Optimal Strong Stability Preserving Time Discretizations

Traditional stability concepts for ODE solvers typically deal with linear equations and/or bounds involving inner-product norms only. Modern problems of interest are typically nonlinear and in many casesthe relevant bounds for the problem involve more general boundedness properties such as positivity or or the total variation diminishing property. Strong stability preserving (SSP) methods (referred to also as contractivity preserving, monotonicity preserving, or total variation diminishing methods) provide such boundedness properties whenever the desired property is satisfied under forward Euler integration. After reviewing the relationship between strong stability preservation and other stability concepts, I will discuss work on finding SSP Runge-Kutta methods that preserve stability under the largest possible timestep. Investigation of implicit SSP Runge-Kutta methods has revealed some remarkable properties, as well as barriers to their efficiency. This work spurred a new search for optimal explicit methods, leading to new methods that are preferable to existing ones in terms of both memory and computational efficiency. I will also mention some current work on SSP properties of general linear methods and spectral deferred correction methods.