static char help[] = "Solves the van der Pol equation and demonstrate IMEX.\n\ Input parameters include:\n\ -mu : stiffness parameter\n\n"; /* ------------------------------------------------------------------------ This program solves the van der Pol equation y'' - \mu ((1-y^2)*y' - y) = 0 (1) on the domain 0 <= x <= 1, with the boundary conditions y(0) = 2, y'(0) = - 2/3 +10/(81*\mu) - 292/(2187*\mu^2), 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 two variants of linear problems, u_t = f(u,t), namely turning (1) into a system of first order differential equations, [ y' ] = [ z ] [ z' ] [ \mu ((1 - y^2) z - y) ] which then we can write as a vector equation [ u_1' ] = [ u_2 ] (2) [ u_2' ] [ \mu (1 - u_1^2) u_2 - u_1 ] which is now in the desired form of u_t = f(u,t). One way that we can split f(u,t) in (2) is to split by component, [ u_1' ] = [ u_2 ] + [ 0 ] [ u_2' ] [ 0 ] [ \mu ((1 - u_1^2) u_2 - u_1) ] where [ G(u,t) ] = [ u_2 ] [ 0 ] and [ F(u',u,t) ] = [ u_1' ] - [ 0 ] [ u_2' ] [ \mu ((1 - u_1^2) 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 ; 1 ] dF [ 0 ; 0 ] -- = [ ] du [ -\mu (2*u_1*u_2 + 1); \mu (1 - u_1^2) ] Hence, [ a ; 0 ] J(F) = [ ] [ \mu (2*u_1*u_2 + 1); a - \mu (1 - u_1^2) ] ------------------------------------------------------------------------- */ #include typedef struct _n_User *User; struct _n_User { PetscReal mu; PetscBool imex; 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] = (user->imex ? x[1] : 0); f[1] = 0.0; 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] + (user->imex ? 0 : x[1]); f[1] = xdot[1] - user->mu * ((1. - 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; PetscReal mu = user->mu; PetscInt rowcol[] = {0, 1}; const PetscScalar *x; PetscScalar J[2][2]; PetscFunctionBeginUser; PetscCall(VecGetArrayRead(X, &x)); J[0][0] = a; J[0][1] = (user->imex ? 0 : 1.); J[1][0] = mu * (2. * x[0] * x[1] + 1.); J[1][1] = a - mu * (1. - 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); } 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, 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; 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.mu = 1000.0; user.imex = PETSC_TRUE; user.next_output = 0.0; PetscCall(PetscOptionsGetReal(NULL, NULL, "-mu", &user.mu, NULL)); PetscCall(PetscOptionsGetBool(NULL, NULL, "-imex", &user.imex, NULL)); 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(TSSetRHSFunction(ts, NULL, RHSFunction, &user)); 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] = 2.0; 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)); PetscCall(TSSetTimeStep(ts, 0.01)); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 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, "mu %g, steps %" PetscInt_FMT ", ftime %g\n", (double)user.mu, 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: args: -ts_type arkimex -ts_arkimex_type myark2 -ts_adapt_type none requires: !single TEST*/