Actual source code: ex20.c

petsc-3.5.2 2014-09-08
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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: */
 89: static PetscErrorCode RHSFunction(TS ts,PetscReal t,Vec X,Vec F,void *ctx)
 90: {
 92:   User           user = (User)ctx;
 93:   PetscScalar    *x,*f;

 96:   VecGetArray(X,&x);
 97:   VecGetArray(F,&f);
 98:   f[0] = (user->imex ? x[1] : 0.0);
 99:   f[1] = 0.0;
100:   VecRestoreArray(X,&x);
101:   VecRestoreArray(F,&f);
102:   return(0);
103: }

107: static PetscErrorCode IFunction(TS ts,PetscReal t,Vec X,Vec Xdot,Vec F,void *ctx)
108: {
110:   User           user = (User)ctx;
111:   PetscScalar    *x,*xdot,*f;

114:   VecGetArray(X,&x);
115:   VecGetArray(Xdot,&xdot);
116:   VecGetArray(F,&f);
117:   f[0] = xdot[0] - (user->imex ? 0 : x[1]);
118:   f[1] = xdot[1] - user->mu*((1.0-x[0]*x[0])*x[1] - x[0]);
119:   VecRestoreArray(X,&x);
120:   VecRestoreArray(Xdot,&xdot);
121:   VecRestoreArray(F,&f);
122:   return(0);
123: }

127: static PetscErrorCode IJacobian(TS ts,PetscReal t,Vec X,Vec Xdot,PetscReal a,Mat A,Mat B,void *ctx)
128: {
130:   User           user     = (User)ctx;
131:   PetscInt       rowcol[] = {0,1};
132:   PetscScalar    *x,J[2][2];

135:   VecGetArray(X,&x);
136:   J[0][0] = a;     J[0][1] = (user->imex ? 0 : -1.0);
137:   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]);
138:   MatSetValues(B,2,rowcol,2,rowcol,&J[0][0],INSERT_VALUES);
139:   VecRestoreArray(X,&x);

141:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
142:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
143:   if (A != B) {
144:     MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);
145:     MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);
146:   }
147:   return(0);
148: }

152: /* This is an example of registering an user-provided ARKIMEX scheme */
153: static PetscErrorCode RegisterMyARK2(void)
154: {

158:   {
159:     const PetscReal
160:       A[3][3] = {{0,0,0},
161:                  {0.41421356237309504880,0,0},
162:                  {0.75,0.25,0}},
163:       At[3][3] = {{0,0,0},
164:                   {0.12132034355964257320,0.29289321881345247560,0},
165:                   {0.20710678118654752440,0.50000000000000000000,0.29289321881345247560}};
166:     TSARKIMEXRegister("myark2",2,3,&At[0][0],NULL,NULL,&A[0][0],NULL,NULL,NULL,NULL,0,NULL,NULL);
167:   }
168:   return(0);
169: }

173: /* Monitor timesteps and use interpolation to output at integer multiples of 0.1 */
174: static PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal t,Vec X,void *ctx)
175: {
176:   PetscErrorCode    ierr;
177:   const PetscScalar *x;
178:   PetscReal         tfinal, dt;
179:   User              user = (User)ctx;
180:   Vec               interpolatedX;

183:   TSGetTimeStep(ts,&dt);
184:   TSGetDuration(ts,NULL,&tfinal);

186:   while (user->next_output <= t && user->next_output <= tfinal) {
187:     VecDuplicate(X,&interpolatedX);
188:     TSInterpolate(ts,user->next_output,interpolatedX);
189:     VecGetArrayRead(interpolatedX,&x);
190:     PetscPrintf(PETSC_COMM_WORLD,"[%.1f] %D TS %.6f (dt = %.6f) X % 12.6e % 12.6e\n",
191:                        user->next_output,step,t,dt,(double)PetscRealPart(x[0]),
192:                        (double)PetscRealPart(x[1]));
193:     VecRestoreArrayRead(interpolatedX,&x);
194:     VecDestroy(&interpolatedX);
195:     user->next_output += 0.1;
196:   }
197:   return(0);
198: }

202: int main(int argc,char **argv)
203: {
204:   TS             ts;            /* nonlinear solver */
205:   Vec            x;             /* solution, residual vectors */
206:   Mat            A;             /* Jacobian matrix */
207:   PetscInt       steps;
208:   PetscReal      ftime   = 0.5;
209:   PetscBool      monitor = PETSC_FALSE;
210:   PetscScalar    *x_ptr;
211:   PetscMPIInt    size;
212:   struct _n_User user;

215:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
216:      Initialize program
217:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
218:   PetscInitialize(&argc,&argv,NULL,help);

220:   MPI_Comm_size(PETSC_COMM_WORLD,&size);
221:   if (size != 1) SETERRQ(PETSC_COMM_SELF,1,"This is a uniprocessor example only!");

223:   /* Register user-specified ARKIMEX method */
224:   RegisterMyARK2();

226:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
227:     Set runtime options
228:     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
229:   user.imex        = PETSC_TRUE;
230:   user.next_output = 0.0;
231:   user.mu          = 1.0e6;
232:   PetscOptionsGetBool(NULL,"-imex",&user.imex,NULL);
233:   PetscOptionsGetBool(NULL,"-monitor",&monitor,NULL);
234:   PetscOptionsReal("-mu","Stiffness parameter","<1.0e6>",user.mu,&user.mu,NULL);

236:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
237:     Create necessary matrix and vectors, solve same ODE on every process
238:     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
239:   MatCreate(PETSC_COMM_WORLD,&A);
240:   MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,2,2);
241:   MatSetFromOptions(A);
242:   MatSetUp(A);

244:   MatGetVecs(A,&x,NULL);

246:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
247:      Create timestepping solver context
248:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
249:   TSCreate(PETSC_COMM_WORLD,&ts);
250:   TSSetType(ts,TSBEULER);
251:   TSSetRHSFunction(ts,NULL,RHSFunction,&user);
252:   TSSetIFunction(ts,NULL,IFunction,&user);
253:   TSSetIJacobian(ts,A,A,IJacobian,&user);

255:   TSSetDuration(ts,PETSC_DEFAULT,ftime);
256:   if (monitor) {
257:     TSMonitorSet(ts,Monitor,&user,NULL);
258:   }

260:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
261:      Set initial conditions
262:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
263:   VecGetArray(x,&x_ptr);
264:   x_ptr[0] = 2.0;   x_ptr[1] = -6.666665432100101e-01;
265:   VecRestoreArray(x,&x_ptr);
266:   TSSetInitialTimeStep(ts,0.0,.001);

268:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
269:      Set runtime options
270:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
271:   TSSetFromOptions(ts);

273:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
274:      Solve nonlinear system
275:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
276:   TSSolve(ts,x);
277:   TSGetSolveTime(ts,&ftime);
278:   TSGetTimeStepNumber(ts,&steps);
279:   PetscPrintf(PETSC_COMM_WORLD,"steps %D, ftime %g\n",steps,(double)ftime);
280:   VecView(x,PETSC_VIEWER_STDOUT_WORLD);

282:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
283:      Free work space.  All PETSc objects should be destroyed when they
284:      are no longer needed.
285:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
286:   MatDestroy(&A);
287:   VecDestroy(&x);
288:   TSDestroy(&ts);

290:   PetscFinalize();
291:   return(0);
292: }