On the use of Mühlbach expansions in the recovery step of ENO methods

Abstract

The recovery step is the most expensive algorithmic ingredient in modern essentially non-oscillatory (ENO) shock capturing methods on triangular meshes for the numerical simulation of compressible fluid flow. While recovery polynomials in Newton form are used in one-dimensional ENO schemes it is a priori not clear whether such useful as well as numerically stable form of polynomials exists in multiple dimensions. As was observed in [1] a very general answer to this question was provided by Muhlbach in two subsequent papers [15] and [16]. We generalise his interpolation theory further to the general recovery problem and outline the use of Muhlbach's expansion in ENO schemes. Numerical examples show the usefulness of this approach in the problem of recovery from cell average data.