Van_der_Pol_implicit#

class vanderpol(u0=None, mu=5.0, newton_maxiter=100, newton_tol=1e-09, stop_at_nan=True, crash_at_maxiter=True, relative_tolerance=False)[source]#

Bases: Problem

Stiff Van der Pol oscillator as a system of two first-order ODEs, fully implicit with Newton.

It is given by the equation

\[\frac{d^2 u(t)}{d t^2} - \mu (1 - u(t)^2) \frac{d u(t)}{dt} + u(t) = 0.\]
Parameters:

  • u0 (sequence of array_like, optional) – Initial condition.

  • mu (float, optional) – Stiff parameter \(\mu\).

  • newton_maxiter (int, optional) – Maximum number of iterations for Newton’s method to terminate.

  • newton_tol (float, optional) – Tolerance for Newton to terminate.

  • stop_at_nan (bool, optional) – Indicate whether Newton’s method should stop if nan values arise.

  • crash_at_maxiter (bool, optional) – Indicates whether Newton’s method should stop if maximum number of iterations newton_maxiter is reached.

  • relative_tolerance (bool, optional) – Use a relative or absolute tolerance for the Newton solver

Variables:

work_counters (WorkCounter) – Counts different things, here: Number of evaluations of the right-hand side in eval_f and number of Newton calls in each Newton iterations are counted.

dtype_f#

alias of mesh

dtype_u#

alias of mesh

eval_f(u, t)[source]#

Routine to compute the right-hand side for both components simultaneously.

Parameters:

  • u (dtype_u) – Current values of the numerical solution.

  • t (float) – Current time at which the numerical solution is computed (not used here).

Returns:

f (dtype_f) – The right-hand side (contains 2 components).

solve_jacobian(rhs, dt, u, **kwargs)[source]#

Solve the linear system with the Jacobian of the Newton function \(g(u) = u - dt f(u) - rhs\) of solve_system, by applying the analytically computed inverse of the \(2 \times 2\) Jacobian at u.

Parameters:

  • rhs (np.1darray) – Right-hand side of the linear system, i.e., the Newton residual.

  • dt (float) – Abbrev. for the node-to-node stepsize (or any other factor required).

  • u (dtype_u) – Current Newton iterate, at which the Jacobian is evaluated.

  • **kwargs – Not used.

Returns:

du (np.1darray) – The Newton update.

solve_system(rhs, dt, u0, t)[source]#

Simple Newton solver for the nonlinear system.

Parameters:

  • rhs (dtype_f) – Right-hand side for the nonlinear system.

  • dt (float) – Abbrev. for the node-to-node stepsize (or any other factor required).

  • u0 (dtype_u) – Initial guess for the iterative solver.

  • t (float) – Current time (e.g. for time-dependent BCs).

Returns:

u (dtype_u) – The solution u.

u_exact(t, u_init=None, t_init=None)[source]#

Routine to approximate the exact solution at time t by SciPy or give initial conditions when called at \(t=0\).

Parameters:

  • t (float) – Current time.

  • u_init (pySDC.problem.vanderpol.dtype_u) – Initial conditions for getting the exact solution.

  • t_init (float) – The starting time.

Returns:

me (dtype_u) – Approximate exact solution.