The dominant wave-capturing flux: A finite-volume scheme without decomposition for systems of hyperbolic conservation laws

Abstract

More robust developments of schemes for hyperbolic systems, that avoid dependence upon a characteristic decomposition have been achieved by employing schemes that are based on a Rusanov flux. Such schemes permit the construction of higher order approximations without recourse to characteristic decomposition. This is achieved by using the maximum eigenvalue of the hyperbolic system within the definition of the numerical flux. In recent literature the Rusanov flux has been embedded in a local Lax–Friedrichs flux. The current literature on these schemes only appears to indicate success in this regard, with no investigation of the effect of the additional numerical diffusion that is inherent in such formulations. In this paper the foundation for a new scheme is proposed which relies on the detection of the dominant wave in the system. This scheme is designed to permit the construction of lower and higher order approximations without recourse to characteristic decomposition while avoiding the excessive numerical diffusion that is inherent in the Rusanov and local Lax–Friedrichs fluxes.