On stabilized integration for time-dependent PDEs

Abstract:
An integration method is discussed which has been designed to treat parabolic and hy-
perbolic terms explicitly and sti® 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 vis-
cous wave equations, the integration of coupled sound and heat °ow equations, etc.
For our class of methods it also simpli¯es 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.

2000 Mathematics Subject Classi¯cation: Primary: 65M12, 65M20.

1998 ACM Computing Classi¯cation System: G.1.1, G.1.7 and G.1.8.

Keywords and Phrases: Numerical Integration, Stabilized Explicit Integration, Runge-
Kutta-Chebyshev Methods, Reactive Flow Problems, Damped Wave Equations, Cou-
pled Sound and Heat Flow.