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;
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];
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;
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));

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));
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])));
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*/
```