High order entropy stable schemes for the quasi-one-dimensional shallow water and compressible Euler equations

TL;DR

This paper develops high order entropy stable numerical schemes for quasi-one-dimensional shallow water and Euler equations, ensuring stability and accuracy in modeling flow through channels with varying cross-sections.

Contribution

It introduces new non-symmetric entropy conservative fluxes and extends entropy stability to quasi-1D flow equations, including well-balanced schemes for shallow water.

Findings

Schemes are high order accurate and conservative.

Schemes are semi-discretely entropy stable.

For shallow water, schemes are well-balanced.

Abstract

High order schemes are known to be unstable in the presence of shock discontinuities or under-resolved solution features for nonlinear conservation laws. Entropy stable schemes address this instability by ensuring that physically relevant solutions satisfy a semi-discrete entropy inequality independently of discretization parameters. This work extends high order entropy stable schemes to the quasi-1D shallow water equations and the quasi-1D compressible Euler equations, which model one-dimensional flows through channels or nozzles with varying width. We introduce new non-symmetric entropy conservative finite volume fluxes for both sets of quasi-1D equations, as well as a generalization of the entropy conservation condition to non-symmetric fluxes. When combined with an entropy stable interface flux, the resulting schemes are high order accurate, conservative, and semi-discretely entropy stable. For the quasi-1D shallow water equations, the resulting schemes are also well-balanced.