Asymptotic preserving schemes for hyperbolic systems with relaxation

This paper presents the construction of two numerical schemes for the solution of hyperbolic systems with relaxation source terms. The methods are built by considering the relaxation system as a whole, without separating the resolution of the convective part from that of the source term. The first scheme combines the centered FORCE approach of Toro and co-authors with the unsplit strategy proposed by B{\'e}reux and Sainsaulieu. The second scheme consists of an approximate Riemann solver which carefully handles the source term approximation. The two schemes are built to be asymptotic preserving, in the sense that their limit schemes are consistent with the equilibrium model as the relaxation parameter tends to zero, without any CFL restriction. For specific models, it is possible to prove that they preserve invariant domains and admit a discrete entropy inequality.