On Affordable High-Order Entropy-Conservative/Stable and Well-Balanced Methods for Nonconservative Hyperbolic Systems

TL;DR

This paper introduces new, simple-to-implement entropy-preserving numerical methods for nonconservative hyperbolic systems, extending existing theories to higher-order, well-balanced, and multi-dimensional schemes with demonstrated robustness.

Contribution

It proposes specific entropy-preserving fluctuations for nonconservative systems, enabling an algorithmic construction of high-order, well-balanced, and multi-dimensional entropy-preserving methods.

Findings

New entropy-preserving schemes for Euler and shallow-water equations.

Numerical experiments confirm robustness and accuracy.

Extension of entropy-preserving methods to multi-dimensional SBP frameworks.

Abstract

Many entropy-conservative and entropy-stable (summarized as entropy-preserving) methods for hyperbolic conservation laws rely on Tadmor's theory for two-point entropy-preserving numerical fluxes and its higher-order extension via flux differencing using summation-by-parts (SBP) operators, e.g., in discontinuous Galerkin spectral element methods (DGSEMs). The underlying two-point formulations have been extended to nonconservative systems using fluctuations by Castro et al. (2013, doi:10.1137/110845379) with follow-up generalizations to SBP methods. We propose specific forms of entropy-preserving fluctuations for nonconservative hyperbolic systems that are simple to interpret and allow an algorithmic construction of entropy-preserving methods. We analyze necessary and sufficient conditions, and obtain a full characterization of entropy-preserving three-point methods within the finite volume framework. This formulation is extended to SBP methods in multiple space dimensions on Cartesian and curvilinear meshes. Additional properties such as well-balancedness extend naturally from the underlying finite volume method to the SBP framework. We use the algorithmic construction enabled by the chosen formulation to derive several new entropy-preserving schemes for nonconservative hyperbolic systems, e.g., the compressible Euler equations of an ideal gas using the internal energy equation and a dispersive shallow-water model. Numerical experiments show the robustness and accuracy of the proposed schemes.