Structure-preserving schemes conserving entropy and kinetic energy

TL;DR

This paper introduces a new numerical scheme for Euler equations that preserves entropy and kinetic energy, improving stability and accuracy in simulations through innovative flux functions and hybrid methods.

Contribution

The paper develops a novel structure-preserving scheme that simultaneously conserves entropy and kinetic energy, integrating entropy-stable shock capturing and hybrid approaches within finite-volume methods.

Findings

Effective entropy and kinetic energy conservation demonstrated

Enhanced stability with entropy-stable shock capturing

Accurate results across various test cases

Abstract

This paper presents a novel structure-preserving scheme for Euler equations, focusing on the numerical conservation of entropy and kinetic energy. Explicit flux functions engineered to conserve entropy are introduced within the finite-volume framework. Further, discrete kinetic energy conservation too is introduced. A systematic inquiry is presented, commencing with an overview of numerical entropy conservation and formulation of entropy-conserving and kinetic energy-preserving fluxes, followed by the study of their properties and efficacy. A novelty introduced is to associate numerical entropy conservation to the discretization of the energy conservation equation. Furthermore, an entropy-stable shock-capturing diffusion method and a hybrid approach utilizing the entropy distance to manage smooth regions effectively are also introduced. The addition of artificial viscosity in appropriate regions ensures entropy generation sufficient to prevent numerical instabilities. Various test cases, showcasing the efficacy and stability of the proposed methodology, are presented.