Actual source code: ex19.c

petsc-master 2015-07-03
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  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:    [ G(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 G (from the PETSc user manual),
 42:    in the equation G(u',u,t) = F(u,t),

 44:               dG   dG
 45:    J(G) = a * -- - --
 46:               du'  du

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

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

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

 58:    Hence,

 60:           [ a ;    -1     ]
 61:    J(G) = [               ]
 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: */

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

 86:   VecGetArrayRead(X,&x);
 87:   VecGetArrayRead(Xdot,&xdot);
 88:   VecGetArray(F,&f);
 89:   f[0] = xdot[0] + x[1];
 90:   f[1] = (x[1]*x[1]*x[1]/3.0 - x[1])-x[0];
 91:   VecRestoreArrayRead(X,&x);
 92:   VecRestoreArrayRead(Xdot,&xdot);
 93:   VecRestoreArray(F,&f);
 94:   return(0);
 95: }

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

107:   VecGetArrayRead(X,&x);
108:   J[0][0] = a;    J[0][1] = -1.;
109:   J[1][0] = 1.;   J[1][1] = -1. + x[1]*x[1];
110:   MatSetValues(B,2,rowcol,2,rowcol,&J[0][0],INSERT_VALUES);
111:   VecRestoreArrayRead(X,&x);

113:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
114:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
115:   if (A != B) {
116:     MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);
117:     MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);
118:   }
119:   return(0);
120: }

124: static PetscErrorCode RegisterMyARK2(void)
125: {

129:   {
130:     const PetscReal
131:       A[3][3] = {{0,0,0},
132:                  {0.41421356237309504880,0,0},
133:                  {0.75,0.25,0}},
134:       At[3][3] = {{0,0,0},
135:                   {0.12132034355964257320,0.29289321881345247560,0},
136:                   {0.20710678118654752440,0.50000000000000000000,0.29289321881345247560}},
137:     *bembedt = NULL,*bembed = NULL;
138:     TSARKIMEXRegister("myark2",2,3,&At[0][0],NULL,NULL,&A[0][0],NULL,NULL,bembedt,bembed,0,NULL,NULL);
139:   }
140:   return(0);
141: }

145: /* Monitor timesteps and use interpolation to output at integer multiples of 0.1 */
146: static PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal t,Vec X,void *ctx)
147: {
148:   PetscErrorCode    ierr;
149:   const PetscScalar *x;
150:   PetscReal         tfinal, dt;
151:   User              user = (User)ctx;
152:   Vec               interpolatedX;

155:   TSGetTimeStep(ts,&dt);
156:   TSGetDuration(ts,NULL,&tfinal);

158:   while (user->next_output <= t && user->next_output <= tfinal) {
159:     VecDuplicate(X,&interpolatedX);
160:     TSInterpolate(ts,user->next_output,interpolatedX);
161:     VecGetArrayRead(interpolatedX,&x);
162:     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]));
163:     VecRestoreArrayRead(interpolatedX,&x);
164:     VecDestroy(&interpolatedX);
165:     user->next_output += 0.1;
166:   }
167:   return(0);
168: }

172: int main(int argc,char **argv)
173: {
174:   TS             ts;            /* nonlinear solver */
175:   Vec            x;             /* solution, residual vectors */
176:   Mat            A;             /* Jacobian matrix */
177:   PetscInt       steps;
178:   PetscReal      ftime   = 0.5;
179:   PetscBool      monitor = PETSC_FALSE;
180:   PetscScalar    *x_ptr;
181:   PetscMPIInt    size;
182:   struct _n_User user;

185:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
186:      Initialize program
187:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
188:   PetscInitialize(&argc,&argv,NULL,help);

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

193:   RegisterMyARK2();

195:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
196:     Set runtime options
197:     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

202:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
203:     Create necessary matrix and vectors, solve same ODE on every process
204:     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
205:   MatCreate(PETSC_COMM_WORLD,&A);
206:   MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,2,2);
207:   MatSetFromOptions(A);
208:   MatSetUp(A);
209:   MatCreateVecs(A,&x,NULL);

211:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
212:      Create timestepping solver context
213:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
214:   TSCreate(PETSC_COMM_WORLD,&ts);
215:   TSSetType(ts,TSBEULER);
216:   TSSetIFunction(ts,NULL,IFunction,&user);
217:   TSSetIJacobian(ts,A,A,IJacobian,&user);
218:   TSSetDuration(ts,PETSC_DEFAULT,ftime);
219:   if (monitor) {
220:     TSMonitorSet(ts,Monitor,&user,NULL);
221:   }

223:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
224:      Set initial conditions
225:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
226:   VecGetArray(x,&x_ptr);
227:   x_ptr[0] = -2;   x_ptr[1] = -2.355301397608119909925287735864250951918;
228:   VecRestoreArray(x,&x_ptr);
229:   TSSetInitialTimeStep(ts,0.0,.001);

231:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
232:      Set runtime options
233:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
234:   TSSetFromOptions(ts);

236:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
237:      Solve nonlinear system
238:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
239:   TSSolve(ts,x);
240:   TSGetSolveTime(ts,&ftime);
241:   TSGetTimeStepNumber(ts,&steps);
242:   PetscPrintf(PETSC_COMM_WORLD,"steps %D, ftime %g\n",steps,(double)ftime);
243:   VecView(x,PETSC_VIEWER_STDOUT_WORLD);

245:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
246:      Free work space.  All PETSc objects should be destroyed when they
247:      are no longer needed.
248:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
249:   MatDestroy(&A);
250:   VecDestroy(&x);
251:   TSDestroy(&ts);

253:   PetscFinalize();
254:   return(0);
255: }