Strong Stability Preserving Runge–Kutta and Linear Multistep Methods

Abstract

This paper reviews strong stability preserving discrete variable methods for differential systems. The strong stability preserving Runge–Kutta methods have been usually investigated in the literature on the subject, using the so-called Shu–Osher representation of these methods, as a convex combination of first-order steps by forward Euler method. In this paper, we revisit the analysis of strong stability preserving Runge–Kutta methods by reformulating these methods as a subclass of general linear methods for ordinary differential equations, and then using a characterization of monotone general linear methods, which was derived by Spijker in his seminal paper (SIAM J Numer Anal 45:1226–1245, 2007). Using this new approach, explicit and implicit strong stability preserving Runge–Kutta methods up to the order four are derived. These methods are equivalent to explicit and implicit RK methods obtained using Shu–Osher or generalized Shu–Osher representation. We also investigate strong stability preserving linear multistep methods using again monotonicity theory of Spijker.