Actual source code: ex20.c
petsc-3.7.5 2017-01-01
2: static char help[] = "Solves the van der Pol equation.\n\
3: Input parameters include:\n";
5: /*
6: Concepts: TS^time-dependent nonlinear problems
7: Concepts: TS^van der Pol equation DAE equivalent
8: Processors: 1
9: */
10: /* ------------------------------------------------------------------------
12: This program solves the van der Pol DAE ODE equivalent
13: y' = z (1)
14: z' = mu[(1-y^2)z-y]
15: on the domain 0 <= x <= 1, with the boundary conditions
16: y(0) = 2, y'(0) = -6.666665432100101e-01,
17: and
18: mu = 10^6.
19: This is a nonlinear equation.
21: Notes:
22: This code demonstrates the TS solver interface to a variant of
23: linear problems, u_t = f(u,t), namely turning (1) into a system of
24: first order differential equations,
26: [ y' ] = [ z ]
27: [ z' ] [ mu[(1-y^2)z-y] ]
29: which then we can write as a vector equation
31: [ u_1' ] = [ u_2 ] (2)
32: [ u_2' ] [ mu[(1-u_1^2)u_2-u_1] ]
34: which is now in the desired form of u_t = f(u,t). One way that we
35: can split f(u,t) in (2) is to split by component,
37: [ u_1' ] = [ u_2 ] + [ 0 ]
38: [ u_2' ] [ 0 ] [ mu[(1-u_1^2)u_2-u_1] ]
40: where
42: [ F(u,t) ] = [ u_2 ]
43: [ 0 ]
45: and
47: [ G(u',u,t) ] = [ u_1' ] - [ 0 ]
48: [ u_2' ] [ mu[(1-u_1^2)u_2-u_1] ]
50: Using the definition of the Jacobian of G (from the PETSc user manual),
51: in the equation G(u',u,t) = F(u,t),
53: dG dG
54: J(G) = a * -- - --
55: du' du
57: where d is the partial derivative. In this example,
59: dG [ 1 ; 0 ]
60: -- = [ ]
61: du' [ 0 ; 1 ]
63: dG [ 0 ; 0 ]
64: -- = [ ]
65: du [ -mu*(1.0 + 2.0*u_1*u_2) ; mu*(1-u_1*u_1) ]
67: Hence,
69: [ a ; 0 ]
70: J(G) = [ ]
71: [ mu*(1.0 + 2.0*u_1*u_2) ; a - mu*(1-u_1*u_1) ]
73: ------------------------------------------------------------------------- */
75: #include <petscts.h>
77: typedef struct _n_User *User;
78: struct _n_User {
79: PetscReal mu;
80: PetscBool imex;
81: PetscReal next_output;
82: };
84: /*
85: * User-defined routines
86: */
89: static PetscErrorCode RHSFunction(TS ts,PetscReal t,Vec X,Vec F,void *ctx)
90: {
91: PetscErrorCode ierr;
92: User user = (User)ctx;
93: PetscScalar *f;
94: const PetscScalar *x;
97: VecGetArrayRead(X,&x);
98: VecGetArray(F,&f);
99: f[0] = (user->imex ? x[1] : 0.0);
100: f[1] = 0.0;
101: VecRestoreArrayRead(X,&x);
102: VecRestoreArray(F,&f);
103: return(0);
104: }
108: static PetscErrorCode IFunction(TS ts,PetscReal t,Vec X,Vec Xdot,Vec F,void *ctx)
109: {
110: PetscErrorCode ierr;
111: User user = (User)ctx;
112: const PetscScalar *x,*xdot;
113: PetscScalar *f;
116: VecGetArrayRead(X,&x);
117: VecGetArrayRead(Xdot,&xdot);
118: VecGetArray(F,&f);
119: f[0] = xdot[0] - (user->imex ? 0 : x[1]);
120: f[1] = xdot[1] - user->mu*((1.0-x[0]*x[0])*x[1] - x[0]);
121: VecRestoreArrayRead(X,&x);
122: VecRestoreArrayRead(Xdot,&xdot);
123: VecRestoreArray(F,&f);
124: return(0);
125: }
129: static PetscErrorCode IJacobian(TS ts,PetscReal t,Vec X,Vec Xdot,PetscReal a,Mat A,Mat B,void *ctx)
130: {
131: PetscErrorCode ierr;
132: User user = (User)ctx;
133: PetscInt rowcol[] = {0,1};
134: const PetscScalar *x;
135: PetscScalar J[2][2];
138: VecGetArrayRead(X,&x);
139: J[0][0] = a; J[0][1] = (user->imex ? 0 : -1.0);
140: J[1][0] = user->mu*(1.0 + 2.0*x[0]*x[1]); J[1][1] = a - user->mu*(1.0-x[0]*x[0]);
141: MatSetValues(B,2,rowcol,2,rowcol,&J[0][0],INSERT_VALUES);
142: VecRestoreArrayRead(X,&x);
144: MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
145: MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
146: if (A != B) {
147: MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);
148: MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);
149: }
150: return(0);
151: }
155: /* This is an example of registering an user-provided ARKIMEX scheme */
156: static PetscErrorCode RegisterMyARK2(void)
157: {
161: {
162: const PetscReal
163: A[3][3] = {{0,0,0},
164: {0.41421356237309504880,0,0},
165: {0.75,0.25,0}},
166: At[3][3] = {{0,0,0},
167: {0.12132034355964257320,0.29289321881345247560,0},
168: {0.20710678118654752440,0.50000000000000000000,0.29289321881345247560}};
169: TSARKIMEXRegister("myark2",2,3,&At[0][0],NULL,NULL,&A[0][0],NULL,NULL,NULL,NULL,0,NULL,NULL);
170: }
171: return(0);
172: }
176: /* Monitor timesteps and use interpolation to output at integer multiples of 0.1 */
177: static PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal t,Vec X,void *ctx)
178: {
179: PetscErrorCode ierr;
180: const PetscScalar *x;
181: PetscReal tfinal, dt;
182: User user = (User)ctx;
183: Vec interpolatedX;
186: TSGetTimeStep(ts,&dt);
187: TSGetDuration(ts,NULL,&tfinal);
189: while (user->next_output <= t && user->next_output <= tfinal) {
190: VecDuplicate(X,&interpolatedX);
191: TSInterpolate(ts,user->next_output,interpolatedX);
192: VecGetArrayRead(interpolatedX,&x);
193: PetscPrintf(PETSC_COMM_WORLD,"[%.1f] %D TS %.6f (dt = %.6f) X % 12.6e % 12.6e\n",
194: user->next_output,step,t,dt,(double)PetscRealPart(x[0]),
195: (double)PetscRealPart(x[1]));
196: VecRestoreArrayRead(interpolatedX,&x);
197: VecDestroy(&interpolatedX);
198: user->next_output += 0.1;
199: }
200: return(0);
201: }
205: int main(int argc,char **argv)
206: {
207: TS ts; /* nonlinear solver */
208: Vec x; /* solution, residual vectors */
209: Mat A; /* Jacobian matrix */
210: PetscInt steps;
211: PetscReal ftime = 0.5;
212: PetscBool monitor = PETSC_FALSE;
213: PetscScalar *x_ptr;
214: PetscMPIInt size;
215: struct _n_User user;
218: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
219: Initialize program
220: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
221: PetscInitialize(&argc,&argv,NULL,help);
223: MPI_Comm_size(PETSC_COMM_WORLD,&size);
224: if (size != 1) SETERRQ(PETSC_COMM_SELF,1,"This is a uniprocessor example only!");
226: /* Register user-specified ARKIMEX method */
227: RegisterMyARK2();
229: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
230: Set runtime options
231: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
232: user.imex = PETSC_TRUE;
233: user.next_output = 0.0;
234: user.mu = 1.0e6;
235: PetscOptionsGetBool(NULL,NULL,"-imex",&user.imex,NULL);
236: PetscOptionsGetBool(NULL,NULL,"-monitor",&monitor,NULL);
237: PetscOptionsBegin(PETSC_COMM_WORLD,NULL,"Physical parameters",NULL);
238: PetscOptionsReal("-mu","Stiffness parameter","<1.0e6>",user.mu,&user.mu,PETSC_NULL);
239: PetscOptionsEnd();
241: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
242: Create necessary matrix and vectors, solve same ODE on every process
243: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
244: MatCreate(PETSC_COMM_WORLD,&A);
245: MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,2,2);
246: MatSetFromOptions(A);
247: MatSetUp(A);
249: MatCreateVecs(A,&x,NULL);
251: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
252: Create timestepping solver context
253: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
254: TSCreate(PETSC_COMM_WORLD,&ts);
255: TSSetType(ts,TSBEULER);
256: TSSetRHSFunction(ts,NULL,RHSFunction,&user);
257: TSSetIFunction(ts,NULL,IFunction,&user);
258: TSSetIJacobian(ts,A,A,IJacobian,&user);
260: TSSetDuration(ts,PETSC_DEFAULT,ftime);
261: TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);
262: if (monitor) {
263: TSMonitorSet(ts,Monitor,&user,NULL);
264: }
266: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
267: Set initial conditions
268: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
269: VecGetArray(x,&x_ptr);
270: x_ptr[0] = 2.0; x_ptr[1] = -6.666665432100101e-01;
271: VecRestoreArray(x,&x_ptr);
272: TSSetInitialTimeStep(ts,0.0,.001);
274: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
275: Set runtime options
276: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
277: TSSetFromOptions(ts);
279: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
280: Solve nonlinear system
281: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
282: TSSolve(ts,x);
283: TSGetSolveTime(ts,&ftime);
284: TSGetTimeStepNumber(ts,&steps);
285: PetscPrintf(PETSC_COMM_WORLD,"steps %D, ftime %g\n",steps,(double)ftime);
286: VecView(x,PETSC_VIEWER_STDOUT_WORLD);
288: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
289: Free work space. All PETSc objects should be destroyed when they
290: are no longer needed.
291: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
292: MatDestroy(&A);
293: VecDestroy(&x);
294: TSDestroy(&ts);
296: PetscFinalize();
297: return(0);
298: }