A plethora of physical phenomena are modelled by hyperbolic partial differential equations, for which the exact solution is usually not known. Numerical methods are employed to approximate the solution to hyperbolic problems; however, in many cases, it is difficult to satisfy certain physical properties while maintaining high order of accuracy. Strong stability preserving (SSP) time discretizations were developed to ensure that nonlinear stability properties of the solution are maintained when coupled with suitable spatial discretizations.
In the first part of this talk, we review the development of optimal SSP Runge-Kutta and multistep methods for nonlinear problems. We emphasize the usage of an alternative representation of Runge-Kutta methods that reveals the SSP properties of such methods. Numerical examples illustrate the effectiveness and usefulness of SSP methods.
In the second part, we present some recent results related to perturbed methods that use both upwind- and downwind-biased spatial discretizations. We introduce a novel family of third-order implicit Runge–Kutta methods with arbitrarily large SSP coefficient and investigate the stability and accuracy of these methods. Moreover, we extend the analysis of SSP linear multistep methods to semi-discretized problems for which different terms on the right-hand side of the initial value problem satisfy different forward Euler (or circle) conditions. Optimal perturbed and additive monotonicity-preserving linear multistep methods are studied in the context of such problems.