High Order Asymptotic Preserving Deferred Correction Implicit-Explicit Schemes for Kinetic Models

Abstract

This work aims to extend the residual distribution (RD) framework to stiff relaxation problems. The RD is a class of schemes which is used to solve hyperbolic system of partial differential equations. Up to our knowledge, it was used only for systems with mild source terms, such as gravitation problems or shallow water equations. What we propose is an IMEX (implicit--explicit) version of the residual distribution schemes, that can resolve stiff source terms, without refining the discretization up to the stiffness scale. This can be particularly useful in various models, where the stiffness is given by topological or physical quantities, e.g. multiphase flows, kinetic models, viscoelasticity problems. Moreover, the provided scheme is able to catch different relaxation scales automatically, without losing accuracy. The scheme is asymptotic preserving and this guarantees that in the relaxation limit, we recast the expected macroscopic behaviour. To get a high order accuracy, we use an IMEX time discretization combined with a Deferred Correction (DeC) procedure, while naturally RD provides high order space discretization. Finally, we show some numerical tests in 1D and 2D for stiff systems of equations.