One-dimensional flows of a polytropic gas: Lie group classification, conservation laws, invariant and conservative difference schemes

TL;DR

This paper classifies symmetries and conservation laws of one-dimensional polytropic gas flows in Lagrangian coordinates, and develops invariant and conservative numerical schemes for these flows.

Contribution

It performs Lie group classification, derives conservation laws via Noether's theorem, and constructs invariant and conservative difference schemes for various flow symmetries.

Findings

Lie group classification of the governing PDEs

Derivation of conservation laws using variational structure

Development of invariant and conservative difference schemes

Abstract

The paper considers one-dimensional flows of a polytropic gas in the Lagrangian coordinates in three cases: plain one-dimensional flows, radially symmetric flows and spherically symmetric flows. The one-dimensional flow of a polytropic gas is described by one second-order partial differential equation in the Lagrangian variables. Lie group classification of this PDE is performed. Its variational structure allows to construct conservation laws with the help of Noether's theorem. These conservation laws are also recalculated for the gas dynamics variables in the Lagrangian and Eulerian coordinates. Additionally, invariant and conservative difference schemes are provided.