Actual source code: ex16.c

petsc-3.4.4 2014-03-13
  2: static char help[] = "Solves the van der Pol equation.\n\
  3: Input parameters include:\n\
  4:       -mu : stiffness parameter\n\n";

  6: /*
  7:    Concepts: TS^time-dependent nonlinear problems
  8:    Concepts: TS^van der Pol equation
  9:    Processors: 1
 10: */
 11: /* ------------------------------------------------------------------------

 13:    This program solves the van der Pol equation
 14:        y'' - \mu (1-y^2)*y' + y = 0        (1)
 15:    on the domain 0 <= x <= 1, with the boundary conditions
 16:        y(0) = 2, y'(0) = 0,
 17:    This is a nonlinear equation.

 19:    Notes:
 20:    This code demonstrates the TS solver interface to two variants of
 21:    linear problems, u_t = f(u,t), namely turning (1) into a system of
 22:    first order differential equations,

 24:    [ y' ] = [          z          ]
 25:    [ z' ]   [ \mu (1 - y^2) z - y ]

 27:    which then we can write as a vector equation

 29:    [ u_1' ] = [             u_2           ]  (2)
 30:    [ u_2' ]   [ \mu (1 - u_1^2) u_2 - u_1 ]

 32:    which is now in the desired form of u_t = f(u,t). One way that we
 33:    can split f(u,t) in (2) is to split by component,

 35:    [ u_1' ] = [ u_2 ] + [            0              ]
 36:    [ u_2' ]   [  0  ]   [ \mu (1 - u_1^2) u_2 - u_1 ]

 38:    where

 40:    [ F(u,t) ] = [ u_2 ]
 41:                 [  0  ]

 43:    and

 45:    [ G(u',u,t) ] = [ u_1' ] - [            0              ]
 46:                    [ u_2' ]   [ \mu (1 - u_1^2) u_2 - u_1 ]

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

 51:               dG   dG
 52:    J(G) = a * -- - --
 53:               du'  du

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

 57:    dG   [ 1 ; 0 ]
 58:    -- = [       ]
 59:    du'  [ 0 ; 1 ]

 61:    dG   [       0       ;         0        ]
 62:    -- = [                                  ]
 63:    du   [ -2 \mu u_1 - 1;  \mu (1 - u_1^2) ]

 65:    Hence,

 67:           [      a       ;          0          ]
 68:    J(G) = [                                    ]
 69:           [ 2 \mu u_1 + 1; a - \mu (1 - u_1^2) ]

 71:   ------------------------------------------------------------------------- */

 73: #include <petscts.h>

 75: typedef struct _n_User *User;
 76: struct _n_User {
 77:   PetscReal mu;
 78:   PetscBool imex;
 79:   PetscReal next_output;
 80: };

 82: /*
 83: *  User-defined routines
 84: */
 87: static PetscErrorCode RHSFunction(TS ts,PetscReal t,Vec X,Vec F,void *ctx)
 88: {
 90:   User           user = (User)ctx;
 91:   PetscScalar    *x,*f;

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

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

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

125: static PetscErrorCode IJacobian(TS ts,PetscReal t,Vec X,Vec Xdot,PetscReal a,Mat *A,Mat *B,MatStructure *flag,void *ctx)
126: {
128:   User           user     = (User)ctx;
129:   PetscReal      mu       = user->mu;
130:   PetscInt       rowcol[] = {0,1};
131:   PetscScalar    *x,J[2][2];

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

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

152: static PetscErrorCode RegisterMyARK2(void)
153: {

157:   {
158:     const PetscReal
159:       A[3][3] = {{0,0,0},
160:                  {0.41421356237309504880,0,0},
161:                  {0.75,0.25,0}},
162:       At[3][3] = {{0,0,0},
163:                   {0.12132034355964257320,0.29289321881345247560,0},
164:                   {0.20710678118654752440,0.50000000000000000000,0.29289321881345247560}},
165:       *bembedt = NULL,*bembed = NULL;
166:     TSARKIMEXRegister("myark2",2,3,&At[0][0],NULL,NULL,&A[0][0],NULL,NULL,bembedt,bembed,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",user->next_output,step,t,dt,(double)PetscRealPart(x[0]),(double)PetscRealPart(x[1]));
191:     VecRestoreArrayRead(interpolatedX,&x);
192:     VecDestroy(&interpolatedX);

194:     user->next_output += 0.1;
195:   }
196:   return(0);
197: }

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

214:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
215:      Initialize program
216:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
217:   PetscInitialize(&argc,&argv,NULL,help);

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

222:   RegisterMyARK2();

224:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
225:     Set runtime options
226:     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
227:   user.mu          = 1000;
228:   user.imex        = PETSC_TRUE;
229:   user.next_output = 0.0;

231:   PetscOptionsGetReal(NULL,"-mu",&user.mu,NULL);
232:   PetscOptionsGetBool(NULL,"-imex",&user.imex,NULL);
233:   PetscOptionsGetBool(NULL,"-monitor",&monitor,NULL);

235:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
236:     Create necessary matrix and vectors, solve same ODE on every process
237:     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
238:   MatCreate(PETSC_COMM_WORLD,&A);
239:   MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,2,2);
240:   MatSetFromOptions(A);
241:   MatSetUp(A);
242:   MatGetVecs(A,&x,NULL);

244:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
245:      Create timestepping solver context
246:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
247:   TSCreate(PETSC_COMM_WORLD,&ts);
248:   TSSetType(ts,TSBEULER);
249:   TSSetRHSFunction(ts,NULL,RHSFunction,&user);
250:   TSSetIFunction(ts,NULL,IFunction,&user);
251:   TSSetIJacobian(ts,A,A,IJacobian,&user);
252:   TSSetDuration(ts,PETSC_DEFAULT,ftime);
253:   if (monitor) {
254:     TSMonitorSet(ts,Monitor,&user,NULL);
255:   }

257:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
258:      Set initial conditions
259:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
260:   VecGetArray(x,&x_ptr);

262:   x_ptr[0] = 2;   x_ptr[1] = 0.66666654321;

264:   VecRestoreArray(x,&x_ptr);
265:   TSSetInitialTimeStep(ts,0.0,.001);

267:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
268:      Set runtime options
269:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
270:   TSSetFromOptions(ts);

272:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
273:      Solve nonlinear system
274:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
275:   TSSolve(ts,x);
276:   TSGetSolveTime(ts,&ftime);
277:   TSGetTimeStepNumber(ts,&steps);
278:   PetscPrintf(PETSC_COMM_WORLD,"mu %G, steps %D, ftime %G\n",user.mu,steps,ftime);
279:   VecView(x,PETSC_VIEWER_STDOUT_WORLD);

281:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
282:      Free work space.  All PETSc objects should be destroyed when they
283:      are no longer needed.
284:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
285:   MatDestroy(&A);
286:   VecDestroy(&x);
287:   TSDestroy(&ts);

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