Actual source code: ex20.c

petsc-3.11.3 2019-06-26
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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: */
87: static PetscErrorCode RHSFunction(TS ts,PetscReal t,Vec X,Vec F,void *ctx)
88: {
89:   PetscErrorCode    ierr;
90:   User              user = (User)ctx;
91:   PetscScalar       *f;
92:   const PetscScalar *x;

95:   VecGetArrayRead(X,&x);
96:   VecGetArray(F,&f);
97:   f[0] = (user->imex ? x[1] : 0.0);
98:   f[1] = 0.0;
99:   VecRestoreArrayRead(X,&x);
100:   VecRestoreArray(F,&f);
101:   return(0);
102: }

104: static PetscErrorCode IFunction(TS ts,PetscReal t,Vec X,Vec Xdot,Vec F,void *ctx)
105: {
106:   PetscErrorCode    ierr;
107:   User              user = (User)ctx;
108:   const PetscScalar *x,*xdot;
109:   PetscScalar       *f;

112:   VecGetArrayRead(X,&x);
113:   VecGetArrayRead(Xdot,&xdot);
114:   VecGetArray(F,&f);
115:   f[0] = xdot[0] - (user->imex ? 0 : x[1]);
116:   f[1] = xdot[1] - user->mu*((1.0-x[0]*x[0])*x[1] - x[0]);
117:   VecRestoreArrayRead(X,&x);
118:   VecRestoreArrayRead(Xdot,&xdot);
119:   VecRestoreArray(F,&f);
120:   return(0);
121: }

123: static PetscErrorCode IJacobian(TS ts,PetscReal t,Vec X,Vec Xdot,PetscReal a,Mat A,Mat B,void *ctx)
124: {
125:   PetscErrorCode    ierr;
126:   User              user     = (User)ctx;
127:   PetscInt          rowcol[] = {0,1};
128:   const PetscScalar *x;
129:   PetscScalar       J[2][2];

132:   VecGetArrayRead(X,&x);
133:   J[0][0] = a;     J[0][1] = (user->imex ? 0 : -1.0);
134:   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]);
135:   MatSetValues(B,2,rowcol,2,rowcol,&J[0][0],INSERT_VALUES);
136:   VecRestoreArrayRead(X,&x);

138:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
139:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
140:   if (A != B) {
141:     MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);
142:     MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);
143:   }
144:   return(0);
145: }

147: /* This is an example of registering an user-provided ARKIMEX scheme */
148: static PetscErrorCode RegisterMyARK2(void)
149: {

153:   {
154:     const PetscReal
155:       A[3][3] = {{0,0,0},
156:                  {0.41421356237309504880,0,0},
157:                  {0.75,0.25,0}},
158:       At[3][3] = {{0,0,0},
159:                   {0.12132034355964257320,0.29289321881345247560,0},
160:                   {0.20710678118654752440,0.50000000000000000000,0.29289321881345247560}};
161:     TSARKIMEXRegister("myark2",2,3,&At[0][0],NULL,NULL,&A[0][0],NULL,NULL,NULL,NULL,0,NULL,NULL);
162:   }
163:   return(0);
164: }

166: /* Monitor timesteps and use interpolation to output at integer multiples of 0.1 */
167: static PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal t,Vec X,void *ctx)
168: {
169:   PetscErrorCode    ierr;
170:   const PetscScalar *x;
171:   PetscReal         tfinal, dt;
172:   User              user = (User)ctx;
173:   Vec               interpolatedX;

176:   TSGetTimeStep(ts,&dt);
177:   TSGetMaxTime(ts,&tfinal);

179:   while (user->next_output <= t && user->next_output <= tfinal) {
180:     VecDuplicate(X,&interpolatedX);
181:     TSInterpolate(ts,user->next_output,interpolatedX);
182:     VecGetArrayRead(interpolatedX,&x);
183:     PetscPrintf(PETSC_COMM_WORLD,"[%.1f] %D TS %.6f (dt = %.6f) X % 12.6e % 12.6e\n",
184:                        user->next_output,step,t,dt,(double)PetscRealPart(x[0]),
185:                        (double)PetscRealPart(x[1]));
186:     VecRestoreArrayRead(interpolatedX,&x);
187:     VecDestroy(&interpolatedX);
188:     user->next_output += 0.1;
189:   }
190:   return(0);
191: }

193: int main(int argc,char **argv)
194: {
195:   TS             ts;            /* nonlinear solver */
196:   Vec            x;             /* solution, residual vectors */
197:   Mat            A;             /* Jacobian matrix */
198:   PetscInt       steps;
199:   PetscReal      ftime   = 0.5;
200:   PetscBool      monitor = PETSC_FALSE;
201:   PetscScalar    *x_ptr;
202:   PetscMPIInt    size;
203:   struct _n_User user;

206:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
207:      Initialize program
208:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
209:   PetscInitialize(&argc,&argv,NULL,help);if (ierr) return ierr;
210:   MPI_Comm_size(PETSC_COMM_WORLD,&size);
211:   if (size != 1) SETERRQ(PETSC_COMM_SELF,1,"This is a uniprocessor example only!");

213:   /* Register user-specified ARKIMEX method */
214:   RegisterMyARK2();

216:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
217:     Set runtime options
218:     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
219:   user.imex        = PETSC_TRUE;
220:   user.next_output = 0.0;
221:   user.mu          = 1.0e6;
222:   PetscOptionsGetBool(NULL,NULL,"-imex",&user.imex,NULL);
223:   PetscOptionsGetBool(NULL,NULL,"-monitor",&monitor,NULL);
224:   PetscOptionsBegin(PETSC_COMM_WORLD,NULL,"Physical parameters",NULL);
225:   PetscOptionsReal("-mu","Stiffness parameter","<1.0e6>",user.mu,&user.mu,NULL);
226:   PetscOptionsEnd();

228:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
229:     Create necessary matrix and vectors, solve same ODE on every process
230:     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
231:   MatCreate(PETSC_COMM_WORLD,&A);
232:   MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,2,2);
233:   MatSetFromOptions(A);
234:   MatSetUp(A);

236:   MatCreateVecs(A,&x,NULL);

238:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
239:      Create timestepping solver context
240:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
241:   TSCreate(PETSC_COMM_WORLD,&ts);
242:   TSSetType(ts,TSBEULER);
243:   TSSetRHSFunction(ts,NULL,RHSFunction,&user);
244:   TSSetIFunction(ts,NULL,IFunction,&user);
245:   TSSetIJacobian(ts,A,A,IJacobian,&user);

247:   TSSetMaxTime(ts,ftime);
248:   TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);
249:   if (monitor) {
250:     TSMonitorSet(ts,Monitor,&user,NULL);
251:   }

253:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
254:      Set initial conditions
255:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
256:   VecGetArray(x,&x_ptr);
257:   x_ptr[0] = 2.0;   x_ptr[1] = -6.666665432100101e-01;
258:   VecRestoreArray(x,&x_ptr);
259:   TSSetTimeStep(ts,.001);

261:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
262:      Set runtime options
263:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
264:   TSSetFromOptions(ts);

266:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
267:      Solve nonlinear system
268:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
269:   TSSolve(ts,x);
270:   TSGetSolveTime(ts,&ftime);
271:   TSGetStepNumber(ts,&steps);
272:   PetscPrintf(PETSC_COMM_WORLD,"steps %D, ftime %g\n",steps,(double)ftime);
273:   VecView(x,PETSC_VIEWER_STDOUT_WORLD);

275:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
276:      Free work space.  All PETSc objects should be destroyed when they
277:      are no longer needed.
278:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
279:   MatDestroy(&A);
280:   VecDestroy(&x);
281:   TSDestroy(&ts);

283:   PetscFinalize();
284:   return(ierr);
285: }

287: /*TEST

289:     test:
290:       requires: !single

292: TEST*/
```