A robust high-order Lagrange-projection like scheme with large time steps for the isentropic Euler equations

TL;DR

This paper introduces a high-order, robust Lagrange-projection like scheme for the isentropic Euler equations that enables large time steps while maintaining stability, positivity, and entropy conditions through a discontinuous Galerkin approach.

Contribution

It extends a first-order method to high-order accuracy using a discontinuous Galerkin scheme with conditions ensuring positivity and entropy stability at any spatial order.

Findings

Method achieves large time steps without stability loss

Numerical experiments confirm robustness and stability

Positivity and entropy conditions are maintained at all approximation orders

Abstract

We present an extension to high-order of a first-order Lagrange-projection like method for the approximation of the Euler equations introduced in Coquel {\it et al.} (Math. Comput., 79 (2010), pp.~1493--1533). The method is based on a decomposition between acoustic and transport operators associated to an implicit-explicit time integration, thus relaxing the constraint of acoustic waves on the time step. We propose here to use a discontinuous Galerkin method for the space approximation. Considering the isentropic Euler equations, we derive conditions to keep positivity of the mean value of density and satisfy an entropy inequality for the numerical solution in each element of the mesh at any approximation order in space. These results allow to design limiting procedures to restore these properties at nodal values within elements. Numerical experiments support the conclusions of the analysis and highlight stability and robustness of the present method, though it allows the use of large time steps.