Skew-Symmetric Splitting and Stability of High Order Central Schemes

Björn Sjögreen,H.C Yee,Dmitry Kotov

doi:10.1088/1742-6596/837/1/012019

Björn Sjögreen, H.C Yee + Show 1 more

Open Access

https://doi.org/10.1088/1742-6596/837/1/012019

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Abstract

Skew-symmetric splittings of the inviscid flux derivative for high order central schemes are studied and developed to improve their numerical stability without added high order numerical dissipation for long time wave propagations and long time integration of compressible turbulent flows. For flows containing discontinuities and multiscale turbulence fluctuations the Yee & Sjogreen [33] and Kotov et al. [15, 14] high order nonlinear filter approach is utilized in conjunction with the skew-symmetric form of high order central schemes. Due to the incomplete hyperbolic nature of the conservative ideal magnetohydrodynamics (MHD) governing equations, not all of the skew-symmetric splittings for gas dynamics can be extended to the ideal MHD. For the MHD the Ducros et al. [6] variants are constructed. In addition, four formulations of the MHD are considered: (a) the conservative MHD, (b) the Godunov/Powell non-conservative form, (c) the Janhunen MHD with magnetic field source terms [13], and (d) a MHD with source terms of [3]. The different formulation of the equations in conjunction with the variants of Ducros et al. type skew-symmetric splitting will be shown to have a strong effect on the stability of non-dissipative approximations. Representative test cases for both smooth flows and problems containing discontinuities for the ideal MHD are included. The results illustrate the improved stability by using the skew-symmetric splitting as part of the central base scheme instead of the pure high order central scheme.

Highlights

Starting the early 80s skew-symmetric splitting of certain component of the inviscid flux derivatives in conjunction with central schemes was shown to help with numerical stability for long time integration
The objective of this paper is to develop and test four variants of Ducros et al [6] type skew-symmetric splitting methods for equations of compressible ideal MHD, including problems with discontinuities
Note that the perfect gas dynamics conservation laws are straightly hyperbolic equations and the inviscid flux derivatives consist of double product functions of conservative variables terms

Summary

Introduction

Starting the early 80s skew-symmetric splitting of certain component of the inviscid flux derivatives in conjunction with central schemes was shown to help with numerical stability for long time integration. Some of the skew-symmetric splitting for the gas dynamics flux derivatives are not applicable and/or cannot straightforwardly be extended to the ideal MHD [35]. Their construction is dependent on the MHD governing equation formulation. For the Ducros et al type of splitting, the inviscid flux derivatives consist of triple product functions of conservative variables terms. Note that the perfect gas dynamics conservation laws are straightly hyperbolic equations and the inviscid flux derivatives consist of double product functions of conservative variables terms. One Ducros et al split-type approximation of the gas dynamics flux derivative that will be used in this study is given by. (A4,m, Θ(jp+)1/2(A5,m, um) where Ak,m denotes element (k, m) of the matrix valued function A(x), and um denotes the mth componen of the vector u

A Semi-conservative Entropy Splitting of the Euler Flux Derivatives

Numerical results

Conclusions