A Flux-Differencing Formula for Split-Form Summation By Parts Discretizations of Non-Conservative Systems: Applications to Subcell Limiting for magneto-hydrodynamics

TL;DR

This paper introduces a flux-differencing formula for SBP discretizations of non-conservative hyperbolic systems, enabling hybrid high-order and low-order schemes that improve robustness in MHD shock problems.

Contribution

It presents a novel flux-differencing formula for SBP discretizations of non-conservative systems, facilitating hybrid schemes with enhanced shock-capturing capabilities.

Findings

Flux-differencing formula derived for SBP discretizations.

Hybrid schemes effectively handle strong shocks in MHD.

Improved robustness of high-order methods demonstrated.

Abstract

In this paper, we show that diagonal-norm summation by parts (SBP) discretizations of general non-conservative systems of hyperbolic balance laws can be rewritten as a finite-volume-type formula, also known as flux-differencing formula, if the non-conservative terms can be written as the product of a local and a symmetric contribution. Furthermore, we show that the existence of a flux-differencing formula enables the use of recent subcell limiting strategies to improve the robustness of the high-order discretizations. To demonstrate the utility of the novel flux-differencing formula, we construct hybrid schemes that combine high-order SBP methods (the discontinuous Galerkin spectral element method and a high-order SBP finite difference method) with a compatible low-order finite volume (FV) scheme at the subcell level. We apply the hybrid schemes to solve challenging magnetohydrodynamics (MHD) problems featuring strong shocks.