# The notion of conservation for residual distribution schemes (or fluctuation splitting schemes), with some applications

## TL;DR

This paper explores the concept of discrete conservation in hyperbolic conservation laws through fluctuation splitting schemes, demonstrating how existing schemes can be reformulated and extended to satisfy additional conservation laws.

## Contribution

It introduces fluctuation splitting schemes for residual distribution and shows how to reformulate existing schemes in flux form to achieve conservation.

## Key findings

- Most known schemes can be reformulated in flux form
- New schemes can satisfy additional conservation laws
- Provides a unified framework for residual distribution schemes

## Abstract

In this paper, we discuss the notion of discrete conservation for hyperbolic conservation laws. We introduce what we call a fluctuation splitting schemes (or residual distribution, also RDS) and show on several examples how this cal lead to new development. In particular, we show that most, if not all known schemes can be rephrased in flux form, and also show how to satisfy additional conservation laws. This review paper is built on \cite{AbgrallConservation,Abgrall2017,paola,svetlana,ABGRALL2018640}.