Date: 2022-03-31T04:48:45+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 */
     90:   PetscInt       steps,m;
     91:   PetscMPIInt    size;
     92:   PetscBool      flg;
     93:   PetscReal      dt,ftime;

     95:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     96:      Initialize program and set problem parameters
     97:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

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

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

    113:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    114:      Create vector data structures
    115:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    117:   /*
    118:      Create vector data structures for approximate and exact solutions
    119:   */
    120:   VecCreateSeq(PETSC_COMM_SELF,m,&u);
    121:   VecDuplicate(u,&appctx.solution);

    123:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    124:      Set up displays to show graphs of the solution and error
    125:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

    134:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    135:      Create timestepping solver context
    136:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

    141:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    142:      Set optional user-defined monitoring routine
    143:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

    147:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

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

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

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

    178:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    179:      Set solution vector and initial timestep
    180:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

    186:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    187:      Customize timestepping solver:
    188:        - Set the solution method to be the Backward Euler method.
    189:        - Set timestepping duration info
    190:      Then set runtime options, which can override these defaults.
    191:      For example,
    192:           -ts_max_steps <maxsteps> -ts_max_time <maxtime>
    193:      to override the defaults set by TSSetMaxSteps()/TSSetMaxTime().
    194:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

    201:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    202:      Solve the problem
    203:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    205:   /*
    206:      Evaluate initial conditions
    207:   */
    208:   InitialConditions(u,&appctx);

    210:   /*
    211:      Run the timestepping solver
    212:   */
    213:   TSSolve(ts,u);
    214:   TSGetSolveTime(ts,&ftime);
    215:   TSGetStepNumber(ts,&steps);

    217:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    218:      View timestepping solver info
    219:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    221:   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));
    222:   TSView(ts,PETSC_VIEWER_STDOUT_SELF);

    224:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    225:      Free work space.  All PETSc objects should be destroyed when they
    226:      are no longer needed.
    227:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

    236:   /*
    237:      Always call PetscFinalize() before exiting a program.  This routine
    238:        - finalizes the PETSc libraries as well as MPI
    239:        - provides summary and diagnostic information if certain runtime
    240:          options are chosen (e.g., -log_view).
    241:   */
    242:   PetscFinalize();
    243:   return 0;
    244: }
    245: /* --------------------------------------------------------------------- */
    246: /*
    247:    InitialConditions - Computes the solution at the initial time.

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

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

    261:   /*
    262:     Get a pointer to vector data.
    263:     - For default PETSc vectors, VecGetArray() returns a pointer to
    264:       the data array.  Otherwise, the routine is implementation dependent.
    265:     - You MUST call VecRestoreArray() when you no longer need access to
    266:       the array.
    267:     - Note that the Fortran interface to VecGetArray() differs from the
    268:       C version.  See the users manual for details.
    269:   */
    270:   VecGetArray(u,&u_localptr);

    272:   /*
    273:      We initialize the solution array by simply writing the solution
    274:      directly into the array locations.  Alternatively, we could use
    275:      VecSetValues() or VecSetValuesLocal().
    276:   */
    277:   for (i=0; i<appctx->m; i++) u_localptr[i] = PetscCosScalar(PETSC_PI*i*6.*h) + 3.*PetscCosScalar(PETSC_PI*i*2.*h);

    279:   /*
    280:      Restore vector
    281:   */
    282:   VecRestoreArray(u,&u_localptr);

    284:   /*
    285:      Print debugging information if desired
    286:   */
    287:   if (appctx->debug) {
    288:     printf("initial guess vector\n");
    289:     VecView(u,PETSC_VIEWER_STDOUT_SELF);
    290:   }

    292:   return 0;
    293: }
    294: /* --------------------------------------------------------------------- */
    295: /*
    296:    ExactSolution - Computes the exact solution at a given time.

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

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

    311:   /*
    312:      Get a pointer to vector data.
    313:   */
    314:   VecGetArray(solution,&s_localptr);

    316:   /*
    317:      Simply write the solution directly into the array locations.
    318:      Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
    319:   */
    320:   ex1 = PetscExpScalar(-36.*PETSC_PI*PETSC_PI*tc); ex2 = PetscExpScalar(-4.*PETSC_PI*PETSC_PI*tc);
    321:   sc1 = PETSC_PI*6.*h;                 sc2 = PETSC_PI*2.*h;
    322:   for (i=0; i<appctx->m; i++) s_localptr[i] = PetscCosScalar(sc1*(PetscReal)i)*ex1 + 3.*PetscCosScalar(sc2*(PetscReal)i)*ex2;

    324:   /*
    325:      Restore vector
    326:   */
    327:   VecRestoreArray(solution,&s_localptr);
    328:   return 0;
    329: }
    330: /* --------------------------------------------------------------------- */
    331: /*
    332:    Monitor - User-provided routine to monitor the solution computed at
    333:    each timestep.  This example plots the solution and computes the
    334:    error in two different norms.

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

    352:   /*
    353:      View a graph of the current iterate
    354:   */
    355:   VecView(u,appctx->viewer2);

    357:   /*
    358:      Compute the exact solution
    359:   */
    360:   ExactSolution(time,appctx->solution,appctx);

    362:   /*
    363:      Print debugging information if desired
    364:   */
    365:   if (appctx->debug) {
    366:     printf("Computed solution vector\n");
    367:     VecView(u,PETSC_VIEWER_STDOUT_SELF);
    368:     printf("Exact solution vector\n");
    369:     VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
    370:   }

    372:   /*
    373:      Compute the 2-norm and max-norm of the error
    374:   */
    375:   VecAXPY(appctx->solution,-1.0,u);
    376:   VecNorm(appctx->solution,NORM_2,&norm_2);
    377:   norm_2 = PetscSqrtReal(appctx->h)*norm_2;
    378:   VecNorm(appctx->solution,NORM_MAX,&norm_max);
    379:   if (norm_2   < 1e-14) norm_2   = 0;
    380:   if (norm_max < 1e-14) norm_max = 0;

    382:   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);
    383:   appctx->norm_2   += norm_2;
    384:   appctx->norm_max += norm_max;

    386:   /*
    387:      View a graph of the error
    388:   */
    389:   VecView(appctx->solution,appctx->viewer1);

    391:   /*
    392:      Print debugging information if desired
    393:   */
    394:   if (appctx->debug) {
    395:     printf("Error vector\n");
    396:     VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);
    397:   }

    399:   return 0;
    400: }
    401: /* --------------------------------------------------------------------- */
    402: /*
    403:    RHSMatrixHeat - User-provided routine to compute the right-hand-side
    404:    matrix for the heat equation.

    406:    Input Parameters:
    407:    ts - the TS context
    408:    t - current time
    409:    global_in - global input vector
    410:    dummy - optional user-defined context, as set by TSetRHSJacobian()

    412:    Output Parameters:
    413:    AA - Jacobian matrix
    414:    BB - optionally different preconditioning matrix
    415:    str - flag indicating matrix structure

    417:   Notes:
    418:   Recall that MatSetValues() uses 0-based row and column numbers
    419:   in Fortran as well as in C.
    420: */
    421: PetscErrorCode RHSMatrixHeat(TS ts,PetscReal t,Vec X,Mat AA,Mat BB,void *ctx)
    422: {
    423:   Mat            A       = AA;                /* Jacobian matrix */
    424:   AppCtx         *appctx = (AppCtx*)ctx;     /* user-defined application context */
    425:   PetscInt       mstart  = 0;
    426:   PetscInt       mend    = appctx->m;
    427:   PetscInt       i,idx[3];
    428:   PetscScalar    v[3],stwo = -2./(appctx->h*appctx->h),sone = -.5*stwo;

    430:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    431:      Compute entries for the locally owned part of the matrix
    432:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    433:   /*
    434:      Set matrix rows corresponding to boundary data
    435:   */

    437:   mstart = 0;
    438:   v[0]   = 1.0;
    439:   MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
    440:   mstart++;

    442:   mend--;
    443:   v[0] = 1.0;
    444:   MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);

    446:   /*
    447:      Set matrix rows corresponding to interior data.  We construct the
    448:      matrix one row at a time.
    449:   */
    450:   v[0] = sone; v[1] = stwo; v[2] = sone;
    451:   for (i=mstart; i<mend; i++) {
    452:     idx[0] = i-1; idx[1] = i; idx[2] = i+1;
    453:     MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
    454:   }

    456:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    457:      Complete the matrix assembly process and set some options
    458:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    459:   /*
    460:      Assemble matrix, using the 2-step process:
    461:        MatAssemblyBegin(), MatAssemblyEnd()
    462:      Computations can be done while messages are in transition
    463:      by placing code between these two statements.
    464:   */
    465:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
    466:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);

    468:   /*
    469:      Set and option to indicate that we will never add a new nonzero location
    470:      to the matrix. If we do, it will generate an error.
    471:   */
    472:   MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);

    474:   return 0;
    475: }

    477: /*TEST

    479:     test:
    480:       requires: x

    482:     test:
    483:       suffix: nox
    484:       args: -nox

    486: TEST*/