(S)-convergence and approximation of oscillatory solutions in fluid dynamics

TL;DR

This paper introduces (S)-convergence, a novel concept for analyzing the asymptotic behavior of numerical approximations in fluid dynamics, particularly for oscillatory solutions of the Euler system.

Contribution

It presents (S)-convergence as a new framework based on averaging techniques, offering improved insights over traditional Young measures for gas dynamics approximations.

Findings

(S)-convergence captures asymptotic properties more effectively.

Links to ergodic theory are discussed.

Applicable to numerical methods and other approximations.

Abstract

We propose a new concept of (S)-convergence applicable to numerical methods as well as other consistent approximations of the Euler system in gas dynamics. (S)-convergence, based on averaging in the spirit of Strong Law of Large Numbers, reflects the asymptotic properties of a given approximate sequence better than the standard description via Young measures. Similarity with the tools of ergodic theory is discussed.