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";
4
5 /* ------------------------------------------------------------------------
6
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.
12
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,
17
18 [ y' ] = [ z ]
19 [ z' ] [ \mu ((1 - y^2) z - y) ]
20
21 which then we can write as a vector equation
22
23 [ u_1' ] = [ u_2 ] (2)
24 [ u_2' ] [ \mu (1 - u_1^2) u_2 - u_1 ]
25
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,
28
29 [ u_1' ] = [ u_2 ] + [ 0 ]
30 [ u_2' ] [ 0 ] [ \mu ((1 - u_1^2) u_2 - u_1) ]
31
32 where
33
34 [ G(u,t) ] = [ u_2 ]
35 [ 0 ]
36
37 and
38
39 [ F(u',u,t) ] = [ u_1' ] - [ 0 ]
40 [ u_2' ] [ \mu ((1 - u_1^2) u_2 - u_1) ]
41
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),
44
45 dF dF
46 J(F) = a * -- - --
47 du' du
48
49 where d is the partial derivative. In this example,
50
51 dF [ 1 ; 0 ]
52 -- = [ ]
53 du' [ 0 ; 1 ]
54
55 dF [ 0 ; 0 ]
56 -- = [ ]
57 du [ -\mu (2*u_1*u_2 + 1); \mu (1 - u_1^2) ]
58
59 Hence,
60
61 [ a ; 0 ]
62 J(F) = [ ]
63 [ \mu (2*u_1*u_2 + 1); a - \mu (1 - u_1^2) ]
64
65 ------------------------------------------------------------------------- */
66
67 #include <petscts.h>
68
69 typedef struct _n_User *User;
70 struct _n_User {
71 PetscReal mu;
72 PetscBool imex;
73 PetscReal next_output;
74 };
75
76 /*
77 User-defined routines
78 */
RHSFunction(TS ts,PetscReal t,Vec X,Vec F,PetscCtx ctx)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;
84
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 }
94
IFunction(TS ts,PetscReal t,Vec X,Vec Xdot,Vec F,PetscCtx ctx)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;
100
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 }
112
IJacobian(TS ts,PetscReal t,Vec X,Vec Xdot,PetscReal a,Mat A,Mat B,PetscCtx ctx)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];
120
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));
129
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 }
138
RegisterMyARK2(void)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 }
154
155 /* Monitor timesteps and use interpolation to output at integer multiples of 0.1 */
Monitor(TS ts,PetscInt step,PetscReal t,Vec X,PetscCtx ctx)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;
162
163 PetscFunctionBeginUser;
164 PetscCall(TSGetTimeStep(ts, &dt));
165 PetscCall(TSGetMaxTime(ts, &tfinal));
166
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));
174
175 user->next_output += 0.1;
176 }
177 PetscFunctionReturn(PETSC_SUCCESS);
178 }
179
main(int argc,char ** argv)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;
191
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!");
199
200 PetscCall(RegisterMyARK2());
201
202 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
203 Set runtime options
204 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
205 user.mu = 1000.0;
206 user.imex = PETSC_TRUE;
207 user.next_output = 0.0;
208
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));
212
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));
221
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));
233
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));
242
243 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
244 Set runtime options
245 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
246 PetscCall(TSSetFromOptions(ts));
247
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));
256
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));
264
265 PetscCall(PetscFinalize());
266 return 0;
267 }
268
269 /*TEST
270
271 test:
272 args: -ts_type arkimex -ts_arkimex_type myark2 -ts_adapt_type none
273 requires: !single
274
275 TEST*/
276