The Penultimate Scheme for Systems of Conservation Laws: Finite Difference ENO with Marquina’s Flux Splitting1

This paper provides a users’ guide to a new, general finite difference method for the numerical solution of systems of convection dominated conservation laws. We include both extensive motivation for the method design, as well as a detailed formulation suitable for direct implementation. Essentially Non-Oscillatory (ENO) methods are a class of high accuracy, shock capturing numerical methods for hyperbolic systems of conservation laws, based on upwind biased differencing in local characteristic fields. The earliest ENO methods used control volume discretizations, but subsequent work [12] has produced a simpler finite difference form of the ENO method. While this method has achieved excellent results in a great variety of compressible flow problems, there are still special situations where noticeable spurious oscillations develop. Why this occurs is not always understood, and there has been no elegant way to eliminate these problems. Based on the extensive work of Donat and Marquina [1], it appears that these difficulties arise from using a single transformation to local characteristic This paper was presented in ”Solutions of PDE” Conference in honour of Prof. Roe on the occassion of his 60th birthday, July 1998, Arachaon, France