Actual source code: ex16.c


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

  6: /* ------------------------------------------------------------------------

  8:    This program solves the van der Pol equation
  9:        y'' - \mu ((1-y^2)*y' - y) = 0        (1)
 10:    on the domain 0 <= x <= 1, with the boundary conditions
 11:        y(0) = 2, y'(0) = - 2/3 +10/(81*\mu) - 292/(2187*\mu^2),
 12:    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.

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

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

 22:    which then we can write as a vector equation

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

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

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

 33:    where

 35:    [ G(u,t) ] = [ u_2 ]
 36:                 [  0  ]

 38:    and

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

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

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

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

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

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

 60:    Hence,

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

 66:   ------------------------------------------------------------------------- */

 68: #include <petscts.h>

 70: typedef struct _n_User *User;
 71: struct _n_User {
 72:   PetscReal mu;
 73:   PetscBool imex;
 74:   PetscReal next_output;
 75: };

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

 87:   VecGetArrayRead(X, &x);
 88:   VecGetArray(F, &f);
 89:   f[0] = (user->imex ? x[1] : 0);
 90:   f[1] = 0.0;
 91:   VecRestoreArrayRead(X, &x);
 92:   VecRestoreArray(F, &f);
 93:   return 0;
 94: }

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

103:   VecGetArrayRead(X, &x);
104:   VecGetArrayRead(Xdot, &xdot);
105:   VecGetArray(F, &f);
106:   f[0] = xdot[0] + (user->imex ? 0 : x[1]);
107:   f[1] = xdot[1] - user->mu * ((1. - x[0] * x[0]) * x[1] - x[0]);
108:   VecRestoreArrayRead(X, &x);
109:   VecRestoreArrayRead(Xdot, &xdot);
110:   VecRestoreArray(F, &f);
111:   return 0;
112: }

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

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

131:   MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY);
132:   MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY);
133:   if (A != B) {
134:     MatAssemblyBegin(B, MAT_FINAL_ASSEMBLY);
135:     MatAssemblyEnd(B, MAT_FINAL_ASSEMBLY);
136:   }
137:   return 0;
138: }

140: static PetscErrorCode RegisterMyARK2(void)
141: {
143:   {
144:     const PetscReal A[3][3] =
145:       {
146:         {0,                      0,    0},
147:         {0.41421356237309504880, 0,    0},
148:         {0.75,                   0.25, 0}
149:     },
150:                     At[3][3] = {{0, 0, 0}, {0.12132034355964257320, 0.29289321881345247560, 0}, {0.20710678118654752440, 0.50000000000000000000, 0.29289321881345247560}}, *bembedt = NULL, *bembed = NULL;
151:     TSARKIMEXRegister("myark2", 2, 3, &At[0][0], NULL, NULL, &A[0][0], NULL, NULL, bembedt, bembed, 0, NULL, NULL);
152:   }
153:   return 0;
154: }

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

165:   TSGetTimeStep(ts, &dt);
166:   TSGetMaxTime(ts, &tfinal);

168:   while (user->next_output <= t && user->next_output <= tfinal) {
169:     VecDuplicate(X, &interpolatedX);
170:     TSInterpolate(ts, user->next_output, interpolatedX);
171:     VecGetArrayRead(interpolatedX, &x);
172:     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]));
173:     VecRestoreArrayRead(interpolatedX, &x);
174:     VecDestroy(&interpolatedX);

176:     user->next_output += 0.1;
177:   }
178:   return 0;
179: }

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

193:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
194:      Initialize program
195:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
197:   PetscInitialize(&argc, &argv, NULL, help);
198:   MPI_Comm_size(PETSC_COMM_WORLD, &size);

201:   RegisterMyARK2();

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

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

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

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

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

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

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

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

266:   PetscFinalize();
267:   return 0;
268: }

270: /*TEST

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

276: TEST*/