Actual source code: ex16.c

      1: static char help[] = "Solves the van der Pol equation and demonstrate IMEX.\n\
      2: Input parameters include:\n\
      3:       -mu : stiffness parameter\n\n";

      5: /* ------------------------------------------------------------------------

      7:    This program solves the van der Pol equation
      8:        y'' - \mu ((1-y^2)*y' - y) = 0        (1)
      9:    on the domain 0 <= x <= 1, with the boundary conditions
     10:        y(0) = 2, y'(0) = - 2/3 +10/(81*\mu) - 292/(2187*\mu^2),
     11:    This is a nonlinear equation. The well prepared initial condition gives errors that are not dominated by the first few steps of the method when \mu is large.

     13:    Notes:
     14:    This code demonstrates the TS solver interface to two variants of
     15:    linear problems, u_t = f(u,t), namely turning (1) into a system of
     16:    first order differential equations,

     18:    [ y' ] = [          z            ]
     19:    [ z' ]   [ \mu ((1 - y^2) z - y) ]

     21:    which then we can write as a vector equation

     23:    [ u_1' ] = [             u_2           ]  (2)
     24:    [ u_2' ]   [ \mu (1 - u_1^2) u_2 - u_1 ]

     26:    which is now in the desired form of u_t = f(u,t). One way that we
     27:    can split f(u,t) in (2) is to split by component,

     29:    [ u_1' ] = [ u_2 ] + [            0                ]
     30:    [ u_2' ]   [  0  ]   [ \mu ((1 - u_1^2) u_2 - u_1) ]

     32:    where

     34:    [ G(u,t) ] = [ u_2 ]
     35:                 [  0  ]

     37:    and

     39:    [ F(u',u,t) ] = [ u_1' ] - [            0                ]
     40:                    [ u_2' ]   [ \mu ((1 - u_1^2) u_2 - u_1) ]

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

     45:               dF   dF
     46:    J(F) = a * -- - --
     47:               du'  du

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

     51:    dF   [ 1 ; 0 ]
     52:    -- = [       ]
     53:    du'  [ 0 ; 1 ]

     55:    dF   [       0             ;         0        ]
     56:    -- = [                                        ]
     57:    du   [ -\mu (2*u_1*u_2 + 1);  \mu (1 - u_1^2) ]

     59:    Hence,

     61:           [      a             ;          0          ]
     62:    J(F) = [                                          ]
     63:           [ \mu (2*u_1*u_2 + 1); a - \mu (1 - u_1^2) ]

     65:   ------------------------------------------------------------------------- */

     67: #include <petscts.h>

     69: typedef struct _n_User *User;
     70: struct _n_User {
     71:   PetscReal mu;
     72:   PetscBool imex;
     73:   PetscReal next_output;
     74: };

     76: /*
     77:    User-defined routines
     78: */
     79: static PetscErrorCode RHSFunction(TS ts, PetscReal t, Vec X, Vec F, PetscCtx ctx)
     80: {
     81:   User               user = (User)ctx;
     82:   PetscScalar       *f;
     83:   const PetscScalar *x;

     85:   PetscFunctionBeginUser;
     86:   PetscCall(VecGetArrayRead(X, &x));
     87:   PetscCall(VecGetArray(F, &f));
     88:   f[0] = (user->imex ? x[1] : 0);
     89:   f[1] = 0.0;
     90:   PetscCall(VecRestoreArrayRead(X, &x));
     91:   PetscCall(VecRestoreArray(F, &f));
     92:   PetscFunctionReturn(PETSC_SUCCESS);
     93: }

     95: static PetscErrorCode IFunction(TS ts, PetscReal t, Vec X, Vec Xdot, Vec F, PetscCtx ctx)
     96: {
     97:   User               user = (User)ctx;
     98:   const PetscScalar *x, *xdot;
     99:   PetscScalar       *f;

    101:   PetscFunctionBeginUser;
    102:   PetscCall(VecGetArrayRead(X, &x));
    103:   PetscCall(VecGetArrayRead(Xdot, &xdot));
    104:   PetscCall(VecGetArray(F, &f));
    105:   f[0] = xdot[0] + (user->imex ? 0 : x[1]);
    106:   f[1] = xdot[1] - user->mu * ((1. - x[0] * x[0]) * x[1] - x[0]);
    107:   PetscCall(VecRestoreArrayRead(X, &x));
    108:   PetscCall(VecRestoreArrayRead(Xdot, &xdot));
    109:   PetscCall(VecRestoreArray(F, &f));
    110:   PetscFunctionReturn(PETSC_SUCCESS);
    111: }

    113: static PetscErrorCode IJacobian(TS ts, PetscReal t, Vec X, Vec Xdot, PetscReal a, Mat A, Mat B, PetscCtx ctx)
    114: {
    115:   User               user     = (User)ctx;
    116:   PetscReal          mu       = user->mu;
    117:   PetscInt           rowcol[] = {0, 1};
    118:   const PetscScalar *x;
    119:   PetscScalar        J[2][2];

    121:   PetscFunctionBeginUser;
    122:   PetscCall(VecGetArrayRead(X, &x));
    123:   J[0][0] = a;
    124:   J[0][1] = (user->imex ? 0 : 1.);
    125:   J[1][0] = mu * (2. * x[0] * x[1] + 1.);
    126:   J[1][1] = a - mu * (1. - x[0] * x[0]);
    127:   PetscCall(MatSetValues(B, 2, rowcol, 2, rowcol, &J[0][0], INSERT_VALUES));
    128:   PetscCall(VecRestoreArrayRead(X, &x));

    130:   PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
    131:   PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
    132:   if (A != B) {
    133:     PetscCall(MatAssemblyBegin(B, MAT_FINAL_ASSEMBLY));
    134:     PetscCall(MatAssemblyEnd(B, MAT_FINAL_ASSEMBLY));
    135:   }
    136:   PetscFunctionReturn(PETSC_SUCCESS);
    137: }

    139: static PetscErrorCode RegisterMyARK2(void)
    140: {
    141:   PetscFunctionBeginUser;
    142:   {
    143:     const PetscReal A[3][3] =
    144:       {
    145:         {0,                      0,    0},
    146:         {0.41421356237309504880, 0,    0},
    147:         {0.75,                   0.25, 0}
    148:     },
    149:                     At[3][3] = {{0, 0, 0}, {0.12132034355964257320, 0.29289321881345247560, 0}, {0.20710678118654752440, 0.50000000000000000000, 0.29289321881345247560}}, *bembedt = NULL, *bembed = NULL;
    150:     PetscCall(TSARKIMEXRegister("myark2", 2, 3, &At[0][0], NULL, NULL, &A[0][0], NULL, NULL, bembedt, bembed, 0, NULL, NULL));
    151:   }
    152:   PetscFunctionReturn(PETSC_SUCCESS);
    153: }

    155: /* Monitor timesteps and use interpolation to output at integer multiples of 0.1 */
    156: static PetscErrorCode Monitor(TS ts, PetscInt step, PetscReal t, Vec X, PetscCtx ctx)
    157: {
    158:   const PetscScalar *x;
    159:   PetscReal          tfinal, dt;
    160:   User               user = (User)ctx;
    161:   Vec                interpolatedX;

    163:   PetscFunctionBeginUser;
    164:   PetscCall(TSGetTimeStep(ts, &dt));
    165:   PetscCall(TSGetMaxTime(ts, &tfinal));

    167:   while (user->next_output <= t && user->next_output <= tfinal) {
    168:     PetscCall(VecDuplicate(X, &interpolatedX));
    169:     PetscCall(TSInterpolate(ts, user->next_output, interpolatedX));
    170:     PetscCall(VecGetArrayRead(interpolatedX, &x));
    171:     PetscCall(PetscPrintf(PETSC_COMM_WORLD, "[%.1f] %" 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])));
    172:     PetscCall(VecRestoreArrayRead(interpolatedX, &x));
    173:     PetscCall(VecDestroy(&interpolatedX));

    175:     user->next_output += 0.1;
    176:   }
    177:   PetscFunctionReturn(PETSC_SUCCESS);
    178: }

    180: int main(int argc, char **argv)
    181: {
    182:   TS             ts; /* nonlinear solver */
    183:   Vec            x;  /* solution, residual vectors */
    184:   Mat            A;  /* Jacobian matrix */
    185:   PetscInt       steps;
    186:   PetscReal      ftime   = 0.5;
    187:   PetscBool      monitor = PETSC_FALSE;
    188:   PetscScalar   *x_ptr;
    189:   PetscMPIInt    size;
    190:   struct _n_User user;

    192:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    193:      Initialize program
    194:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    195:   PetscFunctionBeginUser;
    196:   PetscCall(PetscInitialize(&argc, &argv, NULL, help));
    197:   PetscCallMPI(MPI_Comm_size(PETSC_COMM_WORLD, &size));
    198:   PetscCheck(size == 1, PETSC_COMM_WORLD, PETSC_ERR_WRONG_MPI_SIZE, "This is a uniprocessor example only!");

    200:   PetscCall(RegisterMyARK2());

    202:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    203:     Set runtime options
    204:     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    205:   user.mu          = 1000.0;
    206:   user.imex        = PETSC_TRUE;
    207:   user.next_output = 0.0;

    209:   PetscCall(PetscOptionsGetReal(NULL, NULL, "-mu", &user.mu, NULL));
    210:   PetscCall(PetscOptionsGetBool(NULL, NULL, "-imex", &user.imex, NULL));
    211:   PetscCall(PetscOptionsGetBool(NULL, NULL, "-monitor", &monitor, NULL));

    213:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    214:     Create necessary matrix and vectors, solve same ODE on every process
    215:     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    216:   PetscCall(MatCreate(PETSC_COMM_WORLD, &A));
    217:   PetscCall(MatSetSizes(A, PETSC_DECIDE, PETSC_DECIDE, 2, 2));
    218:   PetscCall(MatSetFromOptions(A));
    219:   PetscCall(MatSetUp(A));
    220:   PetscCall(MatCreateVecs(A, &x, NULL));

    222:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    223:      Create timestepping solver context
    224:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    225:   PetscCall(TSCreate(PETSC_COMM_WORLD, &ts));
    226:   PetscCall(TSSetType(ts, TSBEULER));
    227:   PetscCall(TSSetRHSFunction(ts, NULL, RHSFunction, &user));
    228:   PetscCall(TSSetIFunction(ts, NULL, IFunction, &user));
    229:   PetscCall(TSSetIJacobian(ts, A, A, IJacobian, &user));
    230:   PetscCall(TSSetMaxTime(ts, ftime));
    231:   PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER));
    232:   if (monitor) PetscCall(TSMonitorSet(ts, Monitor, &user, NULL));

    234:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    235:      Set initial conditions
    236:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    237:   PetscCall(VecGetArray(x, &x_ptr));
    238:   x_ptr[0] = 2.0;
    239:   x_ptr[1] = -2.0 / 3.0 + 10.0 / (81.0 * user.mu) - 292.0 / (2187.0 * user.mu * user.mu);
    240:   PetscCall(VecRestoreArray(x, &x_ptr));
    241:   PetscCall(TSSetTimeStep(ts, 0.01));

    243:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    244:      Set runtime options
    245:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    246:   PetscCall(TSSetFromOptions(ts));

    248:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    249:      Solve nonlinear system
    250:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    251:   PetscCall(TSSolve(ts, x));
    252:   PetscCall(TSGetSolveTime(ts, &ftime));
    253:   PetscCall(TSGetStepNumber(ts, &steps));
    254:   PetscCall(PetscPrintf(PETSC_COMM_WORLD, "mu %g, steps %" PetscInt_FMT ", ftime %g\n", (double)user.mu, steps, (double)ftime));
    255:   PetscCall(VecView(x, PETSC_VIEWER_STDOUT_WORLD));

    257:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    258:      Free work space.  All PETSc objects should be destroyed when they
    259:      are no longer needed.
    260:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    261:   PetscCall(MatDestroy(&A));
    262:   PetscCall(VecDestroy(&x));
    263:   PetscCall(TSDestroy(&ts));

    265:   PetscCall(PetscFinalize());
    266:   return 0;
    267: }

    269: /*TEST

    271:     test:
    272:       args: -ts_type arkimex -ts_arkimex_type myark2 -ts_adapt_type none
    273:       requires: !single

    275: TEST*/