An integration method is discussed which has been designed to treat parabolic and hyperbolic terms explicitly and stiff reaction terms implicitly. The method is a special two-step form of the one-step IMEX (Implicit-Explicit) RKC (Runge-Kutta-Chebyshev) method. The special two-step form is introduced with the aim of getting a non-zero imaginary stability boundary which is zero for the one-step method. Having a non-zero imaginary stability boundary allows, for example, the integration of pure advection equations space-discretized with centered schemes, the integration of damped or viscous wave equations, the integration of coupled sound and heat flow equations, etc. For our class of methods it also simplifies the choice of temporal step sizes satisfying the von Neumann stability criterion, by embedding a thin long rectangle inside the stability region. Embedding rectangles or other tractable domains with this purpose is an idea of Wesseling.

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Academic Press
Journal of Computational Physics
Development of a State-of-the Art Surface-Capturing Method for Two-Fluid Flow
Computational Dynamics

Sommeijer, B.P, & Verwer, J.G. (2007). On stabilized integration for time-dependent PDEs. Journal of Computational Physics, 224(1), 3–16.