Variation entropy: a continuous local generalization of the TVD property using entropy principles

TL;DR

This paper introduces variation entropy, a new entropy-based framework for analyzing the gradient of solutions to conservation laws, providing insights into TVD properties and aiding the development of numerical methods for discontinuities.

Contribution

It proposes the concept of variation entropy as a continuous generalization of TVD, connecting entropy principles with gradient-based analysis in conservation laws.

Findings

All semi-norms are admissible variation entropies.

Provides a new local variation diminishing property in continuous form.

Facilitates design of numerical methods for discontinuous problems.

Abstract

This paper presents the notion of a variation entropy. This concept is an entropy framework for the gradient of the solution of a conservation law instead of on the solution itself. It appears that all semi-norms are admissible variation entropies. This provides insight into the total variation diminishing property and justifies it from entropy principles. The framework allows to derive new local variation diminishing properties in the continuous form. This can facilitate the design of new numerical methods for problems that contain discontinuities.