A Survey of Strong Stability Preserving High Order Time Discretizations

Numerical solution for ordinary di erential equations (ODEs) is an established research area. There are many well established methods, such as Runge-Kutta methods and multi-step methods, for such purposes. There are also many excellent books on this subject, for example [1], [13] and [17]. Special purpose ODE solvers, such as those for sti ODEs, are also well studied. See, e.g., [6]. However, the class of methods surveyed in this article, the so-called strong stability preserving (SSP) methods, is somewhat special. These methods were designed speci cally for solving the ODEs coming from a semi-discrete, spatial discretization of time dependent partial di erential equations (PDEs), especially hyperbolic PDEs. Typically such ODEs are very large (the size of the system depends on the spatial discretization mesh size). More importantly, there are certain stability properties of the original PDE, such as total variation stability or maximum norm stability, which could be maintained by certain special spatial discretizations coupled with simple rst order Euler forward time discretization, that would be desirable to maintain also for the high order time discretizations. SSP methods are designed to achieve such a goal. We can thus highlight the main property of SSP time discretizations: if we assume that the rst order, forward Euler time discretization of a method of lines semi-discrete scheme is stable under a certain norm, then a SSP high order time