An Implicit-Explicit Runge--Kutta--Chebyshev Scheme for Diffusion-Reaction Equations

Abstract

An implicit-explicit (IMEX) extension of the explicit Runge--Kutta--Chebyshev (RKC) scheme designed for parabolic PDEs is proposed for diffusion-reaction problems with severely stiff reaction terms. The IMEX scheme treats these reaction terms implicitly and diffusion terms explicitly. Within the setting of linear stability theory, the new IMEX scheme is unconditionally stable for reaction terms having a Jacobian matrix with a real spectrum. For diffusion terms the stability characteristics remain unchanged. A numerical comparison for a stiff, nonlinear radiation-diffusion problem between an RKC solver, an IMEX-RKC solver, and the popular implicit BDF solver VODPK using the Krylov solver GMRES illustrates the excellent performance of the new scheme.