static char help[] = "Solves the van der Pol DAE.\n\ Input parameters include:\n"; /* ------------------------------------------------------------------------ This program solves the van der Pol DAE y' = -z = f(y,z) (1) 0 = y-(z^3/3 - z) = g(y,z) on the domain 0 <= x <= 1, with the boundary conditions y(0) = -2, y'(0) = -2.355301397608119909925287735864250951918 This is a nonlinear equation. Notes: This code demonstrates the TS solver interface with the Van der Pol DAE, namely it is the case when there is no RHS (meaning the RHS == 0), and the equations are converted to two variants of linear problems, u_t = f(u,t), namely turning (1) into a vector equation in terms of u, [ y' + z ] = [ 0 ] [ (z^3/3 - z) - y ] [ 0 ] which then we can write as a vector equation [ u_1' + u_2 ] = [ 0 ] (2) [ (u_2^3/3 - u_2) - u_1 ] [ 0 ] which is now in the desired form of u_t = f(u,t). As this is a DAE, and there is no u_2', there is no need for a split, so [ F(u',u,t) ] = [ u_1' ] + [ u_2 ] [ 0 ] [ (u_2^3/3 - u_2) - u_1 ] Using the definition of the Jacobian of F (from the PETSc user manual), in the equation F(u',u,t) = G(u,t), dF dF J(F) = a * -- - -- du' du where d is the partial derivative. In this example, dF [ 1 ; 0 ] -- = [ ] du' [ 0 ; 0 ] dF [ 0 ; 1 ] -- = [ ] du [ -1 ; 1 - u_2^2 ] Hence, [ a ; -1 ] J(F) = [ ] [ 1 ; u_2^2 - 1 ] ------------------------------------------------------------------------- */ #include typedef struct _n_User *User; struct _n_User { PetscReal next_output; }; /* User-defined routines */ static PetscErrorCode IFunction(TS ts, PetscReal t, Vec X, Vec Xdot, Vec F, void *ctx) { PetscScalar *f; const PetscScalar *x, *xdot; PetscFunctionBeginUser; PetscCall(VecGetArrayRead(X, &x)); PetscCall(VecGetArrayRead(Xdot, &xdot)); PetscCall(VecGetArray(F, &f)); f[0] = xdot[0] + x[1]; f[1] = (x[1] * x[1] * x[1] / 3.0 - x[1]) - x[0]; PetscCall(VecRestoreArrayRead(X, &x)); PetscCall(VecRestoreArrayRead(Xdot, &xdot)); PetscCall(VecRestoreArray(F, &f)); PetscFunctionReturn(PETSC_SUCCESS); } static PetscErrorCode IJacobian(TS ts, PetscReal t, Vec X, Vec Xdot, PetscReal a, Mat A, Mat B, void *ctx) { PetscInt rowcol[] = {0, 1}; PetscScalar J[2][2]; const PetscScalar *x; PetscFunctionBeginUser; PetscCall(VecGetArrayRead(X, &x)); J[0][0] = a; J[0][1] = -1.; J[1][0] = 1.; J[1][1] = -1. + x[1] * x[1]; PetscCall(MatSetValues(B, 2, rowcol, 2, rowcol, &J[0][0], INSERT_VALUES)); PetscCall(VecRestoreArrayRead(X, &x)); PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY)); PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY)); if (A != B) { PetscCall(MatAssemblyBegin(B, MAT_FINAL_ASSEMBLY)); PetscCall(MatAssemblyEnd(B, MAT_FINAL_ASSEMBLY)); } PetscFunctionReturn(PETSC_SUCCESS); } static PetscErrorCode RegisterMyARK2(void) { PetscFunctionBeginUser; { const PetscReal A[3][3] = { {0, 0, 0}, {0.41421356237309504880, 0, 0}, {0.75, 0.25, 0} }, At[3][3] = {{0, 0, 0}, {0.12132034355964257320, 0.29289321881345247560, 0}, {0.20710678118654752440, 0.50000000000000000000, 0.29289321881345247560}}, *bembedt = NULL, *bembed = NULL; PetscCall(TSARKIMEXRegister("myark2", 2, 3, &At[0][0], NULL, NULL, &A[0][0], NULL, NULL, bembedt, bembed, 0, NULL, NULL)); } PetscFunctionReturn(PETSC_SUCCESS); } /* Monitor timesteps and use interpolation to output at integer multiples of 0.1 */ static PetscErrorCode Monitor(TS ts, PetscInt step, PetscReal t, Vec X, void *ctx) { const PetscScalar *x; PetscReal tfinal, dt; User user = (User)ctx; Vec interpolatedX; PetscFunctionBeginUser; PetscCall(TSGetTimeStep(ts, &dt)); PetscCall(TSGetMaxTime(ts, &tfinal)); while (user->next_output <= t && user->next_output <= tfinal) { PetscCall(VecDuplicate(X, &interpolatedX)); PetscCall(TSInterpolate(ts, user->next_output, interpolatedX)); PetscCall(VecGetArrayRead(interpolatedX, &x)); 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]))); PetscCall(VecRestoreArrayRead(interpolatedX, &x)); PetscCall(VecDestroy(&interpolatedX)); user->next_output += PetscRealConstant(0.1); } PetscFunctionReturn(PETSC_SUCCESS); } int main(int argc, char **argv) { TS ts; /* nonlinear solver */ Vec x; /* solution, residual vectors */ Mat A; /* Jacobian matrix */ PetscInt steps; PetscReal ftime = 0.5; PetscBool monitor = PETSC_FALSE; PetscScalar *x_ptr; PetscMPIInt size; struct _n_User user; /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Initialize program - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ PetscFunctionBeginUser; PetscCall(PetscInitialize(&argc, &argv, NULL, help)); PetscCallMPI(MPI_Comm_size(PETSC_COMM_WORLD, &size)); PetscCheck(size == 1, PETSC_COMM_WORLD, PETSC_ERR_WRONG_MPI_SIZE, "This is a uniprocessor example only!"); PetscCall(RegisterMyARK2()); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Set runtime options - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ user.next_output = 0.0; PetscCall(PetscOptionsGetBool(NULL, NULL, "-monitor", &monitor, NULL)); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Create necessary matrix and vectors, solve same ODE on every process - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ PetscCall(MatCreate(PETSC_COMM_WORLD, &A)); PetscCall(MatSetSizes(A, PETSC_DECIDE, PETSC_DECIDE, 2, 2)); PetscCall(MatSetFromOptions(A)); PetscCall(MatSetUp(A)); PetscCall(MatCreateVecs(A, &x, NULL)); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Create timestepping solver context - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ PetscCall(TSCreate(PETSC_COMM_WORLD, &ts)); PetscCall(TSSetType(ts, TSBEULER)); PetscCall(TSSetIFunction(ts, NULL, IFunction, &user)); PetscCall(TSSetIJacobian(ts, A, A, IJacobian, &user)); PetscCall(TSSetMaxTime(ts, ftime)); PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER)); if (monitor) PetscCall(TSMonitorSet(ts, Monitor, &user, NULL)); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Set initial conditions - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ PetscCall(VecGetArray(x, &x_ptr)); x_ptr[0] = -2; x_ptr[1] = -2.355301397608119909925287735864250951918; PetscCall(VecRestoreArray(x, &x_ptr)); PetscCall(TSSetTimeStep(ts, .001)); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Set runtime options - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ PetscCall(TSSetFromOptions(ts)); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Solve nonlinear system - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ PetscCall(TSSolve(ts, x)); PetscCall(TSGetSolveTime(ts, &ftime)); PetscCall(TSGetStepNumber(ts, &steps)); PetscCall(PetscPrintf(PETSC_COMM_WORLD, "steps %3" PetscInt_FMT ", ftime %g\n", steps, (double)ftime)); PetscCall(VecView(x, PETSC_VIEWER_STDOUT_WORLD)); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Free work space. All PETSc objects should be destroyed when they are no longer needed. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ PetscCall(MatDestroy(&A)); PetscCall(VecDestroy(&x)); PetscCall(TSDestroy(&ts)); PetscCall(PetscFinalize()); return 0; } /*TEST test: requires: !single suffix: a args: -monitor -ts_type bdf -ts_rtol 1e-6 -ts_atol 1e-6 -ts_view -ts_adapt_type dsp output_file: output/ex19_pi42.out test: requires: !single suffix: b args: -monitor -ts_type bdf -ts_rtol 1e-6 -ts_atol 1e-6 -ts_view -ts_adapt_type dsp -ts_adapt_dsp_filter PI42 output_file: output/ex19_pi42.out test: requires: !single suffix: c 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 output_file: output/ex19_pi42.out test: requires: !single suffix: bdf_reject 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 TEST*/