Actual source code: ex74.c

```  1: static char help[] = "Solves the constant-coefficient 1D heat equation \n\
2: with an Implicit Runge-Kutta method using MatKAIJ.                  \n\
3:                                                                     \n\
4:     du      d^2 u                                                   \n\
5:     --  = a ----- ; 0 <= x <= 1;                                    \n\
6:     dt      dx^2                                                    \n\
7:                                                                     \n\
8:   with periodic boundary conditions                                 \n\
9:                                                                     \n\
10: 2nd order central discretization in space:                          \n\
11:                                                                     \n\
12:    [ d^2 u ]     u_{i+1} - 2u_i + u_{i-1}                           \n\
13:    [ ----- ]  =  ------------------------                           \n\
14:    [ dx^2  ]i              h^2                                      \n\
15:                                                                     \n\
16:     i = grid index;    h = x_{i+1}-x_i (Uniform)                    \n\
17:     0 <= i < n         h = 1.0/n                                    \n\
18:                                                                     \n\
19: Thus,                                                               \n\
20:                                                                     \n\
21:    du                                                               \n\
22:    --  = Ju;  J = (a/h^2) tridiagonal(1,-2,1)_n                     \n\
23:    dt                                                               \n\
24:                                                                     \n\
25: This example is a TS version of the KSP ex74.c tutorial.            \n";

27: #include <petscts.h>

29: typedef enum {
30:   PHYSICS_DIFFUSION,
32: } PhysicsType;
33: const char *const PhysicsTypes[] = {"DIFFUSION", "ADVECTION", "PhysicsType", "PHYSICS_", NULL};

35: typedef struct Context {
36:   PetscReal   a;          /* diffusion coefficient      */
37:   PetscReal   xmin, xmax; /* domain bounds              */
38:   PetscInt    imax;       /* number of grid points      */
39:   PhysicsType physics_type;
40: } UserContext;

42: static PetscErrorCode ExactSolution(Vec, void *, PetscReal);
43: static PetscErrorCode RHSJacobian(TS, PetscReal, Vec, Mat, Mat, void *);

45: int main(int argc, char **argv)
46: {
47:   TS          ts;
48:   Mat         A;
49:   Vec         u, uex;
50:   UserContext ctxt;
51:   PetscReal   err, ftime;

53:   PetscFunctionBeginUser;
54:   PetscCall(PetscInitialize(&argc, &argv, (char *)0, help));
55:   /* default value */
56:   ctxt.a            = 0.1;
57:   ctxt.xmin         = 0.0;
58:   ctxt.xmax         = 1.0;
59:   ctxt.imax         = 40;
60:   ctxt.physics_type = PHYSICS_DIFFUSION;

62:   PetscOptionsBegin(PETSC_COMM_WORLD, NULL, "IRK options", "");
63:   PetscCall(PetscOptionsReal("-a", "diffusion coefficient", "<1.0>", ctxt.a, &ctxt.a, NULL));
64:   PetscCall(PetscOptionsInt("-imax", "grid size", "<20>", ctxt.imax, &ctxt.imax, NULL));
65:   PetscCall(PetscOptionsReal("-xmin", "xmin", "<0.0>", ctxt.xmin, &ctxt.xmin, NULL));
66:   PetscCall(PetscOptionsReal("-xmax", "xmax", "<1.0>", ctxt.xmax, &ctxt.xmax, NULL));
67:   PetscCall(PetscOptionsEnum("-physics_type", "Type of process to discretize", "", PhysicsTypes, (PetscEnum)ctxt.physics_type, (PetscEnum *)&ctxt.physics_type, NULL));
68:   PetscOptionsEnd();

70:   /* allocate and initialize solution vector and exact solution */
71:   PetscCall(VecCreate(PETSC_COMM_WORLD, &u));
72:   PetscCall(VecSetSizes(u, PETSC_DECIDE, ctxt.imax));
73:   PetscCall(VecSetFromOptions(u));
74:   PetscCall(VecDuplicate(u, &uex));
75:   /* initial solution */
76:   PetscCall(ExactSolution(u, &ctxt, 0.0));

78:   PetscCall(MatCreate(PETSC_COMM_WORLD, &A));
79:   PetscCall(MatSetType(A, MATAIJ));
80:   PetscCall(MatSetSizes(A, PETSC_DECIDE, PETSC_DECIDE, ctxt.imax, ctxt.imax));
81:   PetscCall(MatSetUp(A));

83:   /* Create and set options for TS */
84:   PetscCall(TSCreate(PETSC_COMM_WORLD, &ts));
85:   PetscCall(TSSetProblemType(ts, TS_LINEAR));
86:   PetscCall(TSSetTimeStep(ts, 0.125));
87:   PetscCall(TSSetSolution(ts, u));
88:   PetscCall(TSSetMaxSteps(ts, 10));
89:   PetscCall(TSSetMaxTime(ts, 1.0));
90:   PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER));
91:   PetscCall(TSSetRHSFunction(ts, NULL, TSComputeRHSFunctionLinear, &ctxt));
92:   PetscCall(RHSJacobian(ts, 0, u, A, A, &ctxt));
93:   PetscCall(TSSetRHSJacobian(ts, A, A, TSComputeRHSJacobianConstant, &ctxt));
94:   PetscCall(TSSetFromOptions(ts));
95:   PetscCall(TSSolve(ts, u));

97:   PetscCall(TSGetSolveTime(ts, &ftime));
98:   /* exact   solution */
99:   PetscCall(ExactSolution(uex, &ctxt, ftime));

101:   /* Calculate error in final solution */
102:   PetscCall(VecAYPX(uex, -1.0, u));
103:   PetscCall(VecNorm(uex, NORM_2, &err));
104:   err = PetscSqrtReal(err * err / ((PetscReal)ctxt.imax));
105:   PetscCall(PetscPrintf(PETSC_COMM_WORLD, "L2 norm of the numerical error = %g (time=%g)\n", (double)err, (double)ftime));

107:   /* Free up memory */
108:   PetscCall(TSDestroy(&ts));
109:   PetscCall(MatDestroy(&A));
110:   PetscCall(VecDestroy(&uex));
111:   PetscCall(VecDestroy(&u));
112:   PetscCall(PetscFinalize());
113:   return 0;
114: }

116: PetscErrorCode ExactSolution(Vec u, void *c, PetscReal t)
117: {
118:   UserContext *ctxt = (UserContext *)c;
119:   PetscInt     i, is, ie;
120:   PetscScalar *uarr;
121:   PetscReal    x, dx, a = ctxt->a, pi = PETSC_PI;

123:   PetscFunctionBeginUser;
124:   dx = (ctxt->xmax - ctxt->xmin) / ((PetscReal)ctxt->imax);
125:   PetscCall(VecGetOwnershipRange(u, &is, &ie));
126:   PetscCall(VecGetArray(u, &uarr));
127:   for (i = is; i < ie; i++) {
128:     x = i * dx;
129:     switch (ctxt->physics_type) {
130:     case PHYSICS_DIFFUSION:
131:       uarr[i - is] = PetscExpScalar(-4.0 * pi * pi * a * t) * PetscSinScalar(2 * pi * x);
132:       break;
134:       uarr[i - is] = PetscSinScalar(2 * pi * (x - a * t));
135:       break;
136:     default:
137:       SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "No support for physics type %s", PhysicsTypes[ctxt->physics_type]);
138:     }
139:   }
140:   PetscCall(VecRestoreArray(u, &uarr));
141:   PetscFunctionReturn(PETSC_SUCCESS);
142: }

144: static PetscErrorCode RHSJacobian(TS ts, PetscReal t, Vec U, Mat J, Mat Jpre, void *ctx)
145: {
146:   UserContext *user = (UserContext *)ctx;
147:   PetscInt     matis, matie, i;
148:   PetscReal    dx, dx2;

150:   PetscFunctionBeginUser;
151:   dx  = (user->xmax - user->xmin) / ((PetscReal)user->imax);
152:   dx2 = dx * dx;
153:   PetscCall(MatGetOwnershipRange(J, &matis, &matie));
154:   for (i = matis; i < matie; i++) {
155:     PetscScalar values[3];
156:     PetscInt    col[3];
157:     switch (user->physics_type) {
158:     case PHYSICS_DIFFUSION:
159:       values[0] = user->a * 1.0 / dx2;
160:       values[1] = -user->a * 2.0 / dx2;
161:       values[2] = user->a * 1.0 / dx2;
162:       break;
164:       values[0] = user->a * .5 / dx;
165:       values[1] = 0.;
166:       values[2] = -user->a * .5 / dx;
167:       break;
168:     default:
169:       SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "No support for physics type %s", PhysicsTypes[user->physics_type]);
170:     }
171:     /* periodic boundaries */
172:     if (i == 0) {
173:       col[0] = user->imax - 1;
174:       col[1] = i;
175:       col[2] = i + 1;
176:     } else if (i == user->imax - 1) {
177:       col[0] = i - 1;
178:       col[1] = i;
179:       col[2] = 0;
180:     } else {
181:       col[0] = i - 1;
182:       col[1] = i;
183:       col[2] = i + 1;
184:     }
185:     PetscCall(MatSetValues(J, 1, &i, 3, col, values, INSERT_VALUES));
186:   }
187:   PetscCall(MatAssemblyBegin(J, MAT_FINAL_ASSEMBLY));
188:   PetscCall(MatAssemblyEnd(J, MAT_FINAL_ASSEMBLY));
189:   PetscFunctionReturn(PETSC_SUCCESS);
190: }

192: /*TEST

194:   test:
195:     requires: double
196:     suffix: 1
197:     nsize: {{1 2}}
198:     args: -ts_max_steps 5 -ts_monitor -ksp_monitor_short -pc_type pbjacobi -ksp_atol 1e-6 -ts_type irk -ts_irk_nstages 2

200:   test:
201:     requires: double
202:     suffix: 2
203:     args: -ts_max_steps 5 -ts_monitor -ksp_monitor_short -pc_type pbjacobi -ksp_atol 1e-6 -ts_type irk -ts_irk_nstages 3

205:   testset:
206:     requires: hpddm
207:     args: -ts_max_steps 5 -ts_monitor -ksp_monitor_short -pc_type pbjacobi -ksp_atol 1e-4 -ts_type irk -ts_irk_nstages 3 -ksp_view_final_residual -ksp_hpddm_type gcrodr -ksp_type hpddm
208:     test:
209:       suffix: 3
210:       requires: double
211:       args: -ksp_hpddm_precision {{single double}shared output}
212:     test:
213:       suffix: 3_single
214:       requires: single
215:       args: -ksp_hpddm_precision {{single double}shared output}
216:     test:
217:       suffix: 3___float128
218:       requires: __float128
219:       output_file: output/ex74_3.out
220:       args: -ksp_hpddm_precision {{double quadruple}shared output}
221: TEST*/
```