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 PetscScalar *f; 74 const PetscScalar *x, *xdot; 75 76 PetscFunctionBeginUser; 77 PetscCall(VecGetArrayRead(X, &x)); 78 PetscCall(VecGetArrayRead(Xdot, &xdot)); 79 PetscCall(VecGetArray(F, &f)); 80 f[0] = xdot[0] + x[1]; 81 f[1] = (x[1] * x[1] * x[1] / 3.0 - x[1]) - x[0]; 82 PetscCall(VecRestoreArrayRead(X, &x)); 83 PetscCall(VecRestoreArrayRead(Xdot, &xdot)); 84 PetscCall(VecRestoreArray(F, &f)); 85 PetscFunctionReturn(0); 86 } 87 88 static PetscErrorCode IJacobian(TS ts, PetscReal t, Vec X, Vec Xdot, PetscReal a, Mat A, Mat B, void *ctx) { 89 PetscInt rowcol[] = {0, 1}; 90 PetscScalar J[2][2]; 91 const PetscScalar *x; 92 93 PetscFunctionBeginUser; 94 PetscCall(VecGetArrayRead(X, &x)); 95 J[0][0] = a; 96 J[0][1] = -1.; 97 J[1][0] = 1.; 98 J[1][1] = -1. + x[1] * x[1]; 99 PetscCall(MatSetValues(B, 2, rowcol, 2, rowcol, &J[0][0], INSERT_VALUES)); 100 PetscCall(VecRestoreArrayRead(X, &x)); 101 102 PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY)); 103 PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY)); 104 if (A != B) { 105 PetscCall(MatAssemblyBegin(B, MAT_FINAL_ASSEMBLY)); 106 PetscCall(MatAssemblyEnd(B, MAT_FINAL_ASSEMBLY)); 107 } 108 PetscFunctionReturn(0); 109 } 110 111 static PetscErrorCode RegisterMyARK2(void) { 112 PetscFunctionBeginUser; 113 { 114 const PetscReal A[3][3] = 115 { 116 {0, 0, 0}, 117 {0.41421356237309504880, 0, 0}, 118 {0.75, 0.25, 0} 119 }, 120 At[3][3] = {{0, 0, 0}, {0.12132034355964257320, 0.29289321881345247560, 0}, {0.20710678118654752440, 0.50000000000000000000, 0.29289321881345247560}}, *bembedt = NULL, *bembed = NULL; 121 PetscCall(TSARKIMEXRegister("myark2", 2, 3, &At[0][0], NULL, NULL, &A[0][0], NULL, NULL, bembedt, bembed, 0, NULL, NULL)); 122 } 123 PetscFunctionReturn(0); 124 } 125 126 /* Monitor timesteps and use interpolation to output at integer multiples of 0.1 */ 127 static PetscErrorCode Monitor(TS ts, PetscInt step, PetscReal t, Vec X, void *ctx) { 128 const PetscScalar *x; 129 PetscReal tfinal, dt; 130 User user = (User)ctx; 131 Vec interpolatedX; 132 133 PetscFunctionBeginUser; 134 PetscCall(TSGetTimeStep(ts, &dt)); 135 PetscCall(TSGetMaxTime(ts, &tfinal)); 136 137 while (user->next_output <= t && user->next_output <= tfinal) { 138 PetscCall(VecDuplicate(X, &interpolatedX)); 139 PetscCall(TSInterpolate(ts, user->next_output, interpolatedX)); 140 PetscCall(VecGetArrayRead(interpolatedX, &x)); 141 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]))); 142 PetscCall(VecRestoreArrayRead(interpolatedX, &x)); 143 PetscCall(VecDestroy(&interpolatedX)); 144 user->next_output += PetscRealConstant(0.1); 145 } 146 PetscFunctionReturn(0); 147 } 148 149 int main(int argc, char **argv) { 150 TS ts; /* nonlinear solver */ 151 Vec x; /* solution, residual vectors */ 152 Mat A; /* Jacobian matrix */ 153 PetscInt steps; 154 PetscReal ftime = 0.5; 155 PetscBool monitor = PETSC_FALSE; 156 PetscScalar *x_ptr; 157 PetscMPIInt size; 158 struct _n_User user; 159 160 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 161 Initialize program 162 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 163 PetscFunctionBeginUser; 164 PetscCall(PetscInitialize(&argc, &argv, NULL, help)); 165 PetscCallMPI(MPI_Comm_size(PETSC_COMM_WORLD, &size)); 166 PetscCheck(size == 1, PETSC_COMM_WORLD, PETSC_ERR_WRONG_MPI_SIZE, "This is a uniprocessor example only!"); 167 168 PetscCall(RegisterMyARK2()); 169 170 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 171 Set runtime options 172 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 173 174 user.next_output = 0.0; 175 PetscCall(PetscOptionsGetBool(NULL, NULL, "-monitor", &monitor, NULL)); 176 177 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 178 Create necessary matrix and vectors, solve same ODE on every process 179 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 180 PetscCall(MatCreate(PETSC_COMM_WORLD, &A)); 181 PetscCall(MatSetSizes(A, PETSC_DECIDE, PETSC_DECIDE, 2, 2)); 182 PetscCall(MatSetFromOptions(A)); 183 PetscCall(MatSetUp(A)); 184 PetscCall(MatCreateVecs(A, &x, NULL)); 185 186 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 187 Create timestepping solver context 188 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 189 PetscCall(TSCreate(PETSC_COMM_WORLD, &ts)); 190 PetscCall(TSSetType(ts, TSBEULER)); 191 PetscCall(TSSetIFunction(ts, NULL, IFunction, &user)); 192 PetscCall(TSSetIJacobian(ts, A, A, IJacobian, &user)); 193 PetscCall(TSSetMaxTime(ts, ftime)); 194 PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER)); 195 if (monitor) { PetscCall(TSMonitorSet(ts, Monitor, &user, NULL)); } 196 197 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 198 Set initial conditions 199 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 200 PetscCall(VecGetArray(x, &x_ptr)); 201 x_ptr[0] = -2; 202 x_ptr[1] = -2.355301397608119909925287735864250951918; 203 PetscCall(VecRestoreArray(x, &x_ptr)); 204 PetscCall(TSSetTimeStep(ts, .001)); 205 206 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 207 Set runtime options 208 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 209 PetscCall(TSSetFromOptions(ts)); 210 211 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 212 Solve nonlinear system 213 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 214 PetscCall(TSSolve(ts, x)); 215 PetscCall(TSGetSolveTime(ts, &ftime)); 216 PetscCall(TSGetStepNumber(ts, &steps)); 217 PetscCall(PetscPrintf(PETSC_COMM_WORLD, "steps %3" PetscInt_FMT ", ftime %g\n", steps, (double)ftime)); 218 PetscCall(VecView(x, PETSC_VIEWER_STDOUT_WORLD)); 219 220 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 221 Free work space. All PETSc objects should be destroyed when they 222 are no longer needed. 223 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 224 PetscCall(MatDestroy(&A)); 225 PetscCall(VecDestroy(&x)); 226 PetscCall(TSDestroy(&ts)); 227 228 PetscCall(PetscFinalize()); 229 return 0; 230 } 231 232 /*TEST 233 234 test: 235 requires: !single 236 suffix: a 237 args: -monitor -ts_type bdf -ts_rtol 1e-6 -ts_atol 1e-6 -ts_view -ts_adapt_type dsp 238 output_file: output/ex19_pi42.out 239 240 test: 241 requires: !single 242 suffix: b 243 args: -monitor -ts_type bdf -ts_rtol 1e-6 -ts_atol 1e-6 -ts_view -ts_adapt_type dsp -ts_adapt_dsp_filter PI42 244 output_file: output/ex19_pi42.out 245 246 test: 247 requires: !single 248 suffix: c 249 args: -monitor -ts_type bdf -ts_rtol 1e-6 -ts_atol 1e-6 -ts_view -ts_adapt_type dsp -ts_adapt_dsp_pid 0.4,0.2 250 output_file: output/ex19_pi42.out 251 252 test: 253 requires: !single 254 suffix: bdf_reject 255 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 256 257 TEST*/ 258