On the asymptotic properties of IMEX Runge–Kutta schemes for hyperbolic balance laws

Implicit–Explicit (IMEX) schemes are a powerful tool in the development of numerical methods for hyperbolic systems with stiff sources. Here we focus our attention on the asymptotic properties of such schemes, like the preservation of steady-states (well-balanced property) and the behavior in presence of small space–time scales (asymptotic preservation property). We analyze conditions under which the standard additive approach based on taking the fluxes explicitly and the sources implicitly yields a well-balanced behavior. In addition, we consider a partitioned strategy which possesses better well-balanced properties. The behavior of the additive and partitioned approaches under classical scaling limits is then studied in the context of asymptotic-preserving schemes. Additional order conditions that guarantee the correct behavior of the schemes in the Navier–Stokes regime are derived. Several examples illustrate these asymptotic behaviors and the performance of the new methods.