*appctx.h/2.0;
    190:   TSSetTimeStep(ts,dt);
    191:   TSSetSolution(ts,u);

    193:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    194:      Customize timestepping solver:
    195:        - Set the solution method to be the Backward Euler method.
    196:        - Set timestepping duration info
    197:      Then set runtime options, which can override these defaults.
    198:      For example,
    199:           -ts_max_steps <maxsteps> -ts_max_time <maxtime>
    200:      to override the defaults set by TSSetMaxSteps()/TSSetMaxTime().
    201:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    203:   TSSetMaxSteps(ts,time_steps_max);
    204:   TSSetMaxTime(ts,time_total_max);
    205:   TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);
    206:   TSSetFromOptions(ts);

    208:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    209:      Solve the problem
    210:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    212:   /*
    213:      Evaluate initial conditions
    214:   */
    215:   InitialConditions(u,&appctx);

    217:   /*
    218:      Run the timestepping solver
    219:   */
    220:   TSSolve(ts,u);
    221:   TSGetSolveTime(ts,&ftime);
    222:   TSGetStepNumber(ts,&steps);

    224:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    225:      View timestepping solver info
    226:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    228:   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));
    229:   TSView(ts,PETSC_VIEWER_STDOUT_SELF);

    231:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    232:      Free work space.  All PETSc objects should be destroyed when they
    233:      are no longer needed.
    234:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    236:   TSDestroy(&ts);
    237:   MatDestroy(&A);
    238:   VecDestroy(&u);
    239:   PetscViewerDestroy(&appctx.viewer1);
    240:   PetscViewerDestroy(&appctx.viewer2);
    241:   VecDestroy(&appctx.solution);

    243:   /*
    244:      Always call PetscFinalize() before exiting a program.  This routine
    245:        - finalizes the PETSc libraries as well as MPI
    246:        - provides summary and diagnostic information if certain runtime
    247:          options are chosen (e.g., -log_view).
    248:   */
    249:   PetscFinalize();
    250:   return 0;
    251: }
    252: /* --------------------------------------------------------------------- */
    253: /*
    254:    InitialConditions - Computes the solution at the initial time.

    256:    Input Parameter:
    257:    u - uninitialized solution vector (global)
    258:    appctx - user-defined application context

    260:    Output Parameter:
    261:    u - vector with solution at initial time (global)
    262: */
    263: PetscErrorCode InitialConditions(Vec u,AppCtx *appctx)
    264: {
    265:   PetscScalar    *u_localptr;
    266:   PetscInt       i;

    268:   /*
    269:     Get a pointer to vector data.
    270:     - For default PETSc vectors, VecGetArray() returns a pointer to
    271:       the data array.  Otherwise, the routine is implementation dependent.
    272:     - You MUST call VecRestoreArray() when you no longer need access to
    273:       the array.
    274:     - Note that the Fortran interface to VecGetArray() differs from the
    275:       C version.  See the users manual for details.
    276:   */
    277:   VecGetArray(u,&u_localptr);

    279:   /*
    280:      We initialize the solution array by simply writing the solution
    281:      directly into the array locations.  Alternatively, we could use
    282:      VecSetValues() or VecSetValuesLocal().
    283:   */
    284:   for (i=0; i<appctx->m; i++) u_localptr[i] = PetscSinReal(PETSC_PI*i*6.*appctx->h) + 3.*PetscSinReal(PETSC_PI*i*2.*appctx->h);

    286:   /*
    287:      Restore vector
    288:   */
    289:   VecRestoreArray(u,&u_localptr);

    291:   /*
    292:      Print debugging information if desired
    293:   */
    294:   if (appctx->debug) {
    295:      VecView(u,PETSC_VIEWER_STDOUT_SELF);
    296:   }

    298:   return 0;
    299: }
    300: /* --------------------------------------------------------------------- */
    301: /*
    302:    ExactSolution - Computes the exact solution at a given time.

    304:    Input Parameters:
    305:    t - current time
    306:    solution - vector in which exact solution will be computed
    307:    appctx - user-defined application context

    309:    Output Parameter:
    310:    solution - vector with the newly computed exact solution
    311: */
    312: PetscErrorCode ExactSolution(PetscReal t,Vec solution,AppCtx *appctx)
    313: {
    314:   PetscScalar    *s_localptr, h = appctx->h, ex1, ex2, sc1, sc2;
    315:   PetscInt       i;

    317:   /*
    318:      Get a pointer to vector data.
    319:   */
    320:   VecGetArray(solution,&s_localptr);

    322:   /*
    323:      Simply write the solution directly into the array locations.
    324:      Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
    325:   */
    326:   ex1 = PetscExpReal(-36.*PETSC_PI*PETSC_PI*t); ex2 = PetscExpReal(-4.*PETSC_PI*PETSC_PI*t);
    327:   sc1 = PETSC_PI*6.*h;                 sc2 = PETSC_PI*2.*h;
    328:   for (i=0; i<appctx->m; i++) s_localptr[i] = PetscSinReal(PetscRealPart(sc1)*(PetscReal)i)*ex1 + 3.*PetscSinReal(PetscRealPart(sc2)*(PetscReal)i)*ex2;

    330:   /*
    331:      Restore vector
    332:   */
    333:   VecRestoreArray(solution,&s_localptr);
    334:   return 0;
    335: }
    336: /* --------------------------------------------------------------------- */
    337: /*
    338:    Monitor - User-provided routine to monitor the solution computed at
    339:    each timestep.  This example plots the solution and computes the
    340:    error in two different norms.

    342:    This example also demonstrates changing the timestep via TSSetTimeStep().

    344:    Input Parameters:
    345:    ts     - the timestep context
    346:    step   - the count of the current step (with 0 meaning the
    347:              initial condition)
    348:    crtime  - the current time
    349:    u      - the solution at this timestep
    350:    ctx    - the user-provided context for this monitoring routine.
    351:             In this case we use the application context which contains
    352:             information about the problem size, workspace and the exact
    353:             solution.
    354: */
    355: PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal crtime,Vec u,void *ctx)
    356: {
    357:   AppCtx         *appctx = (AppCtx*) ctx;   /* user-defined application context */
    358:   PetscReal      norm_2, norm_max, dt, dttol;
    359:   PetscBool      flg;

    361:   /*
    362:      View a graph of the current iterate
    363:   */
    364:   VecView(u,appctx->viewer2);

    366:   /*
    367:      Compute the exact solution
    368:   */
    369:   ExactSolution(crtime,appctx->solution,appctx);

    371:   /*
    372:      Print debugging information if desired
    373:   */
    374:   if (appctx->debug) {
    375:     PetscPrintf(PETSC_COMM_SELF,"Computed solution vector\n");
    376:     VecView(u,PETSC_VIEWER_STDOUT_SELF);
    377:     PetscPrintf(PETSC_COMM_SELF,"Exact solution vector\n");
    378:     VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
    379:   }

    381:   /*
    382:      Compute the 2-norm and max-norm of the error
    383:   */
    384:   VecAXPY(appctx->solution,-1.0,u);
    385:   VecNorm(appctx->solution,NORM_2,&norm_2);
    386:   norm_2 = PetscSqrtReal(appctx->h)*norm_2;
    387:   VecNorm(appctx->solution,NORM_MAX,&norm_max);

    389:   TSGetTimeStep(ts,&dt);
    390:   if (norm_2 > 1.e-2) {
    391:     PetscPrintf(PETSC_COMM_SELF,"Timestep %D: step size = %g, time = %g, 2-norm error = %g, max norm error = %g\n",step,(double)dt,(double)crtime,(double)norm_2,(double)norm_max);
    392:   }
    393:   appctx->norm_2   += norm_2;
    394:   appctx->norm_max += norm_max;

    396:   dttol = .0001;
    397:   PetscOptionsGetReal(NULL,NULL,"-dttol",&dttol,&flg);
    398:   if (dt < dttol) {
    399:     dt  *= .999;
    400:     TSSetTimeStep(ts,dt);
    401:   }

    403:   /*
    404:      View a graph of the error
    405:   */
    406:   VecView(appctx->solution,appctx->viewer1);

    408:   /*
    409:      Print debugging information if desired
    410:   */
    411:   if (appctx->debug) {
    412:     PetscPrintf(PETSC_COMM_SELF,"Error vector\n");
    413:     VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
    414:   }

    416:   return 0;
    417: }
    418: /* --------------------------------------------------------------------- */
    419: /*
    420:    RHSMatrixHeat - User-provided routine to compute the right-hand-side
    421:    matrix for the heat equation.

    423:    Input Parameters:
    424:    ts - the TS context
    425:    t - current time
    426:    global_in - global input vector
    427:    dummy - optional user-defined context, as set by TSetRHSJacobian()

    429:    Output Parameters:
    430:    AA - Jacobian matrix
    431:    BB - optionally different preconditioning matrix
    432:    str - flag indicating matrix structure

    434:    Notes:
    435:    Recall that MatSetValues() uses 0-based row and column numbers
    436:    in Fortran as well as in C.
    437: */
    438: PetscErrorCode RHSMatrixHeat(TS ts,PetscReal t,Vec X,Mat AA,Mat BB,void *ctx)
    439: {
    440:   Mat            A       = AA;                /* Jacobian matrix */
    441:   AppCtx         *appctx = (AppCtx*) ctx;      /* user-defined application context */
    442:   PetscInt       mstart  = 0;
    443:   PetscInt       mend    = appctx->m;
    444:   PetscInt       i, idx[3];
    445:   PetscScalar    v[3], stwo = -2./(appctx->h*appctx->h), sone = -.5*stwo;

    447:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    448:      Compute entries for the locally owned part of the matrix
    449:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    450:   /*
    451:      Set matrix rows corresponding to boundary data
    452:   */

    454:   mstart = 0;
    455:   v[0]   = 1.0;
    456:   MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
    457:   mstart++;

    459:   mend--;
    460:   v[0] = 1.0;
    461:   MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);

    463:   /*
    464:      Set matrix rows corresponding to interior data.  We construct the
    465:      matrix one row at a time.
    466:   */
    467:   v[0] = sone; v[1] = stwo; v[2] = sone;
    468:   for (i=mstart; i<mend; i++) {
    469:     idx[0] = i-1; idx[1] = i; idx[2] = i+1;
    470:     MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
    471:   }

    473:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    474:      Complete the matrix assembly process and set some options
    475:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    476:   /*
    477:      Assemble matrix, using the 2-step process:
    478:        MatAssemblyBegin(), MatAssemblyEnd()
    479:      Computations can be done while messages are in transition
    480:      by placing code between these two statements.
    481:   */
    482:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
    483:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);

    485:   /*
    486:      Set and option to indicate that we will never add a new nonzero location
    487:      to the matrix. If we do, it will generate an error.
    488:   */
    489:   MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);

    491:   return 0;
    492: }
    493: /* --------------------------------------------------------------------- */
    494: /*
    495:    Input Parameters:
    496:    ts - the TS context
    497:    t - current time
    498:    f - function
    499:    ctx - optional user-defined context, as set by TSetBCFunction()
    500:  */
    501: PetscErrorCode MyBCRoutine(TS ts,PetscReal t,Vec f,void *ctx)
    502: {
    503:   AppCtx         *appctx = (AppCtx*) ctx;      /* user-defined application context */
    504:   PetscInt       m = appctx->m;
    505:   PetscScalar    *fa;

    507:   VecGetArray(f,&fa);
    508:   fa[0]   = 0.0;
    509:   fa[m-1] = 1.0;
    510:   VecRestoreArray(f,&fa);
    511:   PetscPrintf(PETSC_COMM_SELF,"t=%g\n",(double)t);

    513:   return 0;
    514: }

    516: /*TEST

    518:     test:
    519:       args: -nox -ts_max_steps 4

    521: TEST*/