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

arXiv (Cornell University)

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., 2 (2025), pp. 87--113], and illustrate its performance on a variety of numerical examples.

Authors 4

1. Shaoshuai Chu Aachen Department of Mathematics

   RWTH Aachen University

   Affiliation as printed

   Department of Mathematics , RWTH Aachen University , 52056 Aachen , Germany;

2. Mingye Na

   Southern University of Science and Technology

   Affiliation as printed

   Department of Mathematics , Southern University of Science and Technology , Shenzhen , 518055 , China;

3. Bao Shan Wang

   Southern University of Science and Technology · Ocean University of China

   Affiliation as printed

   School of Mathematical Sciences & Laboratory of Marine Mathematics , Ocean University of China , Qingdao , 266100 , China;

   Shenzhen International Center for Mathematics , Southern University of Sci- ence and Technology , Shenzhen , 518055 , China;

4. Ruixiao Xin

   Southern University of Science and Technology

   Affiliation as printed

   Department of Mathematics , Southern University of Science and Technology , Shenzhen , 518055 , China;