1 2 static char help[] = "Solves the van der Pol DAE.\n\ 3 Input parameters include:\n"; 4 5 /* ------------------------------------------------------------------------ 6 7 This program solves the van der Pol DAE 8 y' = -z = f(y,z) (1) 9 0 = y-(z^3/3 - z) = g(y,z) 10 on the domain 0 <= x <= 1, with the boundary conditions 11 y(0) = -2, y'(0) = -2.355301397608119909925287735864250951918 12 This is a nonlinear equation. 13 14 Notes: 15 This code demonstrates the TS solver interface with the Van der Pol DAE, 16 namely it is the case when there is no RHS (meaning the RHS == 0), and the 17 equations are converted to two variants of linear problems, u_t = f(u,t), 18 namely turning (1) into a vector equation in terms of u, 19 20 [ y' + z ] = [ 0 ] 21 [ (z^3/3 - z) - y ] [ 0 ] 22 23 which then we can write as a vector equation 24 25 [ u_1' + u_2 ] = [ 0 ] (2) 26 [ (u_2^3/3 - u_2) - u_1 ] [ 0 ] 27 28 which is now in the desired form of u_t = f(u,t). As this is a DAE, and 29 there is no u_2', there is no need for a split, 30 31 so 32 33 [ F(u',u,t) ] = [ u_1' ] + [ u_2 ] 34 [ 0 ] [ (u_2^3/3 - u_2) - u_1 ] 35 36 Using the definition of the Jacobian of F (from the PETSc user manual), 37 in the equation F(u',u,t) = G(u,t), 38 39 dF dF 40 J(F) = a * -- - -- 41 du' du 42 43 where d is the partial derivative. In this example, 44 45 dF [ 1 ; 0 ] 46 -- = [ ] 47 du' [ 0 ; 0 ] 48 49 dF [ 0 ; 1 ] 50 -- = [ ] 51 du [ -1 ; 1 - u_2^2 ] 52 53 Hence, 54 55 [ a ; -1 ] 56 J(F) = [ ] 57 [ 1 ; u_2^2 - 1 ] 58 59 ------------------------------------------------------------------------- */ 60 61 #include <petscts.h> 62 63 typedef struct _n_User *User; 64 struct _n_User { 65 PetscReal next_output; 66 }; 67 68 /* 69 User-defined routines 70 */ 71 72 static PetscErrorCode IFunction(TS ts, PetscReal t, Vec X, Vec Xdot, Vec F, void *ctx) 73 { 74 PetscScalar *f; 75 const PetscScalar *x, *xdot; 76 77 PetscFunctionBeginUser; 78 PetscCall(VecGetArrayRead(X, &x)); 79 PetscCall(VecGetArrayRead(Xdot, &xdot)); 80 PetscCall(VecGetArray(F, &f)); 81 f[0] = xdot[0] + x[1]; 82 f[1] = (x[1] * x[1] * x[1] / 3.0 - x[1]) - x[0]; 83 PetscCall(VecRestoreArrayRead(X, &x)); 84 PetscCall(VecRestoreArrayRead(Xdot, &xdot)); 85 PetscCall(VecRestoreArray(F, &f)); 86 PetscFunctionReturn(PETSC_SUCCESS); 87 } 88 89 static PetscErrorCode IJacobian(TS ts, PetscReal t, Vec X, Vec Xdot, PetscReal a, Mat A, Mat B, void *ctx) 90 { 91 PetscInt rowcol[] = {0, 1}; 92 PetscScalar J[2][2]; 93 const PetscScalar *x; 94 95 PetscFunctionBeginUser; 96 PetscCall(VecGetArrayRead(X, &x)); 97 J[0][0] = a; 98 J[0][1] = -1.; 99 J[1][0] = 1.; 100 J[1][1] = -1. + x[1] * x[1]; 101 PetscCall(MatSetValues(B, 2, rowcol, 2, rowcol, &J[0][0], INSERT_VALUES)); 102 PetscCall(VecRestoreArrayRead(X, &x)); 103 104 PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY)); 105 PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY)); 106 if (A != B) { 107 PetscCall(MatAssemblyBegin(B, MAT_FINAL_ASSEMBLY)); 108 PetscCall(MatAssemblyEnd(B, MAT_FINAL_ASSEMBLY)); 109 } 110 PetscFunctionReturn(PETSC_SUCCESS); 111 } 112 113 static PetscErrorCode RegisterMyARK2(void) 114 { 115 PetscFunctionBeginUser; 116 { 117 const PetscReal A[3][3] = 118 { 119 {0, 0, 0}, 120 {0.41421356237309504880, 0, 0}, 121 {0.75, 0.25, 0} 122 }, 123 At[3][3] = {{0, 0, 0}, {0.12132034355964257320, 0.29289321881345247560, 0}, {0.20710678118654752440, 0.50000000000000000000, 0.29289321881345247560}}, *bembedt = NULL, *bembed = NULL; 124 PetscCall(TSARKIMEXRegister("myark2", 2, 3, &At[0][0], NULL, NULL, &A[0][0], NULL, NULL, bembedt, bembed, 0, NULL, NULL)); 125 } 126 PetscFunctionReturn(PETSC_SUCCESS); 127 } 128 129 /* Monitor timesteps and use interpolation to output at integer multiples of 0.1 */ 130 static PetscErrorCode Monitor(TS ts, PetscInt step, PetscReal t, Vec X, void *ctx) 131 { 132 const PetscScalar *x; 133 PetscReal tfinal, dt; 134 User user = (User)ctx; 135 Vec interpolatedX; 136 137 PetscFunctionBeginUser; 138 PetscCall(TSGetTimeStep(ts, &dt)); 139 PetscCall(TSGetMaxTime(ts, &tfinal)); 140 141 while (user->next_output <= t && user->next_output <= tfinal) { 142 PetscCall(VecDuplicate(X, &interpolatedX)); 143 PetscCall(TSInterpolate(ts, user->next_output, interpolatedX)); 144 PetscCall(VecGetArrayRead(interpolatedX, &x)); 145 PetscCall(PetscPrintf(PETSC_COMM_WORLD, "[%.1f] %3" 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]))); 146 PetscCall(VecRestoreArrayRead(interpolatedX, &x)); 147 PetscCall(VecDestroy(&interpolatedX)); 148 user->next_output += PetscRealConstant(0.1); 149 } 150 PetscFunctionReturn(PETSC_SUCCESS); 151 } 152 153 int main(int argc, char **argv) 154 { 155 TS ts; /* nonlinear solver */ 156 Vec x; /* solution, residual vectors */ 157 Mat A; /* Jacobian matrix */ 158 PetscInt steps; 159 PetscReal ftime = 0.5; 160 PetscBool monitor = PETSC_FALSE; 161 PetscScalar *x_ptr; 162 PetscMPIInt size; 163 struct _n_User user; 164 165 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 166 Initialize program 167 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 168 PetscFunctionBeginUser; 169 PetscCall(PetscInitialize(&argc, &argv, NULL, help)); 170 PetscCallMPI(MPI_Comm_size(PETSC_COMM_WORLD, &size)); 171 PetscCheck(size == 1, PETSC_COMM_WORLD, PETSC_ERR_WRONG_MPI_SIZE, "This is a uniprocessor example only!"); 172 173 PetscCall(RegisterMyARK2()); 174 175 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 176 Set runtime options 177 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 178 179 user.next_output = 0.0; 180 PetscCall(PetscOptionsGetBool(NULL, NULL, "-monitor", &monitor, NULL)); 181 182 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 183 Create necessary matrix and vectors, solve same ODE on every process 184 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 185 PetscCall(MatCreate(PETSC_COMM_WORLD, &A)); 186 PetscCall(MatSetSizes(A, PETSC_DECIDE, PETSC_DECIDE, 2, 2)); 187 PetscCall(MatSetFromOptions(A)); 188 PetscCall(MatSetUp(A)); 189 PetscCall(MatCreateVecs(A, &x, NULL)); 190 191 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 192 Create timestepping solver context 193 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 194 PetscCall(TSCreate(PETSC_COMM_WORLD, &ts)); 195 PetscCall(TSSetType(ts, TSBEULER)); 196 PetscCall(TSSetIFunction(ts, NULL, IFunction, &user)); 197 PetscCall(TSSetIJacobian(ts, A, A, IJacobian, &user)); 198 PetscCall(TSSetMaxTime(ts, ftime)); 199 PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER)); 200 if (monitor) PetscCall(TSMonitorSet(ts, Monitor, &user, NULL)); 201 202 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 203 Set initial conditions 204 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 205 PetscCall(VecGetArray(x, &x_ptr)); 206 x_ptr[0] = -2; 207 x_ptr[1] = -2.355301397608119909925287735864250951918; 208 PetscCall(VecRestoreArray(x, &x_ptr)); 209 PetscCall(TSSetTimeStep(ts, .001)); 210 211 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 212 Set runtime options 213 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 214 PetscCall(TSSetFromOptions(ts)); 215 216 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 217 Solve nonlinear system 218 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 219 PetscCall(TSSolve(ts, x)); 220 PetscCall(TSGetSolveTime(ts, &ftime)); 221 PetscCall(TSGetStepNumber(ts, &steps)); 222 PetscCall(PetscPrintf(PETSC_COMM_WORLD, "steps %3" PetscInt_FMT ", ftime %g\n", steps, (double)ftime)); 223 PetscCall(VecView(x, PETSC_VIEWER_STDOUT_WORLD)); 224 225 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 226 Free work space. All PETSc objects should be destroyed when they 227 are no longer needed. 228 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 229 PetscCall(MatDestroy(&A)); 230 PetscCall(VecDestroy(&x)); 231 PetscCall(TSDestroy(&ts)); 232 233 PetscCall(PetscFinalize()); 234 return 0; 235 } 236 237 /*TEST 238 239 test: 240 requires: !single 241 suffix: a 242 args: -monitor -ts_type bdf -ts_rtol 1e-6 -ts_atol 1e-6 -ts_view -ts_adapt_type dsp 243 output_file: output/ex19_pi42.out 244 245 test: 246 requires: !single 247 suffix: b 248 args: -monitor -ts_type bdf -ts_rtol 1e-6 -ts_atol 1e-6 -ts_view -ts_adapt_type dsp -ts_adapt_dsp_filter PI42 249 output_file: output/ex19_pi42.out 250 251 test: 252 requires: !single 253 suffix: c 254 args: -monitor -ts_type bdf -ts_rtol 1e-6 -ts_atol 1e-6 -ts_view -ts_adapt_type dsp -ts_adapt_dsp_pid 0.4,0.2 255 output_file: output/ex19_pi42.out 256 257 test: 258 requires: !single 259 suffix: bdf_reject 260 args: -ts_type bdf -ts_dt 0.5 -ts_max_steps 1 -ts_max_reject {{0 1 2}separate_output} -ts_error_if_step_fails false -ts_adapt_monitor 261 262 TEST*/ 263