1 2 static char help[] = "Solves the van der Pol equation and demonstrate IMEX.\n\ 3 Input parameters include:\n\ 4 -mu : stiffness parameter\n\n"; 5 6 /* ------------------------------------------------------------------------ 7 8 This program solves the van der Pol equation 9 y'' - \mu ((1-y^2)*y' - y) = 0 (1) 10 on the domain 0 <= x <= 1, with the boundary conditions 11 y(0) = 2, y'(0) = - 2/3 +10/(81*\mu) - 292/(2187*\mu^2), 12 This is a nonlinear equation. The well prepared initial condition gives errors that are not dominated by the first few steps of the method when \mu is large. 13 14 Notes: 15 This code demonstrates the TS solver interface to two variants of 16 linear problems, u_t = f(u,t), namely turning (1) into a system of 17 first order differential equations, 18 19 [ y' ] = [ z ] 20 [ z' ] [ \mu ((1 - y^2) z - y) ] 21 22 which then we can write as a vector equation 23 24 [ u_1' ] = [ u_2 ] (2) 25 [ u_2' ] [ \mu (1 - u_1^2) u_2 - u_1 ] 26 27 which is now in the desired form of u_t = f(u,t). One way that we 28 can split f(u,t) in (2) is to split by component, 29 30 [ u_1' ] = [ u_2 ] + [ 0 ] 31 [ u_2' ] [ 0 ] [ \mu ((1 - u_1^2) u_2 - u_1) ] 32 33 where 34 35 [ G(u,t) ] = [ u_2 ] 36 [ 0 ] 37 38 and 39 40 [ F(u',u,t) ] = [ u_1' ] - [ 0 ] 41 [ u_2' ] [ \mu ((1 - u_1^2) u_2 - u_1) ] 42 43 Using the definition of the Jacobian of F (from the PETSc user manual), 44 in the equation F(u',u,t) = G(u,t), 45 46 dF dF 47 J(F) = a * -- - -- 48 du' du 49 50 where d is the partial derivative. In this example, 51 52 dF [ 1 ; 0 ] 53 -- = [ ] 54 du' [ 0 ; 1 ] 55 56 dF [ 0 ; 0 ] 57 -- = [ ] 58 du [ -\mu (2*u_1*u_2 + 1); \mu (1 - u_1^2) ] 59 60 Hence, 61 62 [ a ; 0 ] 63 J(F) = [ ] 64 [ \mu (2*u_1*u_2 + 1); a - \mu (1 - u_1^2) ] 65 66 ------------------------------------------------------------------------- */ 67 68 #include <petscts.h> 69 70 typedef struct _n_User *User; 71 struct _n_User { 72 PetscReal mu; 73 PetscBool imex; 74 PetscReal next_output; 75 }; 76 77 /* 78 User-defined routines 79 */ 80 static PetscErrorCode RHSFunction(TS ts, PetscReal t, Vec X, Vec F, void *ctx) 81 { 82 User user = (User)ctx; 83 PetscScalar *f; 84 const PetscScalar *x; 85 86 PetscFunctionBeginUser; 87 PetscCall(VecGetArrayRead(X, &x)); 88 PetscCall(VecGetArray(F, &f)); 89 f[0] = (user->imex ? x[1] : 0); 90 f[1] = 0.0; 91 PetscCall(VecRestoreArrayRead(X, &x)); 92 PetscCall(VecRestoreArray(F, &f)); 93 PetscFunctionReturn(PETSC_SUCCESS); 94 } 95 96 static PetscErrorCode IFunction(TS ts, PetscReal t, Vec X, Vec Xdot, Vec F, void *ctx) 97 { 98 User user = (User)ctx; 99 const PetscScalar *x, *xdot; 100 PetscScalar *f; 101 102 PetscFunctionBeginUser; 103 PetscCall(VecGetArrayRead(X, &x)); 104 PetscCall(VecGetArrayRead(Xdot, &xdot)); 105 PetscCall(VecGetArray(F, &f)); 106 f[0] = xdot[0] + (user->imex ? 0 : x[1]); 107 f[1] = xdot[1] - user->mu * ((1. - x[0] * x[0]) * x[1] - x[0]); 108 PetscCall(VecRestoreArrayRead(X, &x)); 109 PetscCall(VecRestoreArrayRead(Xdot, &xdot)); 110 PetscCall(VecRestoreArray(F, &f)); 111 PetscFunctionReturn(PETSC_SUCCESS); 112 } 113 114 static PetscErrorCode IJacobian(TS ts, PetscReal t, Vec X, Vec Xdot, PetscReal a, Mat A, Mat B, void *ctx) 115 { 116 User user = (User)ctx; 117 PetscReal mu = user->mu; 118 PetscInt rowcol[] = {0, 1}; 119 const PetscScalar *x; 120 PetscScalar J[2][2]; 121 122 PetscFunctionBeginUser; 123 PetscCall(VecGetArrayRead(X, &x)); 124 J[0][0] = a; 125 J[0][1] = (user->imex ? 0 : 1.); 126 J[1][0] = mu * (2. * x[0] * x[1] + 1.); 127 J[1][1] = a - mu * (1. - x[0] * x[0]); 128 PetscCall(MatSetValues(B, 2, rowcol, 2, rowcol, &J[0][0], INSERT_VALUES)); 129 PetscCall(VecRestoreArrayRead(X, &x)); 130 131 PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY)); 132 PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY)); 133 if (A != B) { 134 PetscCall(MatAssemblyBegin(B, MAT_FINAL_ASSEMBLY)); 135 PetscCall(MatAssemblyEnd(B, MAT_FINAL_ASSEMBLY)); 136 } 137 PetscFunctionReturn(PETSC_SUCCESS); 138 } 139 140 static PetscErrorCode RegisterMyARK2(void) 141 { 142 PetscFunctionBeginUser; 143 { 144 const PetscReal A[3][3] = 145 { 146 {0, 0, 0}, 147 {0.41421356237309504880, 0, 0}, 148 {0.75, 0.25, 0} 149 }, 150 At[3][3] = {{0, 0, 0}, {0.12132034355964257320, 0.29289321881345247560, 0}, {0.20710678118654752440, 0.50000000000000000000, 0.29289321881345247560}}, *bembedt = NULL, *bembed = NULL; 151 PetscCall(TSARKIMEXRegister("myark2", 2, 3, &At[0][0], NULL, NULL, &A[0][0], NULL, NULL, bembedt, bembed, 0, NULL, NULL)); 152 } 153 PetscFunctionReturn(PETSC_SUCCESS); 154 } 155 156 /* Monitor timesteps and use interpolation to output at integer multiples of 0.1 */ 157 static PetscErrorCode Monitor(TS ts, PetscInt step, PetscReal t, Vec X, void *ctx) 158 { 159 const PetscScalar *x; 160 PetscReal tfinal, dt; 161 User user = (User)ctx; 162 Vec interpolatedX; 163 164 PetscFunctionBeginUser; 165 PetscCall(TSGetTimeStep(ts, &dt)); 166 PetscCall(TSGetMaxTime(ts, &tfinal)); 167 168 while (user->next_output <= t && user->next_output <= tfinal) { 169 PetscCall(VecDuplicate(X, &interpolatedX)); 170 PetscCall(TSInterpolate(ts, user->next_output, interpolatedX)); 171 PetscCall(VecGetArrayRead(interpolatedX, &x)); 172 PetscCall(PetscPrintf(PETSC_COMM_WORLD, "[%.1f] %" PetscInt_FMT " 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]))); 173 PetscCall(VecRestoreArrayRead(interpolatedX, &x)); 174 PetscCall(VecDestroy(&interpolatedX)); 175 176 user->next_output += 0.1; 177 } 178 PetscFunctionReturn(PETSC_SUCCESS); 179 } 180 181 int main(int argc, char **argv) 182 { 183 TS ts; /* nonlinear solver */ 184 Vec x; /* solution, residual vectors */ 185 Mat A; /* Jacobian matrix */ 186 PetscInt steps; 187 PetscReal ftime = 0.5; 188 PetscBool monitor = PETSC_FALSE; 189 PetscScalar *x_ptr; 190 PetscMPIInt size; 191 struct _n_User user; 192 193 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 194 Initialize program 195 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 196 PetscFunctionBeginUser; 197 PetscCall(PetscInitialize(&argc, &argv, NULL, help)); 198 PetscCallMPI(MPI_Comm_size(PETSC_COMM_WORLD, &size)); 199 PetscCheck(size == 1, PETSC_COMM_WORLD, PETSC_ERR_WRONG_MPI_SIZE, "This is a uniprocessor example only!"); 200 201 PetscCall(RegisterMyARK2()); 202 203 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 204 Set runtime options 205 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 206 user.mu = 1000.0; 207 user.imex = PETSC_TRUE; 208 user.next_output = 0.0; 209 210 PetscCall(PetscOptionsGetReal(NULL, NULL, "-mu", &user.mu, NULL)); 211 PetscCall(PetscOptionsGetBool(NULL, NULL, "-imex", &user.imex, NULL)); 212 PetscCall(PetscOptionsGetBool(NULL, NULL, "-monitor", &monitor, NULL)); 213 214 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 215 Create necessary matrix and vectors, solve same ODE on every process 216 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 217 PetscCall(MatCreate(PETSC_COMM_WORLD, &A)); 218 PetscCall(MatSetSizes(A, PETSC_DECIDE, PETSC_DECIDE, 2, 2)); 219 PetscCall(MatSetFromOptions(A)); 220 PetscCall(MatSetUp(A)); 221 PetscCall(MatCreateVecs(A, &x, NULL)); 222 223 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 224 Create timestepping solver context 225 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 226 PetscCall(TSCreate(PETSC_COMM_WORLD, &ts)); 227 PetscCall(TSSetType(ts, TSBEULER)); 228 PetscCall(TSSetRHSFunction(ts, NULL, RHSFunction, &user)); 229 PetscCall(TSSetIFunction(ts, NULL, IFunction, &user)); 230 PetscCall(TSSetIJacobian(ts, A, A, IJacobian, &user)); 231 PetscCall(TSSetMaxTime(ts, ftime)); 232 PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER)); 233 if (monitor) PetscCall(TSMonitorSet(ts, Monitor, &user, NULL)); 234 235 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 236 Set initial conditions 237 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 238 PetscCall(VecGetArray(x, &x_ptr)); 239 x_ptr[0] = 2.0; 240 x_ptr[1] = -2.0 / 3.0 + 10.0 / (81.0 * user.mu) - 292.0 / (2187.0 * user.mu * user.mu); 241 PetscCall(VecRestoreArray(x, &x_ptr)); 242 PetscCall(TSSetTimeStep(ts, 0.01)); 243 244 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 245 Set runtime options 246 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 247 PetscCall(TSSetFromOptions(ts)); 248 249 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 250 Solve nonlinear system 251 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 252 PetscCall(TSSolve(ts, x)); 253 PetscCall(TSGetSolveTime(ts, &ftime)); 254 PetscCall(TSGetStepNumber(ts, &steps)); 255 PetscCall(PetscPrintf(PETSC_COMM_WORLD, "mu %g, steps %" PetscInt_FMT ", ftime %g\n", (double)user.mu, steps, (double)ftime)); 256 PetscCall(VecView(x, PETSC_VIEWER_STDOUT_WORLD)); 257 258 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 259 Free work space. All PETSc objects should be destroyed when they 260 are no longer needed. 261 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 262 PetscCall(MatDestroy(&A)); 263 PetscCall(VecDestroy(&x)); 264 PetscCall(TSDestroy(&ts)); 265 266 PetscCall(PetscFinalize()); 267 return 0; 268 } 269 270 /*TEST 271 272 test: 273 args: -ts_type arkimex -ts_arkimex_type myark2 -ts_adapt_type none 274 requires: !single 275 276 TEST*/ 277