Actual source code: ex16.c

petsc-3.3-p7 2013-05-11
  2: /* Program usage:  ./ex16 [-help] [all PETSc options] */

4: static char help[] = "Solves the van der Pol equation.\n\
5: Input parameters include:\n\
6:       -mu : stiffness parameter\n\n";

8: /*
9:    Concepts: TS^time-dependent nonlinear problems
10:    Concepts: TS^van der Pol equation
11:    Processors: 1
12: */
13: /* ------------------------------------------------------------------------

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

21:    Notes:
22:    This code demonstrates the TS solver interface to two variants 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   [ -2 \mu u_1 - 1;  \mu (1 - u_1^2) ]

67:    Hence,

69:           [      a       ;          0          ]
70:    J(G) = [                                    ]
71:           [ 2 \mu u_1 + 1; a - \mu (1 - u_1^2) ]

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);
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. - 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,MatStructure *flag,void *ctx)
128: {
130:   User user = (User)ctx;
131:   PetscReal mu = user->mu;
132:   PetscInt rowcol[] = {0,1};
133:   PetscScalar *x,J[2][2];

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

142:   MatAssemblyBegin(*A,MAT_FINAL_ASSEMBLY);
143:   MatAssemblyEnd(*A,MAT_FINAL_ASSEMBLY);
144:   if (*A != *B) {
145:     MatAssemblyBegin(*B,MAT_FINAL_ASSEMBLY);
146:     MatAssemblyEnd(*B,MAT_FINAL_ASSEMBLY);
147:   }
148:   *flag = SAME_NONZERO_PATTERN;
149:   return(0);
150: }

154: static PetscErrorCode RegisterMyARK2(void)
155: {

159:   {
160:     const PetscReal
161:       A[3][3] = {{0,0,0},
162:                  {0.41421356237309504880,0,0},
163:                  {0.75,0.25,0}},
164:       At[3][3] = {{0,0,0},
165:                   {0.12132034355964257320,0.29289321881345247560,0},
166:                   {0.20710678118654752440,0.50000000000000000000,0.29289321881345247560}},
167:       *bembedt = PETSC_NULL,*bembed = PETSC_NULL;
168:     TSARKIMEXRegister("myark2",2,3,&At[0][0],PETSC_NULL,PETSC_NULL,&A[0][0],PETSC_NULL,PETSC_NULL,bembedt,bembed,0,PETSC_NULL,PETSC_NULL);
169:   }
170:   return(0);
171: }

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

185:   TSGetTimeStep(ts,&dt);
186:   TSGetDuration(ts,PETSC_NULL,&tfinal);

188:   while (user->next_output <= t && user->next_output <= tfinal) {
189:     VecDuplicate(X,&interpolatedX);
190:     TSInterpolate(ts,user->next_output,interpolatedX);
192:     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]));
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;
213:   PetscErrorCode  ierr;

215:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
216:      Initialize program
217:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
218:   PetscInitialize(&argc,&argv,PETSC_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:   RegisterMyARK2();

225:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
226:     Set runtime options
227:     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
228:   user.mu = 1000;
229:   user.imex = PETSC_TRUE;
230:   user.next_output = 0.0;
231:   PetscOptionsGetReal(PETSC_NULL,"-mu",&user.mu,PETSC_NULL);
232:   PetscOptionsGetBool(PETSC_NULL,"-imex",&user.imex,PETSC_NULL);
233:   PetscOptionsGetBool(PETSC_NULL,"-monitor",&monitor,PETSC_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);

242:   MatGetVecs(A,&x,PETSC_NULL);

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

257:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
258:      Set initial conditions
259:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
260:   VecGetArray(x,&x_ptr);
261:   x_ptr[0] = 2;   x_ptr[1] = 0.66666654321;
262:   VecRestoreArray(x,&x_ptr);
263:   TSSetInitialTimeStep(ts,0.0,.001);

265:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
266:      Set runtime options
267:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
268:   TSSetFromOptions(ts);

270:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
271:      Solve nonlinear system
272:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
273:   TSSolve(ts,x,&ftime);
274:   TSGetTimeStepNumber(ts,&steps);
275:   PetscPrintf(PETSC_COMM_WORLD,"mu %G, steps %D, ftime %G\n",user.mu,steps,ftime);
276:   VecView(x,PETSC_VIEWER_STDOUT_WORLD);

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

286:   PetscFinalize();
287:   return(0);
288: }