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We consider Implicit-Explicit (IMEX) Runge-Kutta (R-K) schemes for hyperbolicsystems with stiff relaxation in the so-called diffusion limit. In such regimethe system relaxes towards a convection-diffusion equation. The first objectiveof the paper is to show that traditional partitioned IMEX R-K schemes willrelax to an explicit scheme for the limit equation with no need of modificationof the original system. Of course the explicit scheme obtained in the limitsuffers from the classical parabolic stability restriction on the time step.The main goal of the paper is to present an approach, based on IMEX R-Kschemes, that in the diffusion limit relaxes to an IMEX R-K scheme for theconvection-diffusion equation, in which the diffusion is treated implicitly.This is achieved by an original reformulation of the problem, and subsequentapplication of IMEX R-K schemes to it. An analysis on such schemes to thereformulated problem shows that the schemes reduce to IMEX R-K schemes for thelimit equation, under the same conditions derived for hyperbolic relaxation.Several numerical examples including neutron transport equations confirm thetheoretical analysis.

Implicit-Explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit