A New Approach for a Wider Class of Entropy Split Methods for Compressible Gas Dynamics and MHD

The high order entropy split methods of Sjögreen & Yee [1, 2] by entropy splitting of
the compressible Euler (inviscid) flux derivatives for a thermally-perfect gas are based on Harten’s
entropy function [3, 4, 5]. Their derivation takes advantage of the homogeneity property of Euler flux, symmetrizable Euler flux derivatives and energy-norm stability in conjunction with high
order classical spatial central, DRP (dispersion relation-preserving) [6, 7, 8] or Padé (compact)
spatial discretizations [9] with summation-by-parts (SBP) operators [10]. Our entropy split methods have been proven entropy conserving and stable [1, 11, 12]. Our proofs do not rely on a
two-point numerical flux, but rather only a linear difference operator is required to derive these
methods. To extend the entropy split method for the MHD, we used the Godunov symmetrizable
non-conservative MHD form [12, 13, 14]. 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. In our previous published work,
extensive error norm comparison with grid refinement was performed to show the high accuracy
performance of these methods. These studies also showed how well the entropy split methods conserve the entropy, momentum and mass, and preserve the kinetic energy for long time integration of the various flows [1, 2, 12, 13, 14].

The objective of the present work is to use a new approach to obtain a wider class of entropy
split methods consisting of a two-point numerical flux portion and a non-conservative portion in
such a way that the homogeneity property of the compressible Euler flux is not required. For high
order classical spatial central, DRP (dispersion relation-preserving) or Padé (compact) spatial discretizations, this new approach can be proven to be entropy conservative with conservative spatial dsicretizations while at the same time allowing a wider class of symmetrizable inviscid flux derivatives. We also use this generalization to derive an entropy split scheme that is entropy conserving for the equations of MHD without the homogeneity property using the Godunov symmetrizable ideal MHD formulation [15].