A general framework to construct schemes satisfying additional conservation relations. Application to entropy conservative and entropy dissipative schemes

TL;DR

This paper presents a general framework for constructing numerical schemes that preserve additional conservation laws, such as entropy, without degrading accuracy, applicable to both scalar and system hyperbolic problems.

Contribution

It introduces a flexible method to build schemes satisfying extra conservation relations, improving upon recent results by avoiding constraints on quadrature formulas.

Findings

Scheme maintains original PDE accuracy with at most one order degradation.

Explicit entropy conservative scheme demonstrated for hyperbolic systems.

Method applicable to discontinuous Galerkin and residual distribution schemes.

Abstract

We are interested in the approximation of a steady hyperbolic problem. In some cases, the solution can satisfy an additional conservation relation, at least when it is smooth. This is the case of an entropy. In this paper, we show, starting from the discretisation of the original PDE, how to construct a scheme that is consistent with the original PDE and the additional conservation relation. Since one interesting example is given by the systems endowed by an entropy, we provide one explicit solution, and show that the accuracy of the new scheme is at most degraded by one order. In the case of a discontinuous Galerkin scheme and a Residual distribution scheme, we show how not to degrade the accuracy. This improves the recent results obtained in [1, 2, 3, 4] in the sense that no particular constraints are set on quadrature formula and that a priori maximum accuracy can still be achieved. We study the behavior of the method on a non linear scalar problem. However, the method is not restricted to scalar problems.