Generalization to a wider class of entropy split methods for compressible ideal MHD

Abstract

The high order entropy split methods of Sjögreen & Yee (2019, 2021), by entropy splitting of the compressible Euler (inviscid) flux derivatives for a thermally-perfect gas are based on Harten’s entropy function (Harten, 1983; Gerritsen and Olsson, 1996; Yee et al., 2000). Their formulation have been proven entropy conserving and stable by taking advantage of the homogeneity property of Euler flux, symmetrizable Euler flux derivatives and energy-norm stability in conjunction with high order classical spatial central, dispersion relation-preserving (DRP) (Bogey and Bailly, 2002; Brambley, 2016; Johansson, 2004) or Padé (compact) spatial discretizations (Hirsh, 1975) with summation-by-parts (SBP) operators (Strand, 1994). These high order entropy split methods not only preserve certain physical properties of the chosen governing equations but are also known to either improve numerical stability, and/or minimize aliasing errors in long time integration of turbulent flow computations without the aid of added numerical dissipation. The present work employs a new approach to obtain a wider class of high order entropy split methods that do not have the homogeneous property. The new method consists of a two-point numerical flux portion and a non-conservative portion in such a way that entropy conservation holds without requiring the homogeneity property of the compressible inviscid flux. For high order classical spatial central, DRP or Padé spatial discretizations, this new approach can be proven to be entropy conservative while at the same time allowing a wider class of symmetrizable inviscid flux derivatives. More importantly, we extend this new approach to derive a new entropy split method for the equations of ideal MHD. Representative test cases are illustrated with comparison among existing methods of the same high order of accuracy.