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; J[0][1] = (user->imex ? 0 : 1.);
125: J[1][0] = mu*(2.*x[0]*x[1]+1.); J[1][1] = a - mu*(1. - x[0]*x[0]);
126: MatSetValues(B,2,rowcol,2,rowcol,&J[0][0],INSERT_VALUES);
127: VecRestoreArrayRead(X,&x);
129: MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
130: MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
131: if (A != B) {
132: MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);
133: MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);
134: }
135: return 0;
136: }
138: static PetscErrorCode RegisterMyARK2(void)
139: {
141: {
142: const PetscReal
143: A[3][3] = {{0,0,0},
144: {0.41421356237309504880,0,0},
145: {0.75,0.25,0}},
146: At[3][3] = {{0,0,0},
147: {0.12132034355964257320,0.29289321881345247560,0},
148: {0.20710678118654752440,0.50000000000000000000,0.29289321881345247560}},
149: *bembedt = NULL,*bembed = NULL;
150: TSARKIMEXRegister("myark2",2,3,&At[0][0],NULL,NULL,&A[0][0],NULL,NULL,bembedt,bembed,0,NULL,NULL);
151: }
152: return 0;
153: }
155: /* Monitor timesteps and use interpolation to output at integer multiples of 0.1 */
156: static PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal t,Vec X,void *ctx)
157: {
158: const PetscScalar *x;
159: PetscReal tfinal, dt;
160: User user = (User)ctx;
161: Vec interpolatedX;
164: TSGetTimeStep(ts,&dt);
165: TSGetMaxTime(ts,&tfinal);
167: while (user->next_output <= t && user->next_output <= tfinal) {
168: VecDuplicate(X,&interpolatedX);
169: TSInterpolate(ts,user->next_output,interpolatedX);
170: VecGetArrayRead(interpolatedX,&x);
171: 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]));
172: VecRestoreArrayRead(interpolatedX,&x);
173: VecDestroy(&interpolatedX);
175: user->next_output += 0.1;
176: }
177: return 0;
178: }
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;
192: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
193: Initialize program
194: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
195: PetscInitialize(&argc,&argv,NULL,help);
196: MPI_Comm_size(PETSC_COMM_WORLD,&size);
199: RegisterMyARK2();
201: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
202: Set runtime options
203: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
204: user.mu = 1000.0;
205: user.imex = PETSC_TRUE;
206: user.next_output = 0.0;
208: PetscOptionsGetReal(NULL,NULL,"-mu",&user.mu,NULL);
209: PetscOptionsGetBool(NULL,NULL,"-imex",&user.imex,NULL);
210: PetscOptionsGetBool(NULL,NULL,"-monitor",&monitor,NULL);
212: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
213: Create necessary matrix and vectors, solve same ODE on every process
214: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
215: MatCreate(PETSC_COMM_WORLD,&A);
216: MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,2,2);
217: MatSetFromOptions(A);
218: MatSetUp(A);
219: MatCreateVecs(A,&x,NULL);
221: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
222: Create timestepping solver context
223: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
224: TSCreate(PETSC_COMM_WORLD,&ts);
225: TSSetType(ts,TSBEULER);
226: TSSetRHSFunction(ts,NULL,RHSFunction,&user);
227: TSSetIFunction(ts,NULL,IFunction,&user);
228: TSSetIJacobian(ts,A,A,IJacobian,&user);
229: TSSetMaxTime(ts,ftime);
230: TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);
231: if (monitor) {
232: TSMonitorSet(ts,Monitor,&user,NULL);
233: }
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 %D, 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*/