High-order In-cell Discontinuous Reconstruction path-conservative methods for nonconservative hyperbolic systems -- DR.MOOD generalization

TL;DR

This paper introduces a novel high-order numerical framework for nonconservative hyperbolic systems, extending the MOOD methodology with in-cell discontinuous reconstruction to accurately capture shocks.

Contribution

It generalizes the MOOD approach using Taylor expansions and in-cell reconstruction, enabling precise shock capturing in nonconservative hyperbolic systems.

Findings

Successfully applied to Modified Shallow Water equations

Accurately captures isolated shocks

Demonstrates high-order accuracy in test cases

Abstract

In this work we develop a new framework to deal numerically with discontinuous solutions in nonconservative hyperbolic systems. First an extension of the MOOD methodology to nonconservative systems based on Taylor expansions is presented. This extension combined with an in-cell discontinuous reconstruction operator are the key points to develop a new family of high-order methods that are able to capture exactly isolated shocks. Several test cases are proposed to validate these methods for the Modified Shallow Water equations and the Two-Layer Shallow Water system.