On strong stability preserving time discretization methods

Over the last few years, great effort has been made to develop high order strong stability preserving (SSP) Runge-Kutta methods. These methods have a nonlinear stability property that makes them suitable for the time integration of ODEs that arise from a method of lines approximation of hyperbolic conservation laws. Basically, this stability property is a monotonicity property for the internal stages and the numerical solution. Recently Ferracina and Spijker have established a link between stepsize restrictions for monotonicity and the already known stepsize restrictions for contractivity. Hence the extensive research on contractivity can be transferred to the SSP context. In this paper we consider monotonicity issues for arbitrary norms and linear and nonlinear problems. We collect and review some known results and relate them with the ones obtained in the SSP context.