On high order strong stability preserving runge-kutta and multi step time discretizations

Abstract

Strong stability preserving (SSP) high order time discretizations were developed for solution of semi-discrete method of lines approximations of hyperbolic partial differential equations. These high order time discretization methods preserve the strong stability properties-in any norm or seminorm-of the spatial discretization coupled with first order Euler time stepping. This paper describes the development of SSP methods and the recently developed theory which connects the timestep restriction on SSP methods with the theory of monotonicity and contractivity. Optimal explicit SSP Runge-Kutta methods for nonlinear problems and for linear problems as well as implicit Runge-Kutta methods and multi step methods will be collected.

Strong stability preserving hybrid methods

“…As pointed out by Gottlieb [6], the research in the field of SSP methods centers around the search for high order SSP methods with a CFL coefficient c as large as possible. When we make a comparison between the CFL coefficients of different methods, however, their computational costs and order should be taken into account too.…”

“…SSP methods for constant coefficient linear systems were studied in [7,9]. For a description of state-of-the-art, we refer the reader to the review papers by Gottlieb, Shu and Tadmor [9], Shu [26], and Gottlieb [6].…”

Semi discrete discontinuous Galerkin methods and stage-exceeding-order, strong-stability-preserving Runge–Kutta time discretizations

“…The design of high-order methods that are nonlinearly stable is possible through the use of a special class of high-order time discretization methods -what have come to be called strong-stability-preserving (SSP) methods (see, for example, the review articles [10,11,13]). Originally referred to as TVD time discretization methods, these time stepping schemes were first introduced by Shu and Osher [24] as a means of obtaining secondorder accurate and higher time discretization methods that preserve the TVD property of a given spatial discretization and the first-order forward Euler method in time.…”

Optimal implicit strong stability preserving Runge–Kutta methods

“…Much of the research in this field is devoted to finding methods that are optimal in terms of their timestep restriction. For this purpose, various implicit extensions and generalizations of the Shu-Osher form have been introduced [10,8,6,14]. The most general of these, and the form we use in this paper, was introduced independently in [6] and [14].…”

“…Our search for new SSP methods is facilitated by known results on contractivity and absolute monotonicity of Runge-Kutta methods [31,22] and their connection to strong stability preservation [13,14,4,6]. For a more detailed description of the Shu-Osher form and the SSP property, we refer the interested reader to [30,9,10,29,8,20].…”