Strong Stability Preserving IMEX Methods for Partitioned Systems of Differential Equations

We investigate strong stability preserving (SSP) implicit-explicit (IMEX) methods for partitioned systems of differential equations with stiff and nonstiff subsystems. Conditions for order p and stage order $$q=p$$ q = p are derived, and characterization of SSP IMEX methods is provided following the recent work by Spijker. Stability properties of these methods with respect to the decoupled linear system with a complex parameter, and a coupled linear system with real parameters are also investigated. Examples of methods up to the order $$p=4$$ p = 4 and stage order $$q=p$$ q = p are provided. Numerical examples on six partitioned test systems confirm that the derived methods achieve the expected order of convergence for large range of stepsizes of integration, and they are also suitable for preserving the accuracy in the stiff limit or preserving the positivity of the numerical solution for large stepsizes.

A framework to evaluate IMEX schemes for atmospheric models

We present a new evaluation framework for implicit and explicit (IMEX) Runge-Kutta timestepping schemes. The new framework uses a linearized nonhydrostatic system of normal modes. We utilize the framework to investigate stability of IMEX methods and their dispersion and dissipation for gravity, Rossby, and acoustic waves. We test the new framework on a variety of IMEX schemes and use it to develop and analyze a set of 2nd order low-storage IMEX Runge-Kutta methods with high CFL. We show that the new framework is more selective than the 2D acoustic system previously used in literature. Schemes that are stable for the 2D acoustic system are not stable for the system of normal modes.