A Flux-Correction Form of the Third-Order Edge-Based Scheme for a General Numerical Flux Function

TL;DR

This paper introduces a flux-correction form for a third-order edge-based scheme that allows the use of general numerical flux functions while maintaining third-order accuracy for Euler equations.

Contribution

It proposes a novel flux-correction approach that enables direct use of various flux functions in a third-order scheme without losing accuracy.

Findings

Preserves third-order accuracy with general flux functions.

Verifies accuracy on irregular tetrahedral grids.

Demonstrates effectiveness with HLLC and LDFSS fluxes.

Abstract

In this short note, we present a flux-correction form of the third-order edge-based scheme for the Euler equations that enables the direct use of a general flux function. The core idea is to replace, without loss of accuracy, the arithmetic average of the flux extrapolations by a general numerical flux evaluated at the edge midpoint, together with a correction term. We show that the proposed flux-correction form preserves third-order accuracy, provided that the general numerical flux is evaluated with the left and right states that are computed exactly for a quadratic function, which can be achieved effectively by the U-MUSCL scheme with {\kappa} = 1/2. Numerical results are presented to verify third-order accuracy with the HLLC and LDFSS flux functions on irregular tetrahedral grids.