Recent advancement of entropy split methods for compressible gas dynamics and MHD

Abstract

The entropy splitting of the compressible Euler flux derivatives based on Harten’s entropy function Harten (1983), Gerritsen and Olsson (1996), Yee et al. (2000) in conjunction with classical spatial central and DRP (dispersion relation-preserving) finite discretizations with summation-by-parts (SBP) operators Strand (1994) for both periodic and non-periodic boundary conditions is proven to be entropy conservative and stable for a thermally-perfect gas by Sjögreen and Yee (2019), Sjögreen et al. (2020), Sjögreen and Yee (2021). The various high order methods resulting from applying classical spatial central, DRP and Padé (compact) methods to the split form of the Euler flux derivative are referred to as entropy split methods as a function of the splitting parameter β. These entropy split methods are entropy conserving and stable but they are usually not conservative numerical methods without additional reformulation; e.g., those proposed in Sjögreen and Yee (2021).This is an expanded version of Sjögreen and Yee (2022) and is intended to present a more in depth study for both the compressible gas dynamics and ideal MHD equations. Here the same Harten entropy function is used for the ideal magnetohydrodynamic (MHD) governing equation set. This approach was not examined in great depth by Sjögreen & Yee and Yee & Sjögreen in two conference proceedings papers Sjögreen and Yee (2022), Yee and Sjögreen (2022), which also included the high order Padé (compact) spatial discretizations Hirsh(1975). For the extension of the entropy split method to the ideal MHD, the Godunov symmetrizable non-conservative form of the formulation is used. Due to the non-conservative portion (commonly referred to as a nonlinear source term vector) of the symmetrizable system Godunov (1972), there are variants in formulating the entropy split methods. Two different numerical treatments for these particular source terms of the symmetrizable MHD governing equation set are presented.Systematic 3D formulation in curvilinear grids with the corresponding numerical fluxes among method comparisons for the Euler and ideal MHD equations is included. The curvilinear formulation includes high order classical central, DRP and compact spatial discretizations. For readers interested in the implementation of these methods into their computer code, it is hope that these formulations and the corresponding numerical fluxes in 3D curvilinear grids will be helpful.The comparative studies concentrate on the Tadmor-type of discrete entropy conserving Tadmor (2003), momentum conserving Ducros et al. (2000), kinetic energy preserving Kennedy and Gruber (2008), Pirozzoli (2010), Coppola et al. (2019), Yee et al., Sjögreen & Yee entropy split methods Yee et al.(2000), Sjögreen and Yee (2018), Sjögreen and Yee (2019), Sjögreen et al. (2020), Sjögreen and Yee (2021), as well as the combination of these physical-preserving methods Ranocha (2020), Yee and Sjögreen (2020). All of these methods are not only preserved 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 for selected compressible flow types. Extensive error norm comparison with grid refinement were performed to show how well this methods conserve the entropy, momentum and mass, and preserve the kinetic energy for long time integration of the various flows. In general the Tadmor-type entropy conserving methods are at least twice the CPU per time step than the rest of the skew-symmetric splitting methods.