Local Characteristic Decomposition of Equilibrium Variables for Hyperbolic Systems of Balance Laws

TL;DR

This paper introduces a new local characteristic decomposition method for equilibrium variables to improve high-order numerical schemes for hyperbolic balance laws, enhancing stability and accuracy in preserving steady states.

Contribution

The paper develops a novel LCD technique for equilibrium variables, integrated with fifth-order Ai-WENO-Z interpolation, to improve well-balanced high-order schemes for hyperbolic systems.

Findings

Enhanced preservation of steady states in numerical simulations.

Reduced oscillations in high-order reconstructions.

Improved accuracy demonstrated on various examples.

Abstract

This paper is concerned with high-order numerical methods for hyperbolic systems of balance laws. Such methods are typically based on high-order piecewise polynomial reconstructions (interpolations) of the computed discrete quantities. However, such reconstructions (interpolations) may be oscillatory unless the reconstruction (interpolation) procedure is applied to the local characteristic variables via the local characteristic decomposition (LCD). Another challenge in designing accurate and stable high-order schemes is related to enforcing a delicate balance between the fluxes, sources, and nonconservative product terms: a good scheme should be well-balanced (WB) in the sense that it should be capable of exactly preserving certain (physically relevant) steady states. One of the ways to ensure that the reconstruction (interpolation) preserves these steady states is to apply the reconstruction (interpolation) to the equilibrium variables, which are supposed to be constant at the steady states. To achieve this goal and to keep the reconstruction (interpolation) non-oscillatory, we introduce a new LCD of equilibrium variables. We apply the developed technique to the fifth-order Ai-WENO-Z interpolation implemented within the WB A-WENO framework recently introduced in [S. Chu, A. Kurganov, and R. Xin, Beijing J. of Pure and Appl. Math., to appear], and illustrate its performance on a variety of numerical examples.