Actual source code: ex19.c


  2: static char help[] = "Solves the van der Pol DAE.\n\
  3: Input parameters include:\n";

  5: /* ------------------------------------------------------------------------

  7:    This program solves the van der Pol DAE
  8:        y' = -z = f(y,z)        (1)
  9:        0  = y-(z^3/3 - z) = g(y,z)
 10:    on the domain 0 <= x <= 1, with the boundary conditions
 11:        y(0) = -2, y'(0) = -2.355301397608119909925287735864250951918
 12:    This is a nonlinear equation.

 14:    Notes:
 15:    This code demonstrates the TS solver interface with the Van der Pol DAE,
 16:    namely it is the case when there is no RHS (meaning the RHS == 0), and the
 17:    equations are converted to two variants of linear problems, u_t = f(u,t),
 18:    namely turning (1) into a vector equation in terms of u,

 20:    [     y' + z      ] = [ 0 ]
 21:    [ (z^3/3 - z) - y ]   [ 0 ]

 23:    which then we can write as a vector equation

 25:    [      u_1' + u_2       ] = [ 0 ]  (2)
 26:    [ (u_2^3/3 - u_2) - u_1 ]   [ 0 ]

 28:    which is now in the desired form of u_t = f(u,t). As this is a DAE, and
 29:    there is no u_2', there is no need for a split,

 31:    so

 33:    [ F(u',u,t) ] = [ u_1' ] + [         u_2           ]
 34:                    [  0   ]   [ (u_2^3/3 - u_2) - u_1 ]

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

 39:               dF   dF
 40:    J(F) = a * -- - --
 41:               du'  du

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

 45:    dF   [ 1 ; 0 ]
 46:    -- = [       ]
 47:    du'  [ 0 ; 0 ]

 49:    dF   [  0 ;      1     ]
 50:    -- = [                 ]
 51:    du   [ -1 ; 1 - u_2^2  ]

 53:    Hence,

 55:           [ a ;    -1     ]
 56:    J(F) = [               ]
 57:           [ 1 ; u_2^2 - 1 ]

 59:   ------------------------------------------------------------------------- */

 61: #include <petscts.h>

 63: typedef struct _n_User *User;
 64: struct _n_User {
 65:   PetscReal next_output;
 66: };

 68: /*
 69:    User-defined routines
 70: */

 72: static PetscErrorCode IFunction(TS ts,PetscReal t,Vec X,Vec Xdot,Vec F,void *ctx)
 73: {
 74:   PetscScalar       *f;
 75:   const PetscScalar *x,*xdot;

 78:   VecGetArrayRead(X,&x);
 79:   VecGetArrayRead(Xdot,&xdot);
 80:   VecGetArray(F,&f);
 81:   f[0] = xdot[0] + x[1];
 82:   f[1] = (x[1]*x[1]*x[1]/3.0 - x[1])-x[0];
 83:   VecRestoreArrayRead(X,&x);
 84:   VecRestoreArrayRead(Xdot,&xdot);
 85:   VecRestoreArray(F,&f);
 86:   return 0;
 87: }

 89: static PetscErrorCode IJacobian(TS ts,PetscReal t,Vec X,Vec Xdot,PetscReal a,Mat A,Mat B,void *ctx)
 90: {
 91:   PetscInt          rowcol[] = {0,1};
 92:   PetscScalar       J[2][2];
 93:   const PetscScalar *x;

 96:   VecGetArrayRead(X,&x);
 97:   J[0][0] = a;    J[0][1] = -1.;
 98:   J[1][0] = 1.;   J[1][1] = -1. + x[1]*x[1];
 99:   MatSetValues(B,2,rowcol,2,rowcol,&J[0][0],INSERT_VALUES);
100:   VecRestoreArrayRead(X,&x);

102:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
103:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
104:   if (A != B) {
105:     MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);
106:     MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);
107:   }
108:   return 0;
109: }

111: static PetscErrorCode RegisterMyARK2(void)
112: {
114:   {
115:     const PetscReal
116:       A[3][3] = {{0,0,0},
117:                  {0.41421356237309504880,0,0},
118:                  {0.75,0.25,0}},
119:       At[3][3] = {{0,0,0},
120:                   {0.12132034355964257320,0.29289321881345247560,0},
121:                   {0.20710678118654752440,0.50000000000000000000,0.29289321881345247560}},
122:     *bembedt = NULL,*bembed = NULL;
123:     TSARKIMEXRegister("myark2",2,3,&At[0][0],NULL,NULL,&A[0][0],NULL,NULL,bembedt,bembed,0,NULL,NULL);
124:   }
125:   return 0;
126: }

128: /* Monitor timesteps and use interpolation to output at integer multiples of 0.1 */
129: static PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal t,Vec X,void *ctx)
130: {
131:   const PetscScalar *x;
132:   PetscReal         tfinal, dt;
133:   User              user = (User)ctx;
134:   Vec               interpolatedX;

137:   TSGetTimeStep(ts,&dt);
138:   TSGetMaxTime(ts,&tfinal);

140:   while (user->next_output <= t && user->next_output <= tfinal) {
141:     VecDuplicate(X,&interpolatedX);
142:     TSInterpolate(ts,user->next_output,interpolatedX);
143:     VecGetArrayRead(interpolatedX,&x);
144:     PetscPrintf(PETSC_COMM_WORLD,"[%.1f] %3D TS %.6f (dt = %.6f) X % 12.6e % 12.6e\n",(double)user->next_output,step,(double)t,(double)dt,(double)PetscRealPart(x[0]),(double)PetscRealPart(x[1]));
145:     VecRestoreArrayRead(interpolatedX,&x);
146:     VecDestroy(&interpolatedX);
147:     user->next_output += PetscRealConstant(0.1);
148:   }
149:   return 0;
150: }

152: int main(int argc,char **argv)
153: {
154:   TS             ts;            /* nonlinear solver */
155:   Vec            x;             /* solution, residual vectors */
156:   Mat            A;             /* Jacobian matrix */
157:   PetscInt       steps;
158:   PetscReal      ftime   = 0.5;
159:   PetscBool      monitor = PETSC_FALSE;
160:   PetscScalar    *x_ptr;
161:   PetscMPIInt    size;
162:   struct _n_User user;

164:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
165:      Initialize program
166:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
167:   PetscInitialize(&argc,&argv,NULL,help);
168:   MPI_Comm_size(PETSC_COMM_WORLD,&size);

171:   RegisterMyARK2();

173:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
174:     Set runtime options
175:     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

180:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
181:     Create necessary matrix and vectors, solve same ODE on every process
182:     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
183:   MatCreate(PETSC_COMM_WORLD,&A);
184:   MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,2,2);
185:   MatSetFromOptions(A);
186:   MatSetUp(A);
187:   MatCreateVecs(A,&x,NULL);

189:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
190:      Create timestepping solver context
191:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
192:   TSCreate(PETSC_COMM_WORLD,&ts);
193:   TSSetType(ts,TSBEULER);
194:   TSSetIFunction(ts,NULL,IFunction,&user);
195:   TSSetIJacobian(ts,A,A,IJacobian,&user);
196:   TSSetMaxTime(ts,ftime);
197:   TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);
198:   if (monitor) {
199:     TSMonitorSet(ts,Monitor,&user,NULL);
200:   }

202:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
203:      Set initial conditions
204:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
205:   VecGetArray(x,&x_ptr);
206:   x_ptr[0] = -2;   x_ptr[1] = -2.355301397608119909925287735864250951918;
207:   VecRestoreArray(x,&x_ptr);
208:   TSSetTimeStep(ts,.001);

210:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
211:      Set runtime options
212:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
213:   TSSetFromOptions(ts);

215:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
216:      Solve nonlinear system
217:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
218:   TSSolve(ts,x);
219:   TSGetSolveTime(ts,&ftime);
220:   TSGetStepNumber(ts,&steps);
221:   PetscPrintf(PETSC_COMM_WORLD,"steps %3D, ftime %g\n",steps,(double)ftime);
222:   VecView(x,PETSC_VIEWER_STDOUT_WORLD);

224:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
225:      Free work space.  All PETSc objects should be destroyed when they
226:      are no longer needed.
227:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
228:   MatDestroy(&A);
229:   VecDestroy(&x);
230:   TSDestroy(&ts);

232:   PetscFinalize();
233:   return 0;
234: }

236: /*TEST

238:    test:
239:       requires: !single
240:       suffix: a
241:       args: -monitor -ts_type bdf -ts_rtol 1e-6 -ts_atol 1e-6 -ts_view -ts_adapt_type dsp
242:       output_file: output/ex19_pi42.out

244:    test:
245:       requires: !single
246:       suffix: b
247:       args: -monitor -ts_type bdf -ts_rtol 1e-6 -ts_atol 1e-6 -ts_view -ts_adapt_type dsp -ts_adapt_dsp_filter PI42
248:       output_file: output/ex19_pi42.out

250:    test:
251:       requires: !single
252:       suffix: c
253:       args: -monitor -ts_type bdf -ts_rtol 1e-6 -ts_atol 1e-6 -ts_view -ts_adapt_type dsp -ts_adapt_dsp_pid 0.4,0.2
254:       output_file: output/ex19_pi42.out

256:    test:
257:       requires: !single
258:       suffix: bdf_reject
259:       args: -ts_type bdf -ts_dt 0.5 -ts_max_steps 1 -ts_max_reject {{0 1 2}separate_output} -ts_error_if_step_fails false -ts_adapt_monitor

261: TEST*/