Stability Properties of Interpolants for Runge–Kutta Methods

Abstract

In solving initial value problems for ordinary differential equations by means of a discrete method, we often need to know the solution on a set of points that differs from the grid (systems with driving equation, dense output, discontinuity problems, delay equations, etc.). The nonnodal approximations are generally obtained by local interpolation, and different features can be requested from the interpolants according to the problem treated. In this paper we introduce the concept of stable interpolant for Runge–Kutta methods. Roughly speaking, a stable interpolant maintains the stability properties of the Runge–Kutta method itself. The lack of such stability makes the continuous extension not reliable, especially in solving stiff equations.