Date: 2022-03-04T17:39:37+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: /*
     10:    Concepts: TS^time-dependent linear problems
     11:    Concepts: TS^heat equation
     12:    Concepts: TS^diffusion equation
     13:    Processors: 1
     14: */

     16: /* ------------------------------------------------------------------------

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

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

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

     38:    Notes:
     39:    This code demonstrates the TS solver interface to two variants of
     40:    linear problems, u_t = f(u,t), namely
     41:      - time-dependent f:   f(u,t) is a function of t
     42:      - time-independent f: f(u,t) is simply just f(u)

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

     46:   ------------------------------------------------------------------------- */

     48: /*
     49:    Include "petscts.h" so that we can use TS solvers.  Note that this file
     50:    automatically includes:
     51:      petscsys.h       - base PETSc routines   petscvec.h  - vectors
     52:      petscmat.h  - matrices
     53:      petscis.h     - index sets            petscksp.h  - Krylov subspace methods
     54:      petscviewer.h - viewers               petscpc.h   - preconditioners
     55:      petscksp.h   - linear solvers        petscsnes.h - nonlinear solvers
     56: */
     57: #include <petscts.h>
     58: #include <petscdraw.h>

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

     73: /*
     74:    User-defined routines
     75: */
     76: extern PetscErrorCode InitialConditions(Vec,AppCtx*);
     77: extern PetscErrorCode RHSMatrixHeat(TS,PetscReal,Vec,Mat,Mat,void*);
     78: extern PetscErrorCode Monitor(TS,PetscInt,PetscReal,Vec,void*);
     79: extern PetscErrorCode ExactSolution(PetscReal,Vec,AppCtx*);

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

     96:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     97:      Initialize program and set problem parameters
     98:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

    104:   m               = 60;
    105:   PetscOptionsGetInt(NULL,NULL,"-m",&m,NULL);
    106:   PetscOptionsHasName(NULL,NULL,"-debug",&appctx.debug);
    107:   appctx.m        = m;
    108:   appctx.h        = 1.0/(m-1.0);
    109:   appctx.norm_2   = 0.0;
    110:   appctx.norm_max = 0.0;

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

    114:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    115:      Create vector data structures
    116:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    118:   /*
    119:      Create vector data structures for approximate and exact solutions
    120:   */
    121:   VecCreateSeq(PETSC_COMM_SELF,m,&u);
    122:   VecDuplicate(u,&appctx.solution);

    124:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    125:      Set up displays to show graphs of the solution and error
    126:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    128:   PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,380,400,160,&appctx.viewer1);
    129:   PetscViewerDrawGetDraw(appctx.viewer1,0,&draw);
    130:   PetscDrawSetDoubleBuffer(draw);
    131:   PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,0,400,160,&appctx.viewer2);
    132:   PetscViewerDrawGetDraw(appctx.viewer2,0,&draw);
    133:   PetscDrawSetDoubleBuffer(draw);

    135:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    136:      Create timestepping solver context
    137:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    139:   TSCreate(PETSC_COMM_SELF,&ts);
    140:   TSSetProblemType(ts,TS_LINEAR);

    142:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    143:      Set optional user-defined monitoring routine
    144:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

    148:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

    150:      Create matrix data structure; set matrix evaluation routine.
    151:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    153:   MatCreate(PETSC_COMM_SELF,&A);
    154:   MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,m,m);
    155:   MatSetFromOptions(A);
    156:   MatSetUp(A);

    158:   PetscOptionsHasName(NULL,NULL,"-time_dependent_rhs",&flg);
    159:   if (flg) {
    160:     /*
    161:        For linear problems with a time-dependent f(u,t) in the equation
    162:        u_t = f(u,t), the user provides the discretized right-hand-side
    163:        as a time-dependent matrix.
    164:     */
    165:     TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
    166:     TSSetRHSJacobian(ts,A,A,RHSMatrixHeat,&appctx);
    167:   } else {
    168:     /*
    169:        For linear problems with a time-independent f(u) in the equation
    170:        u_t = f(u), the user provides the discretized right-hand-side
    171:        as a matrix only once, and then sets a null matrix evaluation
    172:        routine.
    173:     */
    174:     RHSMatrixHeat(ts,0.0,u,A,A,&appctx);
    175:     TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);
    176:     TSSetRHSJacobian(ts,A,A,TSComputeRHSJacobianConstant,&appctx);
    177:   }

    179:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    180:      Set solution vector and initial timestep
    181:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    183:   dt   = appctx.h*appctx.h/2.0;
    184:   TSSetTimeStep(ts,dt);
    185:   TSSetSolution(ts,u);

    187:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    188:      Customize timestepping solver:
    189:        - Set the solution method to be the Backward Euler method.
    190:        - Set timestepping duration info
    191:      Then set runtime options, which can override these defaults.
    192:      For example,
    193:           -ts_max_steps <maxsteps> -ts_max_time <maxtime>
    194:      to override the defaults set by TSSetMaxSteps()/TSSetMaxTime().
    195:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    197:   TSSetMaxSteps(ts,time_steps_max);
    198:   TSSetMaxTime(ts,time_total_max);
    199:   TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);
    200:   TSSetFromOptions(ts);

    202:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    203:      Solve the problem
    204:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    206:   /*
    207:      Evaluate initial conditions
    208:   */
    209:   InitialConditions(u,&appctx);

    211:   /*
    212:      Run the timestepping solver
    213:   */
    214:   TSSolve(ts,u);
    215:   TSGetSolveTime(ts,&ftime);
    216:   TSGetStepNumber(ts,&steps);

    218:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    219:      View timestepping solver info
    220:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    222:   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));
    223:   TSView(ts,PETSC_VIEWER_STDOUT_SELF);

    225:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    226:      Free work space.  All PETSc objects should be destroyed when they
    227:      are no longer needed.
    228:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    230:   TSDestroy(&ts);
    231:   MatDestroy(&A);
    232:   VecDestroy(&u);
    233:   PetscViewerDestroy(&appctx.viewer1);
    234:   PetscViewerDestroy(&appctx.viewer2);
    235:   VecDestroy(&appctx.solution);

    237:   /*
    238:      Always call PetscFinalize() before exiting a program.  This routine
    239:        - finalizes the PETSc libraries as well as MPI
    240:        - provides summary and diagnostic information if certain runtime
    241:          options are chosen (e.g., -log_view).
    242:   */
    243:   PetscFinalize();
    244:   return ierr;
    245: }
    246: /* --------------------------------------------------------------------- */
    247: /*
    248:    InitialConditions - Computes the solution at the initial time.

    250:    Input Parameter:
    251:    u - uninitialized solution vector (global)
    252:    appctx - user-defined application context

    254:    Output Parameter:
    255:    u - vector with solution at initial time (global)
    256: */
    257: PetscErrorCode InitialConditions(Vec u,AppCtx *appctx)
    258: {
    259:   PetscScalar    *u_localptr,h = appctx->h;
    260:   PetscInt       i;

    263:   /*
    264:     Get a pointer to vector data.
    265:     - For default PETSc vectors, VecGetArray() returns a pointer to
    266:       the data array.  Otherwise, the routine is implementation dependent.
    267:     - You MUST call VecRestoreArray() when you no longer need access to
    268:       the array.
    269:     - Note that the Fortran interface to VecGetArray() differs from the
    270:       C version.  See the users manual for details.
    271:   */
    272:   VecGetArray(u,&u_localptr);

    274:   /*
    275:      We initialize the solution array by simply writing the solution
    276:      directly into the array locations.  Alternatively, we could use
    277:      VecSetValues() or VecSetValuesLocal().
    278:   */
    279:   for (i=0; i<appctx->m; i++) u_localptr[i] = PetscCosScalar(PETSC_PI*i*6.*h) + 3.*PetscCosScalar(PETSC_PI*i*2.*h);

    281:   /*
    282:      Restore vector
    283:   */
    284:   VecRestoreArray(u,&u_localptr);

    286:   /*
    287:      Print debugging information if desired
    288:   */
    289:   if (appctx->debug) {
    290:     printf("initial guess vector\n");
    291:     VecView(u,PETSC_VIEWER_STDOUT_SELF);
    292:   }

    294:   return 0;
    295: }
    296: /* --------------------------------------------------------------------- */
    297: /*
    298:    ExactSolution - Computes the exact solution at a given time.

    300:    Input Parameters:
    301:    t - current time
    302:    solution - vector in which exact solution will be computed
    303:    appctx - user-defined application context

    305:    Output Parameter:
    306:    solution - vector with the newly computed exact solution
    307: */
    308: PetscErrorCode ExactSolution(PetscReal t,Vec solution,AppCtx *appctx)
    309: {
    310:   PetscScalar    *s_localptr,h = appctx->h,ex1,ex2,sc1,sc2,tc = t;
    311:   PetscInt       i;

    314:   /*
    315:      Get a pointer to vector data.
    316:   */
    317:   VecGetArray(solution,&s_localptr);

    319:   /*
    320:      Simply write the solution directly into the array locations.
    321:      Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
    322:   */
    323:   ex1 = PetscExpScalar(-36.*PETSC_PI*PETSC_PI*tc); ex2 = PetscExpScalar(-4.*PETSC_PI*PETSC_PI*tc);
    324:   sc1 = PETSC_PI*6.*h;                 sc2 = PETSC_PI*2.*h;
    325:   for (i=0; i<appctx->m; i++) s_localptr[i] = PetscCosScalar(sc1*(PetscReal)i)*ex1 + 3.*PetscCosScalar(sc2*(PetscReal)i)*ex2;

    327:   /*
    328:      Restore vector
    329:   */
    330:   VecRestoreArray(solution,&s_localptr);
    331:   return 0;
    332: }
    333: /* --------------------------------------------------------------------- */
    334: /*
    335:    Monitor - User-provided routine to monitor the solution computed at
    336:    each timestep.  This example plots the solution and computes the
    337:    error in two different norms.

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

    356:   /*
    357:      View a graph of the current iterate
    358:   */
    359:   VecView(u,appctx->viewer2);

    361:   /*
    362:      Compute the exact solution
    363:   */
    364:   ExactSolution(time,appctx->solution,appctx);

    366:   /*
    367:      Print debugging information if desired
    368:   */
    369:   if (appctx->debug) {
    370:     printf("Computed solution vector\n");
    371:     VecView(u,PETSC_VIEWER_STDOUT_SELF);
    372:     printf("Exact solution vector\n");
    373:     VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
    374:   }

    376:   /*
    377:      Compute the 2-norm and max-norm of the error
    378:   */
    379:   VecAXPY(appctx->solution,-1.0,u);
    380:   VecNorm(appctx->solution,NORM_2,&norm_2);
    381:   norm_2 = PetscSqrtReal(appctx->h)*norm_2;
    382:   VecNorm(appctx->solution,NORM_MAX,&norm_max);
    383:   if (norm_2   < 1e-14) norm_2   = 0;
    384:   if (norm_max < 1e-14) norm_max = 0;

    386:   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);
    387:   appctx->norm_2   += norm_2;
    388:   appctx->norm_max += norm_max;

    390:   /*
    391:      View a graph of the error
    392:   */
    393:   VecView(appctx->solution,appctx->viewer1);

    395:   /*
    396:      Print debugging information if desired
    397:   */
    398:   if (appctx->debug) {
    399:     printf("Error vector\n");
    400:     VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
    401:   }

    403:   return 0;
    404: }
    405: /* --------------------------------------------------------------------- */
    406: /*
    407:    RHSMatrixHeat - User-provided routine to compute the right-hand-side
    408:    matrix for the heat equation.

    410:    Input Parameters:
    411:    ts - the TS context
    412:    t - current time
    413:    global_in - global input vector
    414:    dummy - optional user-defined context, as set by TSetRHSJacobian()

    416:    Output Parameters:
    417:    AA - Jacobian matrix
    418:    BB - optionally different preconditioning matrix
    419:    str - flag indicating matrix structure

    421:   Notes:
    422:   Recall that MatSetValues() uses 0-based row and column numbers
    423:   in Fortran as well as in C.
    424: */
    425: PetscErrorCode RHSMatrixHeat(TS ts,PetscReal t,Vec X,Mat AA,Mat BB,void *ctx)
    426: {
    427:   Mat            A       = AA;                /* Jacobian matrix */
    428:   AppCtx         *appctx = (AppCtx*)ctx;     /* user-defined application context */
    429:   PetscInt       mstart  = 0;
    430:   PetscInt       mend    = appctx->m;
    432:   PetscInt       i,idx[3];
    433:   PetscScalar    v[3],stwo = -2./(appctx->h*appctx->h),sone = -.5*stwo;

    435:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    436:      Compute entries for the locally owned part of the matrix
    437:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    438:   /*
    439:      Set matrix rows corresponding to boundary data
    440:   */

    442:   mstart = 0;
    443:   v[0]   = 1.0;
    444:   MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
    445:   mstart++;

    447:   mend--;
    448:   v[0] = 1.0;
    449:   MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);

    451:   /*
    452:      Set matrix rows corresponding to interior data.  We construct the
    453:      matrix one row at a time.
    454:   */
    455:   v[0] = sone; v[1] = stwo; v[2] = sone;
    456:   for (i=mstart; i<mend; i++) {
    457:     idx[0] = i-1; idx[1] = i; idx[2] = i+1;
    458:     MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
    459:   }

    461:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    462:      Complete the matrix assembly process and set some options
    463:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    464:   /*
    465:      Assemble matrix, using the 2-step process:
    466:        MatAssemblyBegin(), MatAssemblyEnd()
    467:      Computations can be done while messages are in transition
    468:      by placing code between these two statements.
    469:   */
    470:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
    471:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);

    473:   /*
    474:      Set and option to indicate that we will never add a new nonzero location
    475:      to the matrix. If we do, it will generate an error.
    476:   */
    477:   MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);

    479:   return 0;
    480: }

    482: /*TEST

    484:     test:
    485:       requires: x

    487:     test:
    488:       suffix: nox
    489:       args: -nox

    491: TEST*/