Actual source code: ex1.c
petsc-3.3-p7 2013-05-11
2: static char help[] ="Solves the time dependent Bratu problem using pseudo-timestepping.";
4: /*
5: Concepts: TS^pseudo-timestepping
6: Concepts: pseudo-timestepping
7: Concepts: 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/examples/tutorials/ex4.c[ex4f.F] and
24: snes/examples/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*,MatStructure*,void*),
50: FormFunction(TS,PetscReal,Vec,Vec,void*),
51: FormInitialGuess(Vec,AppCtx*);
55: int main(int argc,char **argv)
56: {
57: TS ts; /* timestepping context */
58: Vec x,r; /* solution, residual vectors */
59: Mat J; /* Jacobian matrix */
60: AppCtx user; /* user-defined work context */
61: PetscInt its,N; /* iterations for convergence */
63: PetscReal param_max = 6.81,param_min = 0.,dt;
64: PetscReal ftime;
65: PetscMPIInt size;
67: PetscInitialize(&argc,&argv,PETSC_NULL,help);
68: MPI_Comm_size(PETSC_COMM_WORLD,&size);
69: if (size != 1) SETERRQ(PETSC_COMM_WORLD,PETSC_ERR_SUP,"This is a uniprocessor example only");
71: user.mx = 4;
72: user.my = 4;
73: user.param = 6.0;
74:
75: /*
76: Allow user to set the grid dimensions and nonlinearity parameter at run-time
77: */
78: PetscOptionsGetInt(PETSC_NULL,"-mx",&user.mx,PETSC_NULL);
79: PetscOptionsGetInt(PETSC_NULL,"-my",&user.my,PETSC_NULL);
80: PetscOptionsGetReal(PETSC_NULL,"-param",&user.param,PETSC_NULL);
81: if (user.param >= param_max || user.param <= param_min) SETERRQ(PETSC_COMM_SELF,1,"Parameter is out of range");
82: dt = .5/PetscMax(user.mx,user.my);
83: PetscOptionsGetReal(PETSC_NULL,"-dt",&dt,PETSC_NULL);
84: N = user.mx*user.my;
85:
86: /*
87: Create vectors to hold the solution and function value
88: */
89: VecCreateSeq(PETSC_COMM_SELF,N,&x);
90: VecDuplicate(x,&r);
92: /*
93: Create matrix to hold Jacobian. Preallocate 5 nonzeros per row
94: in the sparse matrix. Note that this is not the optimal strategy; see
95: the Performance chapter of the users manual for information on
96: preallocating memory in sparse matrices.
97: */
98: MatCreateSeqAIJ(PETSC_COMM_SELF,N,N,5,0,&J);
100: /*
101: Create timestepper context
102: */
103: TSCreate(PETSC_COMM_WORLD,&ts);
104: TSSetProblemType(ts,TS_NONLINEAR);
106: /*
107: Tell the timestepper context where to compute solutions
108: */
109: TSSetSolution(ts,x);
111: /*
112: Provide the call-back for the nonlinear function we are
113: evaluating. Thus whenever the timestepping routines need the
114: function they will call this routine. Note the final argument
115: is the application context used by the call-back functions.
116: */
117: TSSetRHSFunction(ts,PETSC_NULL,FormFunction,&user);
119: /*
120: Set the Jacobian matrix and the function used to compute
121: Jacobians.
122: */
123: TSSetRHSJacobian(ts,J,J,FormJacobian,&user);
125: /*
126: For the initial guess for the problem
127: */
128: FormInitialGuess(x,&user);
130: /*
131: This indicates that we are using pseudo timestepping to
132: find a steady state solution to the nonlinear problem.
133: */
134: TSSetType(ts,TSPSEUDO);
136: /*
137: Set the initial time to start at (this is arbitrary for
138: steady state problems; and the initial timestep given above
139: */
140: TSSetInitialTimeStep(ts,0.0,dt);
142: /*
143: Set a large number of timesteps and final duration time
144: to insure convergence to steady state.
145: */
146: TSSetDuration(ts,1000,1.e12);
148: /*
149: Use the default strategy for increasing the timestep
150: */
151: TSPseudoSetTimeStep(ts,TSPseudoDefaultTimeStep,0);
153: /*
154: Set any additional options from the options database. This
155: includes all options for the nonlinear and linear solvers used
156: internally the the timestepping routines.
157: */
158: TSSetFromOptions(ts);
160: TSSetUp(ts);
162: /*
163: Perform the solve. This is where the timestepping takes place.
164: */
165: TSSolve(ts,x,&ftime);
167: /*
168: Get the number of steps
169: */
170: TSGetTimeStepNumber(ts,&its);
172: printf("Number of pseudo timesteps = %d final time %4.2e\n",(int)its,ftime);
174: /*
175: Free the data structures constructed above
176: */
177: VecDestroy(&x);
178: VecDestroy(&r);
179: MatDestroy(&J);
180: TSDestroy(&ts);
181: PetscFinalize();
183: return 0;
184: }
185: /* ------------------------------------------------------------------ */
186: /* Bratu (Solid Fuel Ignition) Test Problem */
187: /* ------------------------------------------------------------------ */
189: /* -------------------- Form initial approximation ----------------- */
193: PetscErrorCode FormInitialGuess(Vec X,AppCtx *user)
194: {
195: PetscInt i,j,row,mx,my;
197: PetscReal one = 1.0,lambda;
198: PetscReal temp1,temp,hx,hy;
199: PetscScalar *x;
201: mx = user->mx;
202: my = user->my;
203: lambda = user->param;
205: hx = one / (PetscReal)(mx-1);
206: hy = one / (PetscReal)(my-1);
208: VecGetArray(X,&x);
209: temp1 = lambda/(lambda + one);
210: for (j=0; j<my; j++) {
211: temp = (PetscReal)(PetscMin(j,my-j-1))*hy;
212: for (i=0; i<mx; i++) {
213: row = i + j*mx;
214: if (i == 0 || j == 0 || i == mx-1 || j == my-1) {
215: x[row] = 0.0;
216: continue;
217: }
218: x[row] = temp1*PetscSqrtReal(PetscMin((PetscReal)(PetscMin(i,mx-i-1))*hx,temp));
219: }
220: }
221: VecRestoreArray(X,&x);
222: return 0;
223: }
224: /* -------------------- Evaluate Function F(x) --------------------- */
228: PetscErrorCode FormFunction(TS ts,PetscReal t,Vec X,Vec F,void *ptr)
229: {
230: AppCtx *user = (AppCtx*)ptr;
232: PetscInt i,j,row,mx,my;
233: PetscReal two = 2.0,one = 1.0,lambda;
234: PetscReal hx,hy,hxdhy,hydhx;
235: PetscScalar ut,ub,ul,ur,u,uxx,uyy,sc,*x,*f;
237: mx = user->mx;
238: my = user->my;
239: lambda = user->param;
241: hx = one / (PetscReal)(mx-1);
242: hy = one / (PetscReal)(my-1);
243: sc = hx*hy;
244: hxdhy = hx/hy;
245: hydhx = hy/hx;
247: VecGetArray(X,&x);
248: VecGetArray(F,&f);
249: for (j=0; j<my; j++) {
250: for (i=0; i<mx; i++) {
251: row = i + j*mx;
252: if (i == 0 || j == 0 || i == mx-1 || j == my-1) {
253: f[row] = x[row];
254: continue;
255: }
256: u = x[row];
257: ub = x[row - mx];
258: ul = x[row - 1];
259: ut = x[row + mx];
260: ur = x[row + 1];
261: uxx = (-ur + two*u - ul)*hydhx;
262: uyy = (-ut + two*u - ub)*hxdhy;
263: f[row] = -uxx + -uyy + sc*lambda*PetscExpScalar(u);
264: }
265: }
266: VecRestoreArray(X,&x);
267: VecRestoreArray(F,&f);
268: return 0;
269: }
270: /* -------------------- Evaluate Jacobian F'(x) -------------------- */
274: /*
275: Calculate the Jacobian matrix J(X,t).
277: Note: We put the Jacobian in the preconditioner storage B instead of J. This
278: way we can give the -snes_mf_operator flag to check our work. This replaces
279: J with a finite difference approximation, using our analytic Jacobian B for
280: the preconditioner.
281: */
282: PetscErrorCode FormJacobian(TS ts,PetscReal t,Vec X,Mat *J,Mat *B,MatStructure *flag,void *ptr)
283: {
284: AppCtx *user = (AppCtx*)ptr;
285: Mat jac = *B;
286: PetscInt i,j,row,mx,my,col[5];
288: PetscScalar two = 2.0,one = 1.0,lambda,v[5],sc,*x;
289: PetscReal hx,hy,hxdhy,hydhx;
292: mx = user->mx;
293: my = user->my;
294: lambda = user->param;
296: hx = 1.0 / (PetscReal)(mx-1);
297: hy = 1.0 / (PetscReal)(my-1);
298: sc = hx*hy;
299: hxdhy = hx/hy;
300: hydhx = hy/hx;
302: VecGetArray(X,&x);
303: for (j=0; j<my; j++) {
304: for (i=0; i<mx; i++) {
305: row = i + j*mx;
306: if (i == 0 || j == 0 || i == mx-1 || j == my-1) {
307: MatSetValues(jac,1,&row,1,&row,&one,INSERT_VALUES);
308: continue;
309: }
310: v[0] = hxdhy; col[0] = row - mx;
311: v[1] = hydhx; col[1] = row - 1;
312: v[2] = -two*(hydhx + hxdhy) + sc*lambda*PetscExpScalar(x[row]); col[2] = row;
313: v[3] = hydhx; col[3] = row + 1;
314: v[4] = hxdhy; col[4] = row + mx;
315: MatSetValues(jac,1,&row,5,col,v,INSERT_VALUES);
316: }
317: }
318: MatAssemblyBegin(jac,MAT_FINAL_ASSEMBLY);
319: VecRestoreArray(X,&x);
320: MatAssemblyEnd(jac,MAT_FINAL_ASSEMBLY);
321: *flag = SAME_NONZERO_PATTERN;
322: return 0;
323: }