Actual source code: ex19.c


  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:    [ F(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 F (from the PETSc user manual),
 42:    in the equation F(u',u,t) = G(u,t),

 44:               dF   dF
 45:    J(F) = a * -- - --
 46:               du'  du

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

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

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

 58:    Hence,

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

 77: static PetscErrorCode IFunction(TS ts,PetscReal t,Vec X,Vec Xdot,Vec F,void *ctx)
 78: {
 79:   PetscScalar       *f;
 80:   const PetscScalar *x,*xdot;

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

 94: static PetscErrorCode IJacobian(TS ts,PetscReal t,Vec X,Vec Xdot,PetscReal a,Mat A,Mat B,void *ctx)
 95: {
 96:   PetscInt          rowcol[] = {0,1};
 97:   PetscScalar       J[2][2];
 98:   const PetscScalar *x;

101:   VecGetArrayRead(X,&x);
102:   J[0][0] = a;    J[0][1] = -1.;
103:   J[1][0] = 1.;   J[1][1] = -1. + x[1]*x[1];
104:   MatSetValues(B,2,rowcol,2,rowcol,&J[0][0],INSERT_VALUES);
105:   VecRestoreArrayRead(X,&x);

107:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
108:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
109:   if (A != B) {
110:     MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);
111:     MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);
112:   }
113:   return 0;
114: }

116: static PetscErrorCode RegisterMyARK2(void)
117: {
119:   {
120:     const PetscReal
121:       A[3][3] = {{0,0,0},
122:                  {0.41421356237309504880,0,0},
123:                  {0.75,0.25,0}},
124:       At[3][3] = {{0,0,0},
125:                   {0.12132034355964257320,0.29289321881345247560,0},
126:                   {0.20710678118654752440,0.50000000000000000000,0.29289321881345247560}},
127:     *bembedt = NULL,*bembed = NULL;
128:     TSARKIMEXRegister("myark2",2,3,&At[0][0],NULL,NULL,&A[0][0],NULL,NULL,bembedt,bembed,0,NULL,NULL);
129:   }
130:   return 0;
131: }

133: /* Monitor timesteps and use interpolation to output at integer multiples of 0.1 */
134: static PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal t,Vec X,void *ctx)
135: {
136:   const PetscScalar *x;
137:   PetscReal         tfinal, dt;
138:   User              user = (User)ctx;
139:   Vec               interpolatedX;

142:   TSGetTimeStep(ts,&dt);
143:   TSGetMaxTime(ts,&tfinal);

145:   while (user->next_output <= t && user->next_output <= tfinal) {
146:     VecDuplicate(X,&interpolatedX);
147:     TSInterpolate(ts,user->next_output,interpolatedX);
148:     VecGetArrayRead(interpolatedX,&x);
149:     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]));
150:     VecRestoreArrayRead(interpolatedX,&x);
151:     VecDestroy(&interpolatedX);
152:     user->next_output += PetscRealConstant(0.1);
153:   }
154:   return 0;
155: }

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

169:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
170:      Initialize program
171:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
172:   PetscInitialize(&argc,&argv,NULL,help);
173:   MPI_Comm_size(PETSC_COMM_WORLD,&size);

176:   RegisterMyARK2();

178:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
179:     Set runtime options
180:     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

185:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
186:     Create necessary matrix and vectors, solve same ODE on every process
187:     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
188:   MatCreate(PETSC_COMM_WORLD,&A);
189:   MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,2,2);
190:   MatSetFromOptions(A);
191:   MatSetUp(A);
192:   MatCreateVecs(A,&x,NULL);

194:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
195:      Create timestepping solver context
196:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
197:   TSCreate(PETSC_COMM_WORLD,&ts);
198:   TSSetType(ts,TSBEULER);
199:   TSSetIFunction(ts,NULL,IFunction,&user);
200:   TSSetIJacobian(ts,A,A,IJacobian,&user);
201:   TSSetMaxTime(ts,ftime);
202:   TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);
203:   if (monitor) {
204:     TSMonitorSet(ts,Monitor,&user,NULL);
205:   }

207:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
208:      Set initial conditions
209:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
210:   VecGetArray(x,&x_ptr);
211:   x_ptr[0] = -2;   x_ptr[1] = -2.355301397608119909925287735864250951918;
212:   VecRestoreArray(x,&x_ptr);
213:   TSSetTimeStep(ts,.001);

215:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
216:      Set runtime options
217:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
218:   TSSetFromOptions(ts);

220:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
221:      Solve nonlinear system
222:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
223:   TSSolve(ts,x);
224:   TSGetSolveTime(ts,&ftime);
225:   TSGetStepNumber(ts,&steps);
226:   PetscPrintf(PETSC_COMM_WORLD,"steps %3D, ftime %g\n",steps,(double)ftime);
227:   VecView(x,PETSC_VIEWER_STDOUT_WORLD);

229:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
230:      Free work space.  All PETSc objects should be destroyed when they
231:      are no longer needed.
232:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
233:   MatDestroy(&A);
234:   VecDestroy(&x);
235:   TSDestroy(&ts);

237:   PetscFinalize();
238:   return 0;
239: }

241: /*TEST

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

249:    test:
250:       requires: !single
251:       suffix: b
252:       args: -monitor -ts_type bdf -ts_rtol 1e-6 -ts_atol 1e-6 -ts_view -ts_adapt_type dsp -ts_adapt_dsp_filter PI42
253:       output_file: output/ex19_pi42.out

255:    test:
256:       requires: !single
257:       suffix: c
258:       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
259:       output_file: output/ex19_pi42.out

261:    test:
262:       requires: !single
263:       suffix: bdf_reject
264:       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

266: TEST*/