Entropy dissipation of moving mesh adaptation

TL;DR

This paper introduces a mathematical analysis method for mesh adaptation schemes in numerical simulations, demonstrating how proper mesh adaptation can ensure entropy stability and improve solution quality.

Contribution

It provides a new analytical framework that incorporates both mesh evolution and solution dynamics, offering conditions for entropy stability in adaptive mesh methods.

Findings

Derived sufficient conditions for entropy stability with mesh adaptation

Established a framework for analyzing mesh reconstruction and solution evolution

Demonstrated improved numerical stability through entropy dissipation

Abstract

Non-uniform grids and mesh adaptation have been a growing part of numerical simulation over the past years. It has been experimentally noted that mesh adaptation leads not only to locally improved solution but also to numerical stability of the underlying method. There have been though only few results on the mathematical analysis of these schemes due to the lack of proper tools that incorporate both the time evolution and the mesh adaptation step of the overall algorithm. In this paper we provide a method to perform the analysis of the mesh adaptation method, including both the mesh reconstruction and evolution of the solution. We moreover employ this method to extract sufficient conditions -on the adaptation of the mesh- that stabilize a numerical scheme in the sense of the entropy dissipation.