Actual source code: ex19.c

```
2: static char help[] = "Solves the van der Pol DAE.\n\
3: Input parameters include:\n";

5: /*
6:    Concepts: TS^time-dependent nonlinear problems
7:    Concepts: TS^van der Pol DAE
8:    Processors: 1
9: */
10: /* ------------------------------------------------------------------------

12:    This program solves the van der Pol DAE
13:        y' = -z = f(y,z)        (1)
14:        0  = y-(z^3/3 - z) = g(y,z)
15:    on the domain 0 <= x <= 1, with the boundary conditions
16:        y(0) = -2, y'(0) = -2.355301397608119909925287735864250951918
17:    This is a nonlinear equation.

19:    Notes:
20:    This code demonstrates the TS solver interface with the Van der Pol DAE,
21:    namely it is the case when there is no RHS (meaning the RHS == 0), and the
22:    equations are converted to two variants of linear problems, u_t = f(u,t),
23:    namely turning (1) into a vector equation in terms of u,

25:    [     y' + z      ] = [ 0 ]
26:    [ (z^3/3 - z) - y ]   [ 0 ]

28:    which then we can write as a vector equation

30:    [      u_1' + u_2       ] = [ 0 ]  (2)
31:    [ (u_2^3/3 - u_2) - u_1 ]   [ 0 ]

33:    which is now in the desired form of u_t = f(u,t). As this is a DAE, and
34:    there is no u_2', there is no need for a split,

36:    so

38:    [ F(u',u,t) ] = [ u_1' ] + [         u_2           ]
39:                    [  0   ]   [ (u_2^3/3 - u_2) - u_1 ]

41:    Using the definition of the Jacobian of F (from the PETSc user manual),
42:    in the equation F(u',u,t) = G(u,t),

44:               dF   dF
45:    J(F) = a * -- - --
46:               du'  du

48:    where d is the partial derivative. In this example,

50:    dF   [ 1 ; 0 ]
51:    -- = [       ]
52:    du'  [ 0 ; 0 ]

54:    dF   [  0 ;      1     ]
55:    -- = [                 ]
56:    du   [ -1 ; 1 - u_2^2  ]

58:    Hence,

60:           [ a ;    -1     ]
61:    J(F) = [               ]
62:           [ 1 ; u_2^2 - 1 ]

64:   ------------------------------------------------------------------------- */

66: #include <petscts.h>

68: typedef struct _n_User *User;
69: struct _n_User {
70:   PetscReal next_output;
71: };

73: /*
74: *  User-defined routines
75: */

77: static PetscErrorCode IFunction(TS ts,PetscReal t,Vec X,Vec Xdot,Vec F,void *ctx)
78: {
79:   PetscErrorCode    ierr;
80:   PetscScalar       *f;
81:   const PetscScalar *x,*xdot;

86:   VecGetArray(F,&f);
87:   f[0] = xdot[0] + x[1];
88:   f[1] = (x[1]*x[1]*x[1]/3.0 - x[1])-x[0];
91:   VecRestoreArray(F,&f);
92:   return(0);
93: }

95: static PetscErrorCode IJacobian(TS ts,PetscReal t,Vec X,Vec Xdot,PetscReal a,Mat A,Mat B,void *ctx)
96: {
97:   PetscErrorCode    ierr;
98:   PetscInt          rowcol[] = {0,1};
99:   PetscScalar       J[2][2];
100:   const PetscScalar *x;

104:   J[0][0] = a;    J[0][1] = -1.;
105:   J[1][0] = 1.;   J[1][1] = -1. + x[1]*x[1];
106:   MatSetValues(B,2,rowcol,2,rowcol,&J[0][0],INSERT_VALUES);

109:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
110:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
111:   if (A != B) {
112:     MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);
113:     MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);
114:   }
115:   return(0);
116: }

118: static PetscErrorCode RegisterMyARK2(void)
119: {

123:   {
124:     const PetscReal
125:       A[3][3] = {{0,0,0},
126:                  {0.41421356237309504880,0,0},
127:                  {0.75,0.25,0}},
128:       At[3][3] = {{0,0,0},
129:                   {0.12132034355964257320,0.29289321881345247560,0},
130:                   {0.20710678118654752440,0.50000000000000000000,0.29289321881345247560}},
131:     *bembedt = NULL,*bembed = NULL;
132:     TSARKIMEXRegister("myark2",2,3,&At[0][0],NULL,NULL,&A[0][0],NULL,NULL,bembedt,bembed,0,NULL,NULL);
133:   }
134:   return(0);
135: }

137: /* Monitor timesteps and use interpolation to output at integer multiples of 0.1 */
138: static PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal t,Vec X,void *ctx)
139: {
140:   PetscErrorCode    ierr;
141:   const PetscScalar *x;
142:   PetscReal         tfinal, dt;
143:   User              user = (User)ctx;
144:   Vec               interpolatedX;

147:   TSGetTimeStep(ts,&dt);
148:   TSGetMaxTime(ts,&tfinal);

150:   while (user->next_output <= t && user->next_output <= tfinal) {
151:     VecDuplicate(X,&interpolatedX);
152:     TSInterpolate(ts,user->next_output,interpolatedX);
154:     PetscPrintf(PETSC_COMM_WORLD,"[%.1f] %3D 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]));
156:     VecDestroy(&interpolatedX);
157:     user->next_output += PetscRealConstant(0.1);
158:   }
159:   return(0);
160: }

162: int main(int argc,char **argv)
163: {
164:   TS             ts;            /* nonlinear solver */
165:   Vec            x;             /* solution, residual vectors */
166:   Mat            A;             /* Jacobian matrix */
167:   PetscInt       steps;
168:   PetscReal      ftime   = 0.5;
169:   PetscBool      monitor = PETSC_FALSE;
170:   PetscScalar    *x_ptr;
171:   PetscMPIInt    size;
172:   struct _n_User user;

175:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
176:      Initialize program
177:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
178:   PetscInitialize(&argc,&argv,NULL,help);if (ierr) return ierr;
179:   MPI_Comm_size(PETSC_COMM_WORLD,&size);
180:   if (size != 1) SETERRQ(PETSC_COMM_WORLD,PETSC_ERR_WRONG_MPI_SIZE,"This is a uniprocessor example only!");

182:   RegisterMyARK2();

184:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
185:     Set runtime options
186:     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

188:   user.next_output = 0.0;
189:   PetscOptionsGetBool(NULL,NULL,"-monitor",&monitor,NULL);

191:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
192:     Create necessary matrix and vectors, solve same ODE on every process
193:     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
194:   MatCreate(PETSC_COMM_WORLD,&A);
195:   MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,2,2);
196:   MatSetFromOptions(A);
197:   MatSetUp(A);
198:   MatCreateVecs(A,&x,NULL);

200:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
201:      Create timestepping solver context
202:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
203:   TSCreate(PETSC_COMM_WORLD,&ts);
204:   TSSetType(ts,TSBEULER);
205:   TSSetIFunction(ts,NULL,IFunction,&user);
206:   TSSetIJacobian(ts,A,A,IJacobian,&user);
207:   TSSetMaxTime(ts,ftime);
208:   TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);
209:   if (monitor) {
210:     TSMonitorSet(ts,Monitor,&user,NULL);
211:   }

213:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
214:      Set initial conditions
215:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
216:   VecGetArray(x,&x_ptr);
217:   x_ptr[0] = -2;   x_ptr[1] = -2.355301397608119909925287735864250951918;
218:   VecRestoreArray(x,&x_ptr);
219:   TSSetTimeStep(ts,.001);

221:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
222:      Set runtime options
223:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
224:   TSSetFromOptions(ts);

226:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
227:      Solve nonlinear system
228:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
229:   TSSolve(ts,x);
230:   TSGetSolveTime(ts,&ftime);
231:   TSGetStepNumber(ts,&steps);
232:   PetscPrintf(PETSC_COMM_WORLD,"steps %3D, ftime %g\n",steps,(double)ftime);
233:   VecView(x,PETSC_VIEWER_STDOUT_WORLD);

235:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
236:      Free work space.  All PETSc objects should be destroyed when they
237:      are no longer needed.
238:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
239:   MatDestroy(&A);
240:   VecDestroy(&x);
241:   TSDestroy(&ts);

243:   PetscFinalize();
244:   return ierr;
245: }

247: /*TEST

249:    test:
250:       requires: !single
251:       suffix: a
252:       args: -monitor -ts_type bdf -ts_rtol 1e-6 -ts_atol 1e-6 -ts_view -ts_adapt_type dsp
253:       output_file: output/ex19_pi42.out

255:    test:
256:       requires: !single
257:       suffix: b
258:       args: -monitor -ts_type bdf -ts_rtol 1e-6 -ts_atol 1e-6 -ts_view -ts_adapt_type dsp -ts_adapt_dsp_filter PI42
259:       output_file: output/ex19_pi42.out

261:    test:
262:       requires: !single
263:       suffix: c
264:       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
265:       output_file: output/ex19_pi42.out

267:    test:
268:       requires: !single
269:       suffix: bdf_reject
270:       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

272: TEST*/
```