Runge-Kutta methods are stable

Abstract

We prove that Runge-Kutta (RK) methods for numerical integration of arbitrarily large systems of Ordinary Differential Equations are linearly stable. Standard stability arguments -- based on spectral analysis, resolvent condition or strong stability, fail to secure the stability of arbitrarily large RK systems. We explain the failure of different approaches, offer a new stability theory and demonstrate a few examples.