Runge–Kutta Solutions of Stiff Differential Equations Near Stationary Points

Abstract

Runge–Kutta methods applied to stiff systems in singular perturbation form are shown to give accurate approximations of phase portraits near hyperbolic stationary points. Over arbitrarily long time intervals, Runge–Kutta solutions shadow solutions of the differential equation and vice versa. Precise error bounds are derived. The proof uses attractive invariant manifolds to reduce the problem to the nonstiff case, which was previously studied by Beyn.