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: /*
7: Concepts: TS^time-dependent nonlinear problems
8: Concepts: TS^van der Pol equation
9: Processors: 1
10: */
11: /* ------------------------------------------------------------------------
13: This program solves the van der Pol equation
14: y'' - \mu ((1-y^2)*y' - y) = 0 (1)
15: on the domain 0 <= x <= 1, with the boundary conditions
16: y(0) = 2, y'(0) = - 2/3 +10/(81*\mu) - 292/(2187*\mu^2),
17: 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.
19: Notes:
20: This code demonstrates the TS solver interface to two variants of
21: linear problems, u_t = f(u,t), namely turning (1) into a system of
22: first order differential equations,
24: [ y' ] = [ z ]
25: [ z' ] [ \mu ((1 - y^2) z - y) ]
27: which then we can write as a vector equation
29: [ u_1' ] = [ u_2 ] (2)
30: [ u_2' ] [ \mu (1 - u_1^2) u_2 - u_1 ]
32: which is now in the desired form of u_t = f(u,t). One way that we
33: can split f(u,t) in (2) is to split by component,
35: [ u_1' ] = [ u_2 ] + [ 0 ]
36: [ u_2' ] [ 0 ] [ \mu ((1 - u_1^2) u_2 - u_1) ]
38: where
40: [ G(u,t) ] = [ u_2 ]
41: [ 0 ]
43: and
45: [ F(u',u,t) ] = [ u_1' ] - [ 0 ]
46: [ u_2' ] [ \mu ((1 - u_1^2) u_2 - u_1) ]
48: Using the definition of the Jacobian of F (from the PETSc user manual),
49: in the equation F(u',u,t) = G(u,t),
51: dF dF
52: J(F) = a * -- - --
53: du' du
55: where d is the partial derivative. In this example,
57: dF [ 1 ; 0 ]
58: -- = [ ]
59: du' [ 0 ; 1 ]
61: dF [ 0 ; 0 ]
62: -- = [ ]
63: du [ -\mu (2*u_1*u_2 + 1); \mu (1 - u_1^2) ]
65: Hence,
67: [ a ; 0 ]
68: J(F) = [ ]
69: [ \mu (2*u_1*u_2 + 1); a - \mu (1 - u_1^2) ]
71: ------------------------------------------------------------------------- */
73: #include <petscts.h>
75: typedef struct _n_User *User;
76: struct _n_User {
77: PetscReal mu;
78: PetscBool imex;
79: PetscReal next_output;
80: };
82: /*
83: User-defined routines
84: */
85: static PetscErrorCode RHSFunction(TS ts,PetscReal t,Vec X,Vec F,void *ctx)
86: {
87: User user = (User)ctx;
88: PetscScalar *f;
89: const PetscScalar *x;
92: VecGetArrayRead(X,&x);
93: VecGetArray(F,&f);
94: f[0] = (user->imex ? x[1] : 0);
95: f[1] = 0.0;
96: VecRestoreArrayRead(X,&x);
97: VecRestoreArray(F,&f);
98: return 0;
99: }
101: static PetscErrorCode IFunction(TS ts,PetscReal t,Vec X,Vec Xdot,Vec F,void *ctx)
102: {
103: User user = (User)ctx;
104: const PetscScalar *x,*xdot;
105: PetscScalar *f;
108: VecGetArrayRead(X,&x);
109: VecGetArrayRead(Xdot,&xdot);
110: VecGetArray(F,&f);
111: f[0] = xdot[0] + (user->imex ? 0 : x[1]);
112: f[1] = xdot[1] - user->mu*((1. - x[0]*x[0])*x[1] - x[0]);
113: VecRestoreArrayRead(X,&x);
114: VecRestoreArrayRead(Xdot,&xdot);
115: VecRestoreArray(F,&f);
116: return 0;
117: }
119: static PetscErrorCode IJacobian(TS ts,PetscReal t,Vec X,Vec Xdot,PetscReal a,Mat A,Mat B,void *ctx)
120: {
121: User user = (User)ctx;
122: PetscReal mu = user->mu;
123: PetscInt rowcol[] = {0,1};
124: const PetscScalar *x;
125: PetscScalar J[2][2];
128: VecGetArrayRead(X,&x);
129: J[0][0] = a; J[0][1] = (user->imex ? 0 : 1.);
130: J[1][0] = mu*(2.*x[0]*x[1]+1.); J[1][1] = a - mu*(1. - x[0]*x[0]);
131: MatSetValues(B,2,rowcol,2,rowcol,&J[0][0],INSERT_VALUES);
132: VecRestoreArrayRead(X,&x);
134: MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
135: MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
136: if (A != B) {
137: MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);
138: MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);
139: }
140: return 0;
141: }
143: static PetscErrorCode RegisterMyARK2(void)
144: {
146: {
147: const PetscReal
148: A[3][3] = {{0,0,0},
149: {0.41421356237309504880,0,0},
150: {0.75,0.25,0}},
151: At[3][3] = {{0,0,0},
152: {0.12132034355964257320,0.29289321881345247560,0},
153: {0.20710678118654752440,0.50000000000000000000,0.29289321881345247560}},
154: *bembedt = NULL,*bembed = NULL;
155: TSARKIMEXRegister("myark2",2,3,&At[0][0],NULL,NULL,&A[0][0],NULL,NULL,bembedt,bembed,0,NULL,NULL);
156: }
157: return 0;
158: }
160: /* Monitor timesteps and use interpolation to output at integer multiples of 0.1 */
161: static PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal t,Vec X,void *ctx)
162: {
163: const PetscScalar *x;
164: PetscReal tfinal, dt;
165: User user = (User)ctx;
166: Vec interpolatedX;
169: TSGetTimeStep(ts,&dt);
170: TSGetMaxTime(ts,&tfinal);
172: while (user->next_output <= t && user->next_output <= tfinal) {
173: VecDuplicate(X,&interpolatedX);
174: TSInterpolate(ts,user->next_output,interpolatedX);
175: VecGetArrayRead(interpolatedX,&x);
176: PetscPrintf(PETSC_COMM_WORLD,"[%.1f] %D TS %.6f (dt = %.6f) X % 12.6e % 12.6e\n",user->next_output,step,t,dt,(double)PetscRealPart(x[0]),(double)PetscRealPart(x[1]));
177: VecRestoreArrayRead(interpolatedX,&x);
178: VecDestroy(&interpolatedX);
180: user->next_output += 0.1;
181: }
182: return 0;
183: }
185: int main(int argc,char **argv)
186: {
187: TS ts; /* nonlinear solver */
188: Vec x; /* solution, residual vectors */
189: Mat A; /* Jacobian matrix */
190: PetscInt steps;
191: PetscReal ftime = 0.5;
192: PetscBool monitor = PETSC_FALSE;
193: PetscScalar *x_ptr;
194: PetscMPIInt size;
195: struct _n_User user;
197: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
198: Initialize program
199: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
200: PetscInitialize(&argc,&argv,NULL,help);
201: MPI_Comm_size(PETSC_COMM_WORLD,&size);
204: RegisterMyARK2();
206: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
207: Set runtime options
208: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
209: user.mu = 1000.0;
210: user.imex = PETSC_TRUE;
211: user.next_output = 0.0;
213: PetscOptionsGetReal(NULL,NULL,"-mu",&user.mu,NULL);
214: PetscOptionsGetBool(NULL,NULL,"-imex",&user.imex,NULL);
215: PetscOptionsGetBool(NULL,NULL,"-monitor",&monitor,NULL);
217: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
218: Create necessary matrix and vectors, solve same ODE on every process
219: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
220: MatCreate(PETSC_COMM_WORLD,&A);
221: MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,2,2);
222: MatSetFromOptions(A);
223: MatSetUp(A);
224: MatCreateVecs(A,&x,NULL);
226: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
227: Create timestepping solver context
228: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
229: TSCreate(PETSC_COMM_WORLD,&ts);
230: TSSetType(ts,TSBEULER);
231: TSSetRHSFunction(ts,NULL,RHSFunction,&user);
232: TSSetIFunction(ts,NULL,IFunction,&user);
233: TSSetIJacobian(ts,A,A,IJacobian,&user);
234: TSSetMaxTime(ts,ftime);
235: TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);
236: if (monitor) {
237: TSMonitorSet(ts,Monitor,&user,NULL);
238: }
240: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
241: Set initial conditions
242: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
243: VecGetArray(x,&x_ptr);
244: x_ptr[0] = 2.0;
245: x_ptr[1] = -2.0/3.0 + 10.0/(81.0*user.mu) - 292.0/(2187.0*user.mu*user.mu);
246: VecRestoreArray(x,&x_ptr);
247: TSSetTimeStep(ts,0.01);
249: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
250: Set runtime options
251: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
252: TSSetFromOptions(ts);
254: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
255: Solve nonlinear system
256: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
257: TSSolve(ts,x);
258: TSGetSolveTime(ts,&ftime);
259: TSGetStepNumber(ts,&steps);
260: PetscPrintf(PETSC_COMM_WORLD,"mu %g, steps %D, ftime %g\n",(double)user.mu,steps,(double)ftime);
261: VecView(x,PETSC_VIEWER_STDOUT_WORLD);
263: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
264: Free work space. All PETSc objects should be destroyed when they
265: are no longer needed.
266: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
267: MatDestroy(&A);
268: VecDestroy(&x);
269: TSDestroy(&ts);
271: PetscFinalize();
272: return 0;
273: }
275: /*TEST
277: test:
278: args: -ts_type arkimex -ts_arkimex_type myark2 -ts_adapt_type none
279: requires: !single
281: TEST*/