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