Abstract
In this paper, we discuss the notion of discrete conservation for hyperbolic conservation laws. We introduce what we call fluctuation splitting schemes (or residual distribution, also RDS) and show through several examples how these schemes lead to new developments. In particular, we show that most, if not all, known schemes can be rephrased in flux form and also how to satisfy additional conservation laws. This review paper is built on Abgrall et al. (Computers and Fluids 169:10–22, 2018), Abgrall and Tokareva (SIAM SISC 39(5):A2345–A2364, 2017), Abgrall (J Sci Comput 73:461–494, 2017), Abgrall (Methods Appl Math 18(3):327–351, 2018a) and Abgrall (J Comput Phys 372, 640–666, 2018b). This paper is also a direct consequence of the work of Roe, in particular Deconinck et al. (Comput Fluids 22(2/3):215–222, 1993) and Roe (J Comput Phys 43:357–372, 1981) where the notion of conservation was first introduced. In [26], Roe mentioned the Hermes project and the role of Dassault Aviation. Bruno Stoufflet, Vice President R&D and advanced business of this company, proposed me to have a detailed look at Deconinck et al. (Comput Fluids 22(2/3):215–222, 1993). To be honest, at the time, I did not understand anything, and this was the case for several years. I was lucky to work with Katherine Mer, who at the time was a postdoc, and is now research engineer at CEA. She helped me a lot in understanding the notion of conservation. The present contribution can be seen as the result of my understanding after many years of playing around with the notion of residual distribution schemes (or fluctuation-splitting schemes) introduced by Roe.
Abgrall, Rémi