static char help[] = "Solves the van der Pol equation.\n\ Input parameters include:\n"; /* ------------------------------------------------------------------------ This program solves the van der Pol DAE ODE equivalent y' = z (1) z' = \mu ((1-y^2)z-y) on the domain 0 <= x <= 1, with the boundary conditions y(0) = 2, y'(0) = - 2/3 +10/(81*\mu) - 292/(2187*\mu^2), and \mu = 10^6 ( y'(0) ~ -0.6666665432100101). 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. Notes: This code demonstrates the TS solver interface to an ODE -- RHSFunction for explicit form and IFunction for implicit form. ------------------------------------------------------------------------- */ #include typedef struct _n_User *User; struct _n_User { PetscReal mu; PetscReal next_output; }; /* User-defined routines */ static PetscErrorCode RHSFunction(TS ts, PetscReal t, Vec X, Vec F, PetscCtx ctx) { User user = (User)ctx; PetscScalar *f; const PetscScalar *x; PetscFunctionBeginUser; PetscCall(VecGetArrayRead(X, &x)); PetscCall(VecGetArray(F, &f)); f[0] = x[1]; f[1] = user->mu * (1. - x[0] * x[0]) * x[1] - x[0]; PetscCall(VecRestoreArrayRead(X, &x)); PetscCall(VecRestoreArray(F, &f)); PetscFunctionReturn(PETSC_SUCCESS); } static PetscErrorCode IFunction(TS ts, PetscReal t, Vec X, Vec Xdot, Vec F, PetscCtx ctx) { User user = (User)ctx; const PetscScalar *x, *xdot; PetscScalar *f; PetscFunctionBeginUser; PetscCall(VecGetArrayRead(X, &x)); PetscCall(VecGetArrayRead(Xdot, &xdot)); PetscCall(VecGetArray(F, &f)); f[0] = xdot[0] - x[1]; f[1] = xdot[1] - user->mu * ((1.0 - x[0] * x[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, PetscCtx ctx) { User user = (User)ctx; PetscInt rowcol[] = {0, 1}; const PetscScalar *x; PetscScalar J[2][2]; PetscFunctionBeginUser; PetscCall(VecGetArrayRead(X, &x)); J[0][0] = a; J[0][1] = -1.0; J[1][0] = user->mu * (2.0 * x[0] * x[1] + 1.0); J[1][1] = a - user->mu * (1.0 - x[0] * x[0]); 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); } /* Monitor timesteps and use interpolation to output at integer multiples of 0.1 */ static PetscErrorCode Monitor(TS ts, PetscInt step, PetscReal t, Vec X, PetscCtx 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] %" 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 += 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, implicitform = PETSC_TRUE; 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!"); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Set runtime options - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ user.next_output = 0.0; user.mu = 1.0e3; PetscCall(PetscOptionsGetBool(NULL, NULL, "-monitor", &monitor, NULL)); PetscCall(PetscOptionsGetBool(NULL, NULL, "-implicitform", &implicitform, NULL)); PetscOptionsBegin(PETSC_COMM_WORLD, NULL, "Physical parameters", NULL); PetscCall(PetscOptionsReal("-mu", "Stiffness parameter", "<1.0e6>", user.mu, &user.mu, NULL)); PetscOptionsEnd(); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 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)); if (implicitform) { PetscCall(TSSetIFunction(ts, NULL, IFunction, &user)); PetscCall(TSSetIJacobian(ts, A, A, IJacobian, &user)); PetscCall(TSSetType(ts, TSBEULER)); } else { PetscCall(TSSetRHSFunction(ts, NULL, RHSFunction, &user)); PetscCall(TSSetType(ts, TSRK)); } PetscCall(TSSetMaxTime(ts, ftime)); PetscCall(TSSetTimeStep(ts, 0.001)); 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] = 2.0; x_ptr[1] = -2.0 / 3.0 + 10.0 / (81.0 * user.mu) - 292.0 / (2187.0 * user.mu * user.mu); PetscCall(VecRestoreArray(x, &x_ptr)); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 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 %" 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 args: -mu 1e6 test: requires: !single suffix: 2 args: -implicitform false -ts_type rk -ts_rk_type 5dp -ts_adapt_type dsp test: requires: !single suffix: 3 args: -implicitform false -ts_type rk -ts_rk_type 5dp -ts_adapt_type dsp -ts_adapt_dsp_filter H0312 TEST*/