Actual source code: ex1.c


  2: static char help[] ="Solves the time independent Bratu problem using pseudo-timestepping.";

  4: /*
  5:    Concepts: TS^pseudo-timestepping
  6:    Concepts: TS^pseudo-timestepping
  7:    Concepts: TS^nonlinear problems
  8:    Processors: 1

 10: */

 12: /* ------------------------------------------------------------------------

 14:     This code demonstrates how one may solve a nonlinear problem
 15:     with pseudo-timestepping. In this simple example, the pseudo-timestep
 16:     is the same for all grid points, i.e., this is equivalent to using
 17:     the backward Euler method with a variable timestep.

 19:     Note: This example does not require pseudo-timestepping since it
 20:     is an easy nonlinear problem, but it is included to demonstrate how
 21:     the pseudo-timestepping may be done.

 23:     See snes/tutorials/ex4.c[ex4f.F] and
 24:     snes/tutorials/ex5.c[ex5f.F] where the problem is described
 25:     and solved using Newton's method alone.

 27:   ----------------------------------------------------------------------------- */
 28: /*
 29:     Include "petscts.h" to use the PETSc timestepping routines. Note that
 30:     this file automatically includes "petscsys.h" and other lower-level
 31:     PETSc include files.
 32: */
 33: #include <petscts.h>

 35: /*
 36:   Create an application context to contain data needed by the
 37:   application-provided call-back routines, FormJacobian() and
 38:   FormFunction().
 39: */
 40: typedef struct {
 41:   PetscReal param;          /* test problem parameter */
 42:   PetscInt  mx;             /* Discretization in x-direction */
 43:   PetscInt  my;             /* Discretization in y-direction */
 44: } AppCtx;

 46: /*
 47:    User-defined routines
 48: */
 49: extern PetscErrorCode  FormJacobian(TS,PetscReal,Vec,Mat,Mat,void*), FormFunction(TS,PetscReal,Vec,Vec,void*), FormInitialGuess(Vec,AppCtx*);

 51: int main(int argc,char **argv)
 52: {
 53:   TS             ts;                 /* timestepping context */
 54:   Vec            x,r;               /* solution, residual vectors */
 55:   Mat            J;                  /* Jacobian matrix */
 56:   AppCtx         user;               /* user-defined work context */
 57:   PetscInt       its,N;                /* iterations for convergence */
 58:   PetscReal      param_max = 6.81,param_min = 0.,dt;
 59:   PetscReal      ftime;
 60:   PetscMPIInt    size;

 62:   PetscInitialize(&argc,&argv,NULL,help);
 63:   MPI_Comm_size(PETSC_COMM_WORLD,&size);

 66:   user.mx    = 4;
 67:   user.my    = 4;
 68:   user.param = 6.0;

 70:   /*
 71:      Allow user to set the grid dimensions and nonlinearity parameter at run-time
 72:   */
 73:   PetscOptionsGetInt(NULL,NULL,"-mx",&user.mx,NULL);
 74:   PetscOptionsGetInt(NULL,NULL,"-my",&user.my,NULL);
 75:   N  = user.mx*user.my;
 76:   dt = .5/PetscMax(user.mx,user.my);
 77:   PetscOptionsGetReal(NULL,NULL,"-param",&user.param,NULL);

 80:   /*
 81:       Create vectors to hold the solution and function value
 82:   */
 83:   VecCreateSeq(PETSC_COMM_SELF,N,&x);
 84:   VecDuplicate(x,&r);

 86:   /*
 87:     Create matrix to hold Jacobian. Preallocate 5 nonzeros per row
 88:     in the sparse matrix. Note that this is not the optimal strategy; see
 89:     the Performance chapter of the users manual for information on
 90:     preallocating memory in sparse matrices.
 91:   */
 92:   MatCreateSeqAIJ(PETSC_COMM_SELF,N,N,5,0,&J);

 94:   /*
 95:      Create timestepper context
 96:   */
 97:   TSCreate(PETSC_COMM_WORLD,&ts);
 98:   TSSetProblemType(ts,TS_NONLINEAR);

100:   /*
101:      Tell the timestepper context where to compute solutions
102:   */
103:   TSSetSolution(ts,x);

105:   /*
106:      Provide the call-back for the nonlinear function we are
107:      evaluating. Thus whenever the timestepping routines need the
108:      function they will call this routine. Note the final argument
109:      is the application context used by the call-back functions.
110:   */
111:   TSSetRHSFunction(ts,NULL,FormFunction,&user);

113:   /*
114:      Set the Jacobian matrix and the function used to compute
115:      Jacobians.
116:   */
117:   TSSetRHSJacobian(ts,J,J,FormJacobian,&user);

119:   /*
120:        Form the initial guess for the problem
121:   */
122:   FormInitialGuess(x,&user);

124:   /*
125:        This indicates that we are using pseudo timestepping to
126:      find a steady state solution to the nonlinear problem.
127:   */
128:   TSSetType(ts,TSPSEUDO);

130:   /*
131:        Set the initial time to start at (this is arbitrary for
132:      steady state problems); and the initial timestep given above
133:   */
134:   TSSetTimeStep(ts,dt);

136:   /*
137:       Set a large number of timesteps and final duration time
138:      to insure convergence to steady state.
139:   */
140:   TSSetMaxSteps(ts,10000);
141:   TSSetMaxTime(ts,1e12);
142:   TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);

144:   /*
145:       Use the default strategy for increasing the timestep
146:   */
147:   TSPseudoSetTimeStep(ts,TSPseudoTimeStepDefault,0);

149:   /*
150:       Set any additional options from the options database. This
151:      includes all options for the nonlinear and linear solvers used
152:      internally the timestepping routines.
153:   */
154:   TSSetFromOptions(ts);

156:   TSSetUp(ts);

158:   /*
159:       Perform the solve. This is where the timestepping takes place.
160:   */
161:   TSSolve(ts,x);
162:   TSGetSolveTime(ts,&ftime);

164:   /*
165:       Get the number of steps
166:   */
167:   TSGetStepNumber(ts,&its);

169:   PetscPrintf(PETSC_COMM_WORLD,"Number of pseudo timesteps = %D final time %4.2e\n",its,(double)ftime);

171:   /*
172:      Free the data structures constructed above
173:   */
174:   VecDestroy(&x);
175:   VecDestroy(&r);
176:   MatDestroy(&J);
177:   TSDestroy(&ts);
178:   PetscFinalize();
179:   return 0;
180: }
181: /* ------------------------------------------------------------------ */
182: /*           Bratu (Solid Fuel Ignition) Test Problem                 */
183: /* ------------------------------------------------------------------ */

185: /* --------------------  Form initial approximation ----------------- */

187: PetscErrorCode FormInitialGuess(Vec X,AppCtx *user)
188: {
189:   PetscInt       i,j,row,mx,my;
190:   PetscReal      one = 1.0,lambda;
191:   PetscReal      temp1,temp,hx,hy;
192:   PetscScalar    *x;

194:   mx     = user->mx;
195:   my     = user->my;
196:   lambda = user->param;

198:   hx = one / (PetscReal)(mx-1);
199:   hy = one / (PetscReal)(my-1);

201:   VecGetArray(X,&x);
202:   temp1 = lambda/(lambda + one);
203:   for (j=0; j<my; j++) {
204:     temp = (PetscReal)(PetscMin(j,my-j-1))*hy;
205:     for (i=0; i<mx; i++) {
206:       row = i + j*mx;
207:       if (i == 0 || j == 0 || i == mx-1 || j == my-1) {
208:         x[row] = 0.0;
209:         continue;
210:       }
211:       x[row] = temp1*PetscSqrtReal(PetscMin((PetscReal)(PetscMin(i,mx-i-1))*hx,temp));
212:     }
213:   }
214:   VecRestoreArray(X,&x);
215:   return 0;
216: }
217: /* --------------------  Evaluate Function F(x) --------------------- */

219: PetscErrorCode FormFunction(TS ts,PetscReal t,Vec X,Vec F,void *ptr)
220: {
221:   AppCtx            *user = (AppCtx*)ptr;
222:   PetscInt          i,j,row,mx,my;
223:   PetscReal         two = 2.0,one = 1.0,lambda;
224:   PetscReal         hx,hy,hxdhy,hydhx;
225:   PetscScalar       ut,ub,ul,ur,u,uxx,uyy,sc,*f;
226:   const PetscScalar *x;

228:   mx     = user->mx;
229:   my     = user->my;
230:   lambda = user->param;

232:   hx    = one / (PetscReal)(mx-1);
233:   hy    = one / (PetscReal)(my-1);
234:   sc    = hx*hy;
235:   hxdhy = hx/hy;
236:   hydhx = hy/hx;

238:   VecGetArrayRead(X,&x);
239:   VecGetArray(F,&f);
240:   for (j=0; j<my; j++) {
241:     for (i=0; i<mx; i++) {
242:       row = i + j*mx;
243:       if (i == 0 || j == 0 || i == mx-1 || j == my-1) {
244:         f[row] = x[row];
245:         continue;
246:       }
247:       u      = x[row];
248:       ub     = x[row - mx];
249:       ul     = x[row - 1];
250:       ut     = x[row + mx];
251:       ur     = x[row + 1];
252:       uxx    = (-ur + two*u - ul)*hydhx;
253:       uyy    = (-ut + two*u - ub)*hxdhy;
254:       f[row] = -uxx + -uyy + sc*lambda*PetscExpScalar(u);
255:     }
256:   }
257:   VecRestoreArrayRead(X,&x);
258:   VecRestoreArray(F,&f);
259:   return 0;
260: }
261: /* --------------------  Evaluate Jacobian F'(x) -------------------- */

263: /*
264:    Calculate the Jacobian matrix J(X,t).

266:    Note: We put the Jacobian in the preconditioner storage B instead of J. This
267:    way we can give the -snes_mf_operator flag to check our work. This replaces
268:    J with a finite difference approximation, using our analytic Jacobian B for
269:    the preconditioner.
270: */
271: PetscErrorCode FormJacobian(TS ts,PetscReal t,Vec X,Mat J,Mat B,void *ptr)
272: {
273:   AppCtx            *user = (AppCtx*)ptr;
274:   PetscInt          i,j,row,mx,my,col[5];
275:   PetscScalar       two = 2.0,one = 1.0,lambda,v[5],sc;
276:   const PetscScalar *x;
277:   PetscReal         hx,hy,hxdhy,hydhx;

279:   mx     = user->mx;
280:   my     = user->my;
281:   lambda = user->param;

283:   hx    = 1.0 / (PetscReal)(mx-1);
284:   hy    = 1.0 / (PetscReal)(my-1);
285:   sc    = hx*hy;
286:   hxdhy = hx/hy;
287:   hydhx = hy/hx;

289:   VecGetArrayRead(X,&x);
290:   for (j=0; j<my; j++) {
291:     for (i=0; i<mx; i++) {
292:       row = i + j*mx;
293:       if (i == 0 || j == 0 || i == mx-1 || j == my-1) {
294:         MatSetValues(B,1,&row,1,&row,&one,INSERT_VALUES);
295:         continue;
296:       }
297:       v[0] = hxdhy; col[0] = row - mx;
298:       v[1] = hydhx; col[1] = row - 1;
299:       v[2] = -two*(hydhx + hxdhy) + sc*lambda*PetscExpScalar(x[row]); col[2] = row;
300:       v[3] = hydhx; col[3] = row + 1;
301:       v[4] = hxdhy; col[4] = row + mx;
302:       MatSetValues(B,1,&row,5,col,v,INSERT_VALUES);
303:     }
304:   }
305:   VecRestoreArrayRead(X,&x);
306:   MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);
307:   MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);
308:   if (J != B) {
309:     MatAssemblyBegin(J,MAT_FINAL_ASSEMBLY);
310:     MatAssemblyEnd(J,MAT_FINAL_ASSEMBLY);
311:   }
312:   return 0;
313: }

315: /*TEST

317:     test:
318:       args: -ksp_gmres_cgs_refinement_type refine_always -snes_type newtonls -ts_monitor_pseudo -snes_atol 1.e-7 -ts_pseudo_frtol 1.e-5 -ts_view draw:tikz:fig.tex

320:     test:
321:       suffix: 2
322:       args: -ts_monitor_pseudo -ts_pseudo_frtol 1.e-5

324: TEST*/