Bound-Preserving Framework for Central-Upwind Schemes for General Hyperbolic Conservation Laws

TL;DR

This paper introduces a novel bound-preserving framework for central-upwind schemes applied to hyperbolic conservation laws, ensuring solutions stay within physical bounds while maintaining scheme accuracy.

Contribution

The authors develop a new analysis framework for bound-preserving properties of CU schemes, enabling the construction of provably BPCU schemes for complex systems like Euler equations.

Findings

Successfully constructed BPCU schemes for Euler equations

Validated robustness with challenging numerical examples

Achieved bound preservation through minimal scheme modifications

Abstract

Central-upwind (CU) schemes are Riemann-problem-solver-free finite-volume methods widely applied to a variety of hyperbolic systems of PDEs. Exact solutions of these systems typically satisfy certain bounds, and it is highly desirable or even crucial for the numerical schemes to preserve these bounds. In this paper, we develop and analyze bound-preserving (BP) CU schemes for general hyperbolic systems of conservation laws. Unlike many other Godunov-type methods, CU schemes cannot, in general, be recast as convex combinations of first-order BP schemes. Consequently, standard BP analysis techniques are invalidated. We address these challenges by establishing a novel framework for analyzing the BP property of CU schemes. To this end, we discover that the CU schemes can be decomposed as a convex combination of several intermediate solution states. Thanks to this key finding, the goal of designing BPCU schemes is simplified to the enforcement of four more accessible BP conditions, each of which can be achieved with the help of a minor modification of the CU schemes. We employ the proposed approach to construct provably BPCU schemes for the Euler equations of gas dynamics. The robustness and effectiveness of the BPCU schemes are validated by several demanding numerical examples, including high-speed jet problems, flow past a forward-facing step, and a shock diffraction problem.