On the Theoretical Foundation of Overset Grid Methods for Hyperbolic Problems II: Entropy Bounded Formulations for Nonlinear Conservation Laws

TL;DR

This paper develops entropy-based formulations for overset grid methods applied to nonlinear hyperbolic conservation laws, ensuring entropy conservation and dissipation at interfaces for improved numerical stability.

Contribution

It introduces entropy conserving and dissipative overlapping domain formulations with nonlinear penalty functions for hyperbolic systems, extending the theoretical foundation of overset grid methods.

Findings

Entropy conserving formulations with nonlinear penalties.

Entropy dissipative extensions for overlap regions.

Theoretical framework for stable overset mesh methods.

Abstract

We derive entropy conserving and entropy dissipative overlapping domain formulations for systems of nonlinear hyperbolic equations in conservation form, such as would be approximated by overset mesh methods. The entropy conserving formulation imposes two-way coupling at the artificial interface boundaries through nonlinear penalty functions that vanish when the solutions coincide. The penalty functions are expressed in terms of entropy conserving fluxes originally introduced for finite volume schemes. Entropy dissipation and additional coupling in the overlap region are added through the use of linear penalties.