Asymptotic Preserving HLL Schemes

作者：Berthon Christophe^*; Turpault Rodolphe

来源：Numerical Methods for Partial Differential Equations, 2011, 27(6): 1396-1422.

DOI：10.1002/num.20586

• asymptotic preserving scheme
• Euler equations with high friction
• HLL scheme
• hyperbolic system with source term
• M1 model
• radiative transfer
• telegraph equations

This work concerns the derivation of HLL schemes to approximate the solutions of systems of conservation laws supplemented by source terms. Such a system contains many models such as the Euler equations with high friction or the M1 model for radiative transfer. The main difficulty arising from these models comes from a particular asymptotic behavior. Indeed, in the limit of some suitable parameter, the system tends to a diffusion equation. This article is devoted to derive HLL methods able to approximate the associated transport regime but also to restore the suitable asymptotic diffusive regime. To access such an issue, a free parameter is introduced into the source term. This free parameter will be a useful correction to satisfy the expected diffusion equation at the discrete level. The derivation of the HLL scheme for hyperbolic systems with source terms comes from a modification of the HLL scheme for the associated homogeneous hyperbolic system. The resulting numerical procedure is robust as the source term discretization preserves the physical admissible states. The scheme is applied to several models of physical interest. The numerical asymptotic behavior is analyzed and an asymptotic preserving property is systematically exhibited. The scheme is illustrated with numerical experiments.

• 出版日期2011-11