Strong Stability Preserving Properties of Runge-Kutta Time Discretization Methods for Linear Constant Coefficient Operators

Strong stability preserving (SSP) high order Runge-Kutta time discretizations were developed for use with semi-discrete method of lines approximations of hyperbolic partial differential equations, and have proven useful in many other applications. These high order time discretization methods preserve the strong stability properties of first order explicit Euler time stepping. In this paper we analyze the SSP properties of Runge Kutta methods for the ordinary differential equation u ^t = Lu where L is a linear operator. We present optimal SSP Runge-Kutta methods as well as a bound on the optimal timestep restriction. Furthermore, we extend the class of SSP Runge-Kutta methods for linear operators to include the case of time dependent boundary conditions, or a time dependent forcing term.