Embedded discontinuous Galerkin transport schemes with localised limiters

TL;DR

This paper introduces a stable embedded discontinuous Galerkin transport scheme with localized limiters for finite element methods used in weather prediction, ensuring boundedness and stability in numerical simulations.

Contribution

It presents a novel embedded DG scheme with localized limiters for partially-continuous finite element spaces, enhancing stability and boundedness in transport simulations.

Findings

Scheme is stable in L2 norm

Localized flux-correction effectively limits projections

Numerical tests demonstrate scheme's effectiveness

Abstract

Motivated by finite element spaces used for representation of temperature in the compatible finite element approach for numerical weather prediction, we introduce locally bounded transport schemes for (partially-)continuous finite element spaces. The underlying high-order transport scheme is constructed by injecting the partially-continuous field into an embedding discontinuous finite element space, applying a stable upwind discontinuous Galerkin (DG) scheme, and projecting back into the partially-continuous space; we call this an embedded DG scheme. We prove that this scheme is stable in L2 provided that the underlying upwind DG scheme is. We then provide a framework for applying limiters for embedded DG transport schemes. Standard DG limiters are applied during the underlying DG scheme. We introduce a new localised form of element-based flux-correction which we apply to limiting the projection back into the partially-continuous space, so that the whole transport scheme is bounded. We provide details in the specific case of tensor-product finite element spaces on wedge elements that are discontinuous P1/Q1 in the horizontal and continuous P2 in the vertical. The framework is illustrated with numerical tests.