Positivity and Maximum Principle Preserving Discontinuous Galerkin Finite Element Schemes for a Coupled Flow and Transport

TL;DR

This paper develops a new locally conservative flux concept for discontinuous Galerkin schemes that ensures positivity and maximum principle preservation in coupled flow and transport models, backed by theoretical analysis and numerical validation.

Contribution

It introduces a novel locally conservative flux concept that guarantees positivity and maximum principle in DG schemes for coupled flow and transport, a longstanding challenge.

Findings

Positivity and maximum principle are preserved using the new flux concept.

Theoretical stability results are established for the DG scheme.

Numerical experiments confirm the theoretical predictions.

Abstract

We introduce a new concept of the locally conservative flux and investigate its relationship with the compatible discretization pioneered by Dawson, Sun and Wheeler [11]. We then demonstrate how the new concept of the locally conservative flux can play a crucial role in obtaining the L2 norm stability of the discontinuous Galerkin finite element scheme for the transport in the coupled system with flow. In particular, the lowest order discontinuous Galerkin finite element for the transport is shown to inherit the positivity and maximum principle when the locally conservative flux is used, which has been elusive for many years in literature. The theoretical results established in this paper are based on the equivalence between Lesaint-Raviart discontinuous Galerkin scheme and Brezzi-Marini-Suli discontinuous Galerkin scheme for the linear hyperbolic system as well as the relationship between the Lesaint-Raviart discontinuous Galerkin scheme and the characteristic method along the streamline. Sample numerical experiments have also been performed to justify our theoretical findings