Hyperbolic balance laws: residual distribution, local and global fluxes

TL;DR

This paper introduces residual distribution schemes for hyperbolic balance laws, extending Roe's flux, and demonstrates their application to unsteady problems like Euler and Shallow Water equations.

Contribution

It generalizes fluctuation splitting schemes to unsteady hyperbolic systems and illustrates their use in complex conservation law problems.

Findings

Residual distribution schemes effectively handle unsteady hyperbolic systems.

The framework can incorporate prescribed physical constraints.

Examples include applications to Euler and Shallow Water equations.

Abstract

This paper describes a class of scheme named "residual distribution schemes" or "fluctuation splitting schemes". They are a generalization of Roe's numerical flux in fluctuation form. The so-called multidimensional fluctuation schemes have historically first been developed for steady homogeneous hyperbolic systems. Their application to unsteady problems and conservation laws has been really understood only relatively recently. This understanding has allowed to make of the residual distribution framework a powerful playground to develop numerical discretizations embedding some prescribed constraints. This paper describes in some detail these techniques, with several examples, ranging from the compressible Euler equations to the Shallow Water equations.