In-cell Discontinuous Reconstruction path-conservative methods for non conservative hyperbolic systems -- Second-order extension

TL;DR

This paper introduces a second-order in-cell discontinuous reconstruction method for non-conservative hyperbolic systems, improving shock capturing while maintaining exact shock resolution, and lays groundwork for higher-order extensions.

Contribution

It extends a first-order in-cell reconstruction strategy to second-order, enhancing accuracy and shock capturing capabilities in non-conservative hyperbolic systems.

Findings

Successfully captures isolated shocks exactly.

Validates methods on various physical systems.

Lays foundation for high-order shock capturing methods.

Abstract

We are interested in the numerical approximation of discontinuous solutions in non conservative hyperbolic systems. An extension to second-order of a new strategy based on in-cell discontinuous reconstructions to deal with this challenging topic is presented. This extension is based on the combination of the first-order in-cell reconstruction and the MUSCL-Hancock reconstruction. The first-order strategy allowed in particular to capture exactly the isolated shocks and this new second-order extension keep this property. We also set the basis of an extension to high-order methods following this in-cell methodology and a way for capturing exactly more than one shock. Several numerical tests are proposed to validate the methods for the Coupled-Burgers system, Gas dynamics equations in Lagrangian coordinates and the modified shallow water system.