Actual source code: ex22.c
1: static const char help[] = "Time-dependent advection-reaction PDE in 1d, demonstrates IMEX methods.\n";
2: /*
3: u_t + a1*u_x = -k1*u + k2*v + s1
4: v_t + a2*v_x = k1*u - k2*v + s2
5: 0 < x < 1;
6: a1 = 1, k1 = 10^6, s1 = 0,
7: a2 = 0, k2 = 2*k1, s2 = 1
9: Initial conditions:
10: u(x,0) = 1 + s2*x
11: v(x,0) = k0/k1*u(x,0) + s1/k1
13: Upstream boundary conditions:
14: u(0,t) = 1-sin(12*t)^4
16: Useful command-line parameters:
17: -ts_arkimex_fully_implicit - treats advection implicitly, only relevant with -ts_type arkimex (default)
18: -ts_type rosw - use Rosenbrock-W method (linearly implicit IMEX, amortizes assembly and preconditioner setup)
19: */
21: #include <petscdm.h>
22: #include <petscdmda.h>
23: #include <petscts.h>
25: typedef PetscScalar Field[2];
27: typedef struct _User *User;
28: struct _User {
29: PetscReal a[2]; /* Advection speeds */
30: PetscReal k[2]; /* Reaction coefficients */
31: PetscReal s[2]; /* Source terms */
32: };
34: static PetscErrorCode FormRHSFunction(TS, PetscReal, Vec, Vec, void *);
35: static PetscErrorCode FormIFunction(TS, PetscReal, Vec, Vec, Vec, void *);
36: static PetscErrorCode FormIJacobian(TS, PetscReal, Vec, Vec, PetscReal, Mat, Mat, void *);
37: static PetscErrorCode FormInitialSolution(TS, Vec, void *);
39: int main(int argc, char **argv)
40: {
41: TS ts; /* time integrator */
42: SNES snes; /* nonlinear solver */
43: SNESLineSearch linesearch; /* line search */
44: Vec X; /* solution, residual vectors */
45: Mat J; /* Jacobian matrix */
46: PetscInt steps, mx;
47: DM da;
48: PetscReal ftime, dt;
49: struct _User user; /* user-defined work context */
50: TSConvergedReason reason;
53: PetscInitialize(&argc, &argv, (char *)0, help);
55: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
56: Create distributed array (DMDA) to manage parallel grid and vectors
57: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
58: DMDACreate1d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, 11, 2, 2, NULL, &da);
59: DMSetFromOptions(da);
60: DMSetUp(da);
62: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
63: Extract global vectors from DMDA;
64: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
65: DMCreateGlobalVector(da, &X);
67: /* Initialize user application context */
68: PetscOptionsBegin(PETSC_COMM_WORLD, NULL, "Advection-reaction options", "");
69: {
70: user.a[0] = 1;
71: PetscOptionsReal("-a0", "Advection rate 0", "", user.a[0], &user.a[0], NULL);
72: user.a[1] = 0;
73: PetscOptionsReal("-a1", "Advection rate 1", "", user.a[1], &user.a[1], NULL);
74: user.k[0] = 1e6;
75: PetscOptionsReal("-k0", "Reaction rate 0", "", user.k[0], &user.k[0], NULL);
76: user.k[1] = 2 * user.k[0];
77: PetscOptionsReal("-k1", "Reaction rate 1", "", user.k[1], &user.k[1], NULL);
78: user.s[0] = 0;
79: PetscOptionsReal("-s0", "Source 0", "", user.s[0], &user.s[0], NULL);
80: user.s[1] = 1;
81: PetscOptionsReal("-s1", "Source 1", "", user.s[1], &user.s[1], NULL);
82: }
83: PetscOptionsEnd();
85: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
86: Create timestepping solver context
87: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
88: TSCreate(PETSC_COMM_WORLD, &ts);
89: TSSetDM(ts, da);
90: TSSetType(ts, TSARKIMEX);
91: TSSetRHSFunction(ts, NULL, FormRHSFunction, &user);
92: TSSetIFunction(ts, NULL, FormIFunction, &user);
93: DMSetMatType(da, MATAIJ);
94: DMCreateMatrix(da, &J);
95: TSSetIJacobian(ts, J, J, FormIJacobian, &user);
97: /* A line search in the nonlinear solve can fail due to ill-conditioning unless an absolute tolerance is set. Since
98: * this problem is linear, we deactivate the line search. For a linear problem, it is usually recommended to also use
99: * SNESSetType(snes,SNESKSPONLY). */
100: TSGetSNES(ts, &snes);
101: SNESGetLineSearch(snes, &linesearch);
102: SNESLineSearchSetType(linesearch, SNESLINESEARCHBASIC);
104: ftime = .1;
105: TSSetMaxTime(ts, ftime);
106: TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER);
108: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
109: Set initial conditions
110: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
111: FormInitialSolution(ts, X, &user);
112: TSSetSolution(ts, X);
113: VecGetSize(X, &mx);
114: dt = .1 * PetscMax(user.a[0], user.a[1]) / mx; /* Advective CFL, I don't know why it needs so much safety factor. */
115: TSSetTimeStep(ts, dt);
117: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
118: Set runtime options
119: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
120: TSSetFromOptions(ts);
122: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
123: Solve nonlinear system
124: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
125: TSSolve(ts, X);
126: TSGetSolveTime(ts, &ftime);
127: TSGetStepNumber(ts, &steps);
128: TSGetConvergedReason(ts, &reason);
129: PetscPrintf(PETSC_COMM_WORLD, "%s at time %g after %" PetscInt_FMT " steps\n", TSConvergedReasons[reason], (double)ftime, steps);
131: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
132: Free work space.
133: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
134: MatDestroy(&J);
135: VecDestroy(&X);
136: TSDestroy(&ts);
137: DMDestroy(&da);
138: PetscFinalize();
139: return 0;
140: }
142: static PetscErrorCode FormIFunction(TS ts, PetscReal t, Vec X, Vec Xdot, Vec F, void *ptr)
143: {
144: User user = (User)ptr;
145: DM da;
146: DMDALocalInfo info;
147: PetscInt i;
148: Field *f;
149: const Field *x, *xdot;
152: TSGetDM(ts, &da);
153: DMDAGetLocalInfo(da, &info);
155: /* Get pointers to vector data */
156: DMDAVecGetArrayRead(da, X, (void *)&x);
157: DMDAVecGetArrayRead(da, Xdot, (void *)&xdot);
158: DMDAVecGetArray(da, F, &f);
160: /* Compute function over the locally owned part of the grid */
161: for (i = info.xs; i < info.xs + info.xm; i++) {
162: f[i][0] = xdot[i][0] + user->k[0] * x[i][0] - user->k[1] * x[i][1] - user->s[0];
163: f[i][1] = xdot[i][1] - user->k[0] * x[i][0] + user->k[1] * x[i][1] - user->s[1];
164: }
166: /* Restore vectors */
167: DMDAVecRestoreArrayRead(da, X, (void *)&x);
168: DMDAVecRestoreArrayRead(da, Xdot, (void *)&xdot);
169: DMDAVecRestoreArray(da, F, &f);
170: return 0;
171: }
173: static PetscErrorCode FormRHSFunction(TS ts, PetscReal t, Vec X, Vec F, void *ptr)
174: {
175: User user = (User)ptr;
176: DM da;
177: Vec Xloc;
178: DMDALocalInfo info;
179: PetscInt i, j;
180: PetscReal hx;
181: Field *f;
182: const Field *x;
185: TSGetDM(ts, &da);
186: DMDAGetLocalInfo(da, &info);
187: hx = 1.0 / (PetscReal)info.mx;
189: /*
190: Scatter ghost points to local vector,using the 2-step process
191: DMGlobalToLocalBegin(),DMGlobalToLocalEnd().
192: By placing code between these two statements, computations can be
193: done while messages are in transition.
194: */
195: DMGetLocalVector(da, &Xloc);
196: DMGlobalToLocalBegin(da, X, INSERT_VALUES, Xloc);
197: DMGlobalToLocalEnd(da, X, INSERT_VALUES, Xloc);
199: /* Get pointers to vector data */
200: DMDAVecGetArrayRead(da, Xloc, (void *)&x);
201: DMDAVecGetArray(da, F, &f);
203: /* Compute function over the locally owned part of the grid */
204: for (i = info.xs; i < info.xs + info.xm; i++) {
205: const PetscReal *a = user->a;
206: PetscReal u0t[2];
207: u0t[0] = 1.0 - PetscPowRealInt(PetscSinReal(12 * t), 4);
208: u0t[1] = 0.0;
209: for (j = 0; j < 2; j++) {
210: if (i == 0) f[i][j] = a[j] / hx * (1. / 3 * u0t[j] + 0.5 * x[i][j] - x[i + 1][j] + 1. / 6 * x[i + 2][j]);
211: else if (i == 1) f[i][j] = a[j] / hx * (-1. / 12 * u0t[j] + 2. / 3 * x[i - 1][j] - 2. / 3 * x[i + 1][j] + 1. / 12 * x[i + 2][j]);
212: else if (i == info.mx - 2) f[i][j] = a[j] / hx * (-1. / 6 * x[i - 2][j] + x[i - 1][j] - 0.5 * x[i][j] - 1. / 3 * x[i + 1][j]);
213: else if (i == info.mx - 1) f[i][j] = a[j] / hx * (-x[i][j] + x[i - 1][j]);
214: else f[i][j] = a[j] / hx * (-1. / 12 * x[i - 2][j] + 2. / 3 * x[i - 1][j] - 2. / 3 * x[i + 1][j] + 1. / 12 * x[i + 2][j]);
215: }
216: }
218: /* Restore vectors */
219: DMDAVecRestoreArrayRead(da, Xloc, (void *)&x);
220: DMDAVecRestoreArray(da, F, &f);
221: DMRestoreLocalVector(da, &Xloc);
222: return 0;
223: }
225: /* --------------------------------------------------------------------- */
226: /*
227: IJacobian - Compute IJacobian = dF/dU + a dF/dUdot
228: */
229: PetscErrorCode FormIJacobian(TS ts, PetscReal t, Vec X, Vec Xdot, PetscReal a, Mat J, Mat Jpre, void *ptr)
230: {
231: User user = (User)ptr;
232: DMDALocalInfo info;
233: PetscInt i;
234: DM da;
235: const Field *x, *xdot;
238: TSGetDM(ts, &da);
239: DMDAGetLocalInfo(da, &info);
241: /* Get pointers to vector data */
242: DMDAVecGetArrayRead(da, X, (void *)&x);
243: DMDAVecGetArrayRead(da, Xdot, (void *)&xdot);
245: /* Compute function over the locally owned part of the grid */
246: for (i = info.xs; i < info.xs + info.xm; i++) {
247: const PetscReal *k = user->k;
248: PetscScalar v[2][2];
250: v[0][0] = a + k[0];
251: v[0][1] = -k[1];
252: v[1][0] = -k[0];
253: v[1][1] = a + k[1];
254: MatSetValuesBlocked(Jpre, 1, &i, 1, &i, &v[0][0], INSERT_VALUES);
255: }
257: /* Restore vectors */
258: DMDAVecRestoreArrayRead(da, X, (void *)&x);
259: DMDAVecRestoreArrayRead(da, Xdot, (void *)&xdot);
261: MatAssemblyBegin(Jpre, MAT_FINAL_ASSEMBLY);
262: MatAssemblyEnd(Jpre, MAT_FINAL_ASSEMBLY);
263: if (J != Jpre) {
264: MatAssemblyBegin(J, MAT_FINAL_ASSEMBLY);
265: MatAssemblyEnd(J, MAT_FINAL_ASSEMBLY);
266: }
267: return 0;
268: }
270: PetscErrorCode FormInitialSolution(TS ts, Vec X, void *ctx)
271: {
272: User user = (User)ctx;
273: DM da;
274: PetscInt i;
275: DMDALocalInfo info;
276: Field *x;
277: PetscReal hx;
280: TSGetDM(ts, &da);
281: DMDAGetLocalInfo(da, &info);
282: hx = 1.0 / (PetscReal)info.mx;
284: /* Get pointers to vector data */
285: DMDAVecGetArray(da, X, &x);
287: /* Compute function over the locally owned part of the grid */
288: for (i = info.xs; i < info.xs + info.xm; i++) {
289: PetscReal r = (i + 1) * hx, ik = user->k[1] != 0.0 ? 1.0 / user->k[1] : 1.0;
290: x[i][0] = 1 + user->s[1] * r;
291: x[i][1] = user->k[0] * ik * x[i][0] + user->s[1] * ik;
292: }
293: DMDAVecRestoreArray(da, X, &x);
294: return 0;
295: }
297: /*TEST
299: test:
300: args: -nox -da_grid_x 200 -ts_monitor_draw_solution -ts_arkimex_type 4 -ts_adapt_type none -ts_dt .005 -ts_max_time .1
301: requires: !single
303: test:
304: suffix: 2
305: args: -nox -da_grid_x 200 -ts_monitor_draw_solution -ts_type rosw -ts_dt 1e-3 -ts_adapt_type none -ts_dt .005 -ts_max_time .1
306: nsize: 2
308: test:
309: suffix: 3
310: args: -nox -da_grid_x 200 -ts_monitor_draw_solution -ts_type rosw -ts_rosw_type ra34pw2 -ts_dt 5e-3 -ts_max_time .1 -ts_adapt_type none
311: nsize: 2
313: test:
314: suffix: 4
315: args: -ts_type eimex -da_grid_x 200 -ts_eimex_order_adapt true -ts_dt 0.001 -ts_monitor -ts_max_steps 100
316: filter: sed "s/ITS/TIME/g"
317: nsize: 2
319: TEST*/