Actual source code: ex19.c
2: static char help[] = "Solves the van der Pol DAE.\n\
3: Input parameters include:\n";
5: /* ------------------------------------------------------------------------
7: This program solves the van der Pol DAE
8: y' = -z = f(y,z) (1)
9: 0 = y-(z^3/3 - z) = g(y,z)
10: on the domain 0 <= x <= 1, with the boundary conditions
11: y(0) = -2, y'(0) = -2.355301397608119909925287735864250951918
12: This is a nonlinear equation.
14: Notes:
15: This code demonstrates the TS solver interface with the Van der Pol DAE,
16: namely it is the case when there is no RHS (meaning the RHS == 0), and the
17: equations are converted to two variants of linear problems, u_t = f(u,t),
18: namely turning (1) into a vector equation in terms of u,
20: [ y' + z ] = [ 0 ]
21: [ (z^3/3 - z) - y ] [ 0 ]
23: which then we can write as a vector equation
25: [ u_1' + u_2 ] = [ 0 ] (2)
26: [ (u_2^3/3 - u_2) - u_1 ] [ 0 ]
28: which is now in the desired form of u_t = f(u,t). As this is a DAE, and
29: there is no u_2', there is no need for a split,
31: so
33: [ F(u',u,t) ] = [ u_1' ] + [ u_2 ]
34: [ 0 ] [ (u_2^3/3 - u_2) - u_1 ]
36: Using the definition of the Jacobian of F (from the PETSc user manual),
37: in the equation F(u',u,t) = G(u,t),
39: dF dF
40: J(F) = a * -- - --
41: du' du
43: where d is the partial derivative. In this example,
45: dF [ 1 ; 0 ]
46: -- = [ ]
47: du' [ 0 ; 0 ]
49: dF [ 0 ; 1 ]
50: -- = [ ]
51: du [ -1 ; 1 - u_2^2 ]
53: Hence,
55: [ a ; -1 ]
56: J(F) = [ ]
57: [ 1 ; u_2^2 - 1 ]
59: ------------------------------------------------------------------------- */
61: #include <petscts.h>
63: typedef struct _n_User *User;
64: struct _n_User {
65: PetscReal next_output;
66: };
68: /*
69: User-defined routines
70: */
72: static PetscErrorCode IFunction(TS ts, PetscReal t, Vec X, Vec Xdot, Vec F, void *ctx)
73: {
74: PetscScalar *f;
75: const PetscScalar *x, *xdot;
77: PetscFunctionBeginUser;
78: PetscCall(VecGetArrayRead(X, &x));
79: PetscCall(VecGetArrayRead(Xdot, &xdot));
80: PetscCall(VecGetArray(F, &f));
81: f[0] = xdot[0] + x[1];
82: f[1] = (x[1] * x[1] * x[1] / 3.0 - x[1]) - x[0];
83: PetscCall(VecRestoreArrayRead(X, &x));
84: PetscCall(VecRestoreArrayRead(Xdot, &xdot));
85: PetscCall(VecRestoreArray(F, &f));
86: PetscFunctionReturn(PETSC_SUCCESS);
87: }
89: static PetscErrorCode IJacobian(TS ts, PetscReal t, Vec X, Vec Xdot, PetscReal a, Mat A, Mat B, void *ctx)
90: {
91: PetscInt rowcol[] = {0, 1};
92: PetscScalar J[2][2];
93: const PetscScalar *x;
95: PetscFunctionBeginUser;
96: PetscCall(VecGetArrayRead(X, &x));
97: J[0][0] = a;
98: J[0][1] = -1.;
99: J[1][0] = 1.;
100: J[1][1] = -1. + x[1] * x[1];
101: PetscCall(MatSetValues(B, 2, rowcol, 2, rowcol, &J[0][0], INSERT_VALUES));
102: PetscCall(VecRestoreArrayRead(X, &x));
104: PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
105: PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
106: if (A != B) {
107: PetscCall(MatAssemblyBegin(B, MAT_FINAL_ASSEMBLY));
108: PetscCall(MatAssemblyEnd(B, MAT_FINAL_ASSEMBLY));
109: }
110: PetscFunctionReturn(PETSC_SUCCESS);
111: }
113: static PetscErrorCode RegisterMyARK2(void)
114: {
115: PetscFunctionBeginUser;
116: {
117: const PetscReal A[3][3] =
118: {
119: {0, 0, 0},
120: {0.41421356237309504880, 0, 0},
121: {0.75, 0.25, 0}
122: },
123: At[3][3] = {{0, 0, 0}, {0.12132034355964257320, 0.29289321881345247560, 0}, {0.20710678118654752440, 0.50000000000000000000, 0.29289321881345247560}}, *bembedt = NULL, *bembed = NULL;
124: PetscCall(TSARKIMEXRegister("myark2", 2, 3, &At[0][0], NULL, NULL, &A[0][0], NULL, NULL, bembedt, bembed, 0, NULL, NULL));
125: }
126: PetscFunctionReturn(PETSC_SUCCESS);
127: }
129: /* Monitor timesteps and use interpolation to output at integer multiples of 0.1 */
130: static PetscErrorCode Monitor(TS ts, PetscInt step, PetscReal t, Vec X, void *ctx)
131: {
132: const PetscScalar *x;
133: PetscReal tfinal, dt;
134: User user = (User)ctx;
135: Vec interpolatedX;
137: PetscFunctionBeginUser;
138: PetscCall(TSGetTimeStep(ts, &dt));
139: PetscCall(TSGetMaxTime(ts, &tfinal));
141: while (user->next_output <= t && user->next_output <= tfinal) {
142: PetscCall(VecDuplicate(X, &interpolatedX));
143: PetscCall(TSInterpolate(ts, user->next_output, interpolatedX));
144: PetscCall(VecGetArrayRead(interpolatedX, &x));
145: 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(VecRestoreArrayRead(interpolatedX, &x));
147: PetscCall(VecDestroy(&interpolatedX));
148: user->next_output += PetscRealConstant(0.1);
149: }
150: PetscFunctionReturn(PETSC_SUCCESS);
151: }
153: int main(int argc, char **argv)
154: {
155: TS ts; /* nonlinear solver */
156: Vec x; /* solution, residual vectors */
157: Mat A; /* Jacobian matrix */
158: PetscInt steps;
159: PetscReal ftime = 0.5;
160: PetscBool monitor = PETSC_FALSE;
161: PetscScalar *x_ptr;
162: PetscMPIInt size;
163: struct _n_User user;
165: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
166: Initialize program
167: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
168: PetscFunctionBeginUser;
169: PetscCall(PetscInitialize(&argc, &argv, NULL, help));
170: PetscCallMPI(MPI_Comm_size(PETSC_COMM_WORLD, &size));
171: PetscCheck(size == 1, PETSC_COMM_WORLD, PETSC_ERR_WRONG_MPI_SIZE, "This is a uniprocessor example only!");
173: PetscCall(RegisterMyARK2());
175: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
176: Set runtime options
177: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
179: user.next_output = 0.0;
180: PetscCall(PetscOptionsGetBool(NULL, NULL, "-monitor", &monitor, NULL));
182: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
183: Create necessary matrix and vectors, solve same ODE on every process
184: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
185: PetscCall(MatCreate(PETSC_COMM_WORLD, &A));
186: PetscCall(MatSetSizes(A, PETSC_DECIDE, PETSC_DECIDE, 2, 2));
187: PetscCall(MatSetFromOptions(A));
188: PetscCall(MatSetUp(A));
189: PetscCall(MatCreateVecs(A, &x, NULL));
191: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
192: Create timestepping solver context
193: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
194: PetscCall(TSCreate(PETSC_COMM_WORLD, &ts));
195: PetscCall(TSSetType(ts, TSBEULER));
196: PetscCall(TSSetIFunction(ts, NULL, IFunction, &user));
197: PetscCall(TSSetIJacobian(ts, A, A, IJacobian, &user));
198: PetscCall(TSSetMaxTime(ts, ftime));
199: PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER));
200: if (monitor) PetscCall(TSMonitorSet(ts, Monitor, &user, NULL));
202: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
203: Set initial conditions
204: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
205: PetscCall(VecGetArray(x, &x_ptr));
206: x_ptr[0] = -2;
207: x_ptr[1] = -2.355301397608119909925287735864250951918;
208: PetscCall(VecRestoreArray(x, &x_ptr));
209: PetscCall(TSSetTimeStep(ts, .001));
211: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
212: Set runtime options
213: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
214: PetscCall(TSSetFromOptions(ts));
216: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
217: Solve nonlinear system
218: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
219: PetscCall(TSSolve(ts, x));
220: PetscCall(TSGetSolveTime(ts, &ftime));
221: PetscCall(TSGetStepNumber(ts, &steps));
222: PetscCall(PetscPrintf(PETSC_COMM_WORLD, "steps %3" PetscInt_FMT ", ftime %g\n", steps, (double)ftime));
223: PetscCall(VecView(x, PETSC_VIEWER_STDOUT_WORLD));
225: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
226: Free work space. All PETSc objects should be destroyed when they
227: are no longer needed.
228: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
229: PetscCall(MatDestroy(&A));
230: PetscCall(VecDestroy(&x));
231: PetscCall(TSDestroy(&ts));
233: PetscCall(PetscFinalize());
234: return 0;
235: }
237: /*TEST
239: test:
240: requires: !single
241: suffix: a
242: args: -monitor -ts_type bdf -ts_rtol 1e-6 -ts_atol 1e-6 -ts_view -ts_adapt_type dsp
243: output_file: output/ex19_pi42.out
245: test:
246: requires: !single
247: suffix: b
248: args: -monitor -ts_type bdf -ts_rtol 1e-6 -ts_atol 1e-6 -ts_view -ts_adapt_type dsp -ts_adapt_dsp_filter PI42
249: output_file: output/ex19_pi42.out
251: test:
252: requires: !single
253: suffix: c
254: 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
255: output_file: output/ex19_pi42.out
257: test:
258: requires: !single
259: suffix: bdf_reject
260: 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
262: TEST*/