Date: 2022-03-04T17:39:37+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 */
     97:   PetscInt       steps, m;
     98:   PetscMPIInt    size;
     99:   PetscReal      dt;
    100:   PetscReal      ftime;
    101:   PetscBool      flg;
    102:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    103:      Initialize program and set problem parameters
    104:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    106:   PetscInitialize(&argc,&argv,(char*)0,help);if (ierr) return ierr;
    107:   MPI_Comm_size(PETSC_COMM_WORLD,&size);
    108:   if (size != 1) SETERRQ(PETSC_COMM_WORLD,PETSC_ERR_WRONG_MPI_SIZE,"This is a uniprocessor example only!");

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

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

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

    121:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    122:      Create vector data structures
    123:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

    131:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    132:      Set up displays to show graphs of the solution and error
    133:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

    142:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    143:      Create timestepping solver context
    144:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

    149:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    150:      Set optional user-defined monitoring routine
    151:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

    155:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

    157:      Create matrix data structure; set matrix evaluation routine.
    158:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

    165:   PetscOptionsHasName(NULL,NULL,"-time_dependent_rhs",&flg);
    166:   if (flg) {
    167:     /*
    168:        For linear problems with a time-dependent f(u,t) in the equation
    169:        u_t = f(u,t), the user provides the discretized right-hand-side
    170:        as a time-dependent matrix.
    171:     */
    172:     TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
    173:     TSSetRHSJacobian(ts,A,A,RHSMatrixHeat,&appctx);
    174:   } else {
    175:     /*
    176:        For linear problems with a time-independent f(u) in the equation
    177:        u_t = f(u), the user provides the discretized right-hand-side
    178:        as a matrix only once, and then sets a null matrix evaluation
    179:        routine.
    180:     */
    181:     RHSMatrixHeat(ts,0.0,u,A,A,&appctx);
    182:     TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
    183:     TSSetRHSJacobian(ts,A,A,TSComputeRHSJacobianConstant,&appctx);
    184:   }

    186:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    187:      Set solution vector and initial timestep
    188:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

    194:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    195:      Customize timestepping solver:
    196:        - Set the solution method to be the Backward Euler method.
    197:        - Set timestepping duration info
    198:      Then set runtime options, which can override these defaults.
    199:      For example,
    200:           -ts_max_steps <maxsteps> -ts_max_time <maxtime>
    201:      to override the defaults set by TSSetMaxSteps()/TSSetMaxTime().
    202:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

    209:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    210:      Solve the problem
    211:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

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

    225:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    226:      View timestepping solver info
    227:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

    232:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    233:      Free work space.  All PETSc objects should be destroyed when they
    234:      are no longer needed.
    235:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

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

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

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

    270:   /*
    271:     Get a pointer to vector data.
    272:     - For default PETSc vectors, VecGetArray() returns a pointer to
    273:       the data array.  Otherwise, the routine is implementation dependent.
    274:     - You MUST call VecRestoreArray() when you no longer need access to
    275:       the array.
    276:     - Note that the Fortran interface to VecGetArray() differs from the
    277:       C version.  See the users manual for details.
    278:   */
    279:   VecGetArray(u,&u_localptr);

    281:   /*
    282:      We initialize the solution array by simply writing the solution
    283:      directly into the array locations.  Alternatively, we could use
    284:      VecSetValues() or VecSetValuesLocal().
    285:   */
    286:   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);

    288:   /*
    289:      Restore vector
    290:   */
    291:   VecRestoreArray(u,&u_localptr);

    293:   /*
    294:      Print debugging information if desired
    295:   */
    296:   if (appctx->debug) {
    297:      VecView(u,PETSC_VIEWER_STDOUT_SELF);
    298:   }

    300:   return 0;
    301: }
    302: /* --------------------------------------------------------------------- */
    303: /*
    304:    ExactSolution - Computes the exact solution at a given time.

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

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

    320:   /*
    321:      Get a pointer to vector data.
    322:   */
    323:   VecGetArray(solution,&s_localptr);

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

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

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

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

    365:   /*
    366:      View a graph of the current iterate
    367:   */
    368:   VecView(u,appctx->viewer2);

    370:   /*
    371:      Compute the exact solution
    372:   */
    373:   ExactSolution(crtime,appctx->solution,appctx);

    375:   /*
    376:      Print debugging information if desired
    377:   */
    378:   if (appctx->debug) {
    379:     PetscPrintf(PETSC_COMM_SELF,"Computed solution vector\n");
    380:     VecView(u,PETSC_VIEWER_STDOUT_SELF);
    381:     PetscPrintf(PETSC_COMM_SELF,"Exact solution vector\n");
    382:     VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
    383:   }

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

    393:   TSGetTimeStep(ts,&dt);
    394:   if (norm_2 > 1.e-2) {
    395:     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);
    396:   }
    397:   appctx->norm_2   += norm_2;
    398:   appctx->norm_max += norm_max;

    400:   dttol = .0001;
    401:   PetscOptionsGetReal(NULL,NULL,"-dttol",&dttol,&flg);
    402:   if (dt < dttol) {
    403:     dt  *= .999;
    404:     TSSetTimeStep(ts,dt);
    405:   }

    407:   /*
    408:      View a graph of the error
    409:   */
    410:   VecView(appctx->solution,appctx->viewer1);

    412:   /*
    413:      Print debugging information if desired
    414:   */
    415:   if (appctx->debug) {
    416:     PetscPrintf(PETSC_COMM_SELF,"Error vector\n");
    417:     VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
    418:   }

    420:   return 0;
    421: }
    422: /* --------------------------------------------------------------------- */
    423: /*
    424:    RHSMatrixHeat - User-provided routine to compute the right-hand-side
    425:    matrix for the heat equation.

    427:    Input Parameters:
    428:    ts - the TS context
    429:    t - current time
    430:    global_in - global input vector
    431:    dummy - optional user-defined context, as set by TSetRHSJacobian()

    433:    Output Parameters:
    434:    AA - Jacobian matrix
    435:    BB - optionally different preconditioning matrix
    436:    str - flag indicating matrix structure

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

    452:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    453:      Compute entries for the locally owned part of the matrix
    454:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    455:   /*
    456:      Set matrix rows corresponding to boundary data
    457:   */

    459:   mstart = 0;
    460:   v[0]   = 1.0;
    461:   MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
    462:   mstart++;

    464:   mend--;
    465:   v[0] = 1.0;
    466:   MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);

    468:   /*
    469:      Set matrix rows corresponding to interior data.  We construct the
    470:      matrix one row at a time.
    471:   */
    472:   v[0] = sone; v[1] = stwo; v[2] = sone;
    473:   for (i=mstart; i<mend; i++) {
    474:     idx[0] = i-1; idx[1] = i; idx[2] = i+1;
    475:     MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
    476:   }

    478:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    479:      Complete the matrix assembly process and set some options
    480:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    481:   /*
    482:      Assemble matrix, using the 2-step process:
    483:        MatAssemblyBegin(), MatAssemblyEnd()
    484:      Computations can be done while messages are in transition
    485:      by placing code between these two statements.
    486:   */
    487:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
    488:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);

    490:   /*
    491:      Set and option to indicate that we will never add a new nonzero location
    492:      to the matrix. If we do, it will generate an error.
    493:   */
    494:   MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);

    496:   return 0;
    497: }
    498: /* --------------------------------------------------------------------- */
    499: /*
    500:    Input Parameters:
    501:    ts - the TS context
    502:    t - current time
    503:    f - function
    504:    ctx - optional user-defined context, as set by TSetBCFunction()
    505:  */
    506: PetscErrorCode MyBCRoutine(TS ts,PetscReal t,Vec f,void *ctx)
    507: {
    508:   AppCtx         *appctx = (AppCtx*) ctx;      /* user-defined application context */
    510:   PetscInt       m = appctx->m;
    511:   PetscScalar    *fa;

    513:   VecGetArray(f,&fa);
    514:   fa[0]   = 0.0;
    515:   fa[m-1] = 1.0;
    516:   VecRestoreArray(f,&fa);
    517:   PetscPrintf(PETSC_COMM_SELF,"t=%g\n",(double)t);

    519:   return 0;
    520: }

    522: /*TEST

    524:     test:
    525:       args: -nox -ts_max_steps 4

    527: TEST*/