# Convergence of second-order, entropy stable methods for multi-dimensional conservation laws

## TL;DR

This paper proves the convergence of a high-order entropy stable numerical method for multi-dimensional hyperbolic conservation laws, establishing theoretical guarantees for its accuracy and stability.

## Contribution

It demonstrates how entropy stability leads to convergence in multiple dimensions, filling a gap in rigorous analysis of such methods.

## Key findings

- Entropy stability bounds oscillations
- Convergence to weak entropy solutions is proven
- Provides theoretical foundation for high-order methods in multiple dimensions

## Abstract

High-order accurate, $\textit{entropy stable}$ numerical methods for hyperbolic conservation laws have attracted much interest over the last decade, but only a few rigorous convergence results are available, particularly in multiple space dimensions. In this paper we show how the entropy stability of one such method yields a (weak) bound on oscillations, and using compensated compactness we prove convergence to a weak solution satisfying at least one entropy condition.