Some remarks about conservation for residual distribution schemes

TL;DR

This paper unifies various residual distribution schemes within a common framework, highlighting their flux formulation and local conservation properties, and connects entropy stability concepts with recent developments in the field.

Contribution

It introduces a unified framework for residual distribution schemes, demonstrating their flux formulation and conservation properties, and relates entropy stability to recent research.

Findings

All known schemes can be expressed with a flux formulation.

Residual distribution schemes are shown to be locally conservative.

Entropy stability can be incorporated as an additional conservation relation.

Abstract

We are interested in the discretisation of the steady version of hyperbolic problems. We first show that all the known schemes (up to our knowledge) can be rephrased in a common framework. Using this framework, we first show all the known schemes have a flux formulation, with an explicit construction of the flux, and thus are locally conservative. This is well known for the finite volume schemes or the discontinuous Galerkin ones, much less known for the continuous finite element methods. We also show that Tadmor's entropy stability formulation can naturally be rephrased in this framework as an additional conservation relation discretisation, and using this, we show some conenction with the recent papers [1, 2, 3, 4]. This contribution is an enhanced version of [5].