Actual source code: ex19.c

      1: static char help[] = "Solves the van der Pol DAE.\n\
      2: Input parameters include:\n";

      4: /* ------------------------------------------------------------------------

      6:    This program solves the van der Pol DAE
      7:        y' = -z = f(y,z)        (1)
      8:        0  = y-(z^3/3 - z) = g(y,z)
      9:    on the domain 0 <= x <= 1, with the boundary conditions
     10:        y(0) = -2, y'(0) = -2.355301397608119909925287735864250951918
     11:    This is a nonlinear equation.

     13:    Notes:
     14:    This code demonstrates the TS solver interface with the Van der Pol DAE,
     15:    namely it is the case when there is no RHS (meaning the RHS == 0), and the
     16:    equations are converted to two variants of linear problems, u_t = f(u,t),
     17:    namely turning (1) into a vector equation in terms of u,

     19:    [     y' + z      ] = [ 0 ]
     20:    [ (z^3/3 - z) - y ]   [ 0 ]

     22:    which then we can write as a vector equation

     24:    [      u_1' + u_2       ] = [ 0 ]  (2)
     25:    [ (u_2^3/3 - u_2) - u_1 ]   [ 0 ]

     27:    which is now in the desired form of u_t = f(u,t). As this is a DAE, and
     28:    there is no u_2', there is no need for a split,

     30:    so

     32:    [ F(u',u,t) ] = [ u_1' ] + [         u_2           ]
     33:                    [  0   ]   [ (u_2^3/3 - u_2) - u_1 ]

     35:    Using the definition of the Jacobian of F (from the PETSc user manual),
     36:    in the equation F(u',u,t) = G(u,t),

     38:               dF   dF
     39:    J(F) = a * -- - --
     40:               du'  du

     42:    where d is the partial derivative. In this example,

     44:    dF   [ 1 ; 0 ]
     45:    -- = [       ]
     46:    du'  [ 0 ; 0 ]

     48:    dF   [  0 ;      1     ]
     49:    -- = [                 ]
     50:    du   [ -1 ; 1 - u_2^2  ]

     52:    Hence,

     54:           [ a ;    -1     ]
     55:    J(F) = [               ]
     56:           [ 1 ; u_2^2 - 1 ]

     58:   ------------------------------------------------------------------------- */

     60: #include <petscts.h>

     62: typedef struct _n_User *User;
     63: struct _n_User {
     64:   PetscReal next_output;
     65: };

     67: /*
     68:    User-defined routines
     69: */

     71: static PetscErrorCode IFunction(TS ts, PetscReal t, Vec X, Vec Xdot, Vec F, void *ctx)
     72: {
     73:   PetscScalar       *f;
     74:   const PetscScalar *x, *xdot;

     76:   PetscFunctionBeginUser;
     77:   PetscCall(VecGetArrayRead(X, &x));
     78:   PetscCall(VecGetArrayRead(Xdot, &xdot));
     79:   PetscCall(VecGetArray(F, &f));
     80:   f[0] = xdot[0] + x[1];
     81:   f[1] = (x[1] * x[1] * x[1] / 3.0 - x[1]) - x[0];
     82:   PetscCall(VecRestoreArrayRead(X, &x));
     83:   PetscCall(VecRestoreArrayRead(Xdot, &xdot));
     84:   PetscCall(VecRestoreArray(F, &f));
     85:   PetscFunctionReturn(PETSC_SUCCESS);
     86: }

     88: static PetscErrorCode IJacobian(TS ts, PetscReal t, Vec X, Vec Xdot, PetscReal a, Mat A, Mat B, void *ctx)
     89: {
     90:   PetscInt           rowcol[] = {0, 1};
     91:   PetscScalar        J[2][2];
     92:   const PetscScalar *x;

     94:   PetscFunctionBeginUser;
     95:   PetscCall(VecGetArrayRead(X, &x));
     96:   J[0][0] = a;
     97:   J[0][1] = -1.;
     98:   J[1][0] = 1.;
     99:   J[1][1] = -1. + x[1] * x[1];
    100:   PetscCall(MatSetValues(B, 2, rowcol, 2, rowcol, &J[0][0], INSERT_VALUES));
    101:   PetscCall(VecRestoreArrayRead(X, &x));

    103:   PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
    104:   PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
    105:   if (A != B) {
    106:     PetscCall(MatAssemblyBegin(B, MAT_FINAL_ASSEMBLY));
    107:     PetscCall(MatAssemblyEnd(B, MAT_FINAL_ASSEMBLY));
    108:   }
    109:   PetscFunctionReturn(PETSC_SUCCESS);
    110: }

    112: static PetscErrorCode RegisterMyARK2(void)
    113: {
    114:   PetscFunctionBeginUser;
    115:   {
    116:     const PetscReal A[3][3] =
    117:       {
    118:         {0,                      0,    0},
    119:         {0.41421356237309504880, 0,    0},
    120:         {0.75,                   0.25, 0}
    121:     },
    122:                     At[3][3] = {{0, 0, 0}, {0.12132034355964257320, 0.29289321881345247560, 0}, {0.20710678118654752440, 0.50000000000000000000, 0.29289321881345247560}}, *bembedt = NULL, *bembed = NULL;
    123:     PetscCall(TSARKIMEXRegister("myark2", 2, 3, &At[0][0], NULL, NULL, &A[0][0], NULL, NULL, bembedt, bembed, 0, NULL, NULL));
    124:   }
    125:   PetscFunctionReturn(PETSC_SUCCESS);
    126: }

    128: /* Monitor timesteps and use interpolation to output at integer multiples of 0.1 */
    129: static PetscErrorCode Monitor(TS ts, PetscInt step, PetscReal t, Vec X, void *ctx)
    130: {
    131:   const PetscScalar *x;
    132:   PetscReal          tfinal, dt;
    133:   User               user = (User)ctx;
    134:   Vec                interpolatedX;

    136:   PetscFunctionBeginUser;
    137:   PetscCall(TSGetTimeStep(ts, &dt));
    138:   PetscCall(TSGetMaxTime(ts, &tfinal));

    140:   while (user->next_output <= t && user->next_output <= tfinal) {
    141:     PetscCall(VecDuplicate(X, &interpolatedX));
    142:     PetscCall(TSInterpolate(ts, user->next_output, interpolatedX));
    143:     PetscCall(VecGetArrayRead(interpolatedX, &x));
    144:     PetscCall(PetscPrintf(PETSC_COMM_WORLD, "[%.1f] %3" PetscInt_FMT " TS %.6f (dt = %.6f) X % 12.6e % 12.6e\n", (double)user->next_output, step, (double)t, (double)dt, (double)PetscRealPart(x[0]), (double)PetscRealPart(x[1])));
    145:     PetscCall(VecRestoreArrayRead(interpolatedX, &x));
    146:     PetscCall(VecDestroy(&interpolatedX));
    147:     user->next_output += PetscRealConstant(0.1);
    148:   }
    149:   PetscFunctionReturn(PETSC_SUCCESS);
    150: }

    152: int main(int argc, char **argv)
    153: {
    154:   TS             ts; /* nonlinear solver */
    155:   Vec            x;  /* solution, residual vectors */
    156:   Mat            A;  /* Jacobian matrix */
    157:   PetscInt       steps;
    158:   PetscReal      ftime   = 0.5;
    159:   PetscBool      monitor = PETSC_FALSE;
    160:   PetscScalar   *x_ptr;
    161:   PetscMPIInt    size;
    162:   struct _n_User user;

    164:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    165:      Initialize program
    166:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    167:   PetscFunctionBeginUser;
    168:   PetscCall(PetscInitialize(&argc, &argv, NULL, help));
    169:   PetscCallMPI(MPI_Comm_size(PETSC_COMM_WORLD, &size));
    170:   PetscCheck(size == 1, PETSC_COMM_WORLD, PETSC_ERR_WRONG_MPI_SIZE, "This is a uniprocessor example only!");

    172:   PetscCall(RegisterMyARK2());

    174:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    175:     Set runtime options
    176:     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    178:   user.next_output = 0.0;
    179:   PetscCall(PetscOptionsGetBool(NULL, NULL, "-monitor", &monitor, NULL));

    181:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    182:     Create necessary matrix and vectors, solve same ODE on every process
    183:     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    184:   PetscCall(MatCreate(PETSC_COMM_WORLD, &A));
    185:   PetscCall(MatSetSizes(A, PETSC_DECIDE, PETSC_DECIDE, 2, 2));
    186:   PetscCall(MatSetFromOptions(A));
    187:   PetscCall(MatSetUp(A));
    188:   PetscCall(MatCreateVecs(A, &x, NULL));

    190:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    191:      Create timestepping solver context
    192:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    193:   PetscCall(TSCreate(PETSC_COMM_WORLD, &ts));
    194:   PetscCall(TSSetType(ts, TSBEULER));
    195:   PetscCall(TSSetIFunction(ts, NULL, IFunction, &user));
    196:   PetscCall(TSSetIJacobian(ts, A, A, IJacobian, &user));
    197:   PetscCall(TSSetMaxTime(ts, ftime));
    198:   PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER));
    199:   if (monitor) PetscCall(TSMonitorSet(ts, Monitor, &user, NULL));

    201:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    202:      Set initial conditions
    203:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    204:   PetscCall(VecGetArray(x, &x_ptr));
    205:   x_ptr[0] = -2;
    206:   x_ptr[1] = -2.355301397608119909925287735864250951918;
    207:   PetscCall(VecRestoreArray(x, &x_ptr));
    208:   PetscCall(TSSetTimeStep(ts, .001));

    210:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    211:      Set runtime options
    212:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    213:   PetscCall(TSSetFromOptions(ts));

    215:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    216:      Solve nonlinear system
    217:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    218:   PetscCall(TSSolve(ts, x));
    219:   PetscCall(TSGetSolveTime(ts, &ftime));
    220:   PetscCall(TSGetStepNumber(ts, &steps));
    221:   PetscCall(PetscPrintf(PETSC_COMM_WORLD, "steps %3" PetscInt_FMT ", ftime %g\n", steps, (double)ftime));
    222:   PetscCall(VecView(x, PETSC_VIEWER_STDOUT_WORLD));

    224:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    225:      Free work space.  All PETSc objects should be destroyed when they
    226:      are no longer needed.
    227:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    228:   PetscCall(MatDestroy(&A));
    229:   PetscCall(VecDestroy(&x));
    230:   PetscCall(TSDestroy(&ts));

    232:   PetscCall(PetscFinalize());
    233:   return 0;
    234: }

    236: /*TEST

    238:    test:
    239:       requires: !single
    240:       suffix: a
    241:       args: -monitor -ts_type bdf -ts_rtol 1e-6 -ts_atol 1e-6 -ts_view -ts_adapt_type dsp
    242:       output_file: output/ex19_pi42.out

    244:    test:
    245:       requires: !single
    246:       suffix: b
    247:       args: -monitor -ts_type bdf -ts_rtol 1e-6 -ts_atol 1e-6 -ts_view -ts_adapt_type dsp -ts_adapt_dsp_filter PI42
    248:       output_file: output/ex19_pi42.out

    250:    test:
    251:       requires: !single
    252:       suffix: c
    253:       args: -monitor -ts_type bdf -ts_rtol 1e-6 -ts_atol 1e-6 -ts_view -ts_adapt_type dsp -ts_adapt_dsp_pid 0.4,0.2
    254:       output_file: output/ex19_pi42.out

    256:    test:
    257:       requires: !single
    258:       suffix: bdf_reject
    259:       args: -ts_type bdf -ts_dt 0.5 -ts_max_steps 1 -ts_max_reject {{0 1 2}separate_output} -ts_error_if_step_fails false -ts_adapt_monitor

    261: TEST*/