Embedded discontinuous Galerkin transport schemes with localised limiters

C.J Cotter,D Kuzmin

doi:10.1016/j.jcp.2016.02.021

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.

Highlights

We address the problem of finding suitable limiters for the partially continuous finite element spaces for tracers that arise in the framework of compatible finite element methods for numerical weather prediction models (Cotter and Shipton, 2012; Cotter and Thuburn, 2014; Staniforth et al, 2013; McRae and Cotter, 2014)
In this paper we described a limited transport scheme for partially-continuous finite element spaces
Motivated by numerical weather prediction applications, where the finite element space for temperature and other tracers is imposed by hydrostatic balance and wave propagation properties, we focussed on the case of tensor-product elements that are continuous in the vertical direction but discontinuous in the horizontal

Summary

Introduction

There has been a lot of activity in the development of finite element methods for numerical weather prediction (NWP), using continuous (mainly spectral) finite elements as well as discontinuous finite elements (Fournier et al, 2004; Thomas and Loft, 2005; Dennis et al, 2011; Kelly and Giraldo, 2012; Giraldo et al, 2013; Marras et al, 2013; Brdar et al, 2013; Bao et al, 2015); see Marras et al (2015) for a comprehensive review. We address the problem of finding suitable limiters for the partially continuous finite element spaces for tracers that arise in the framework of compatible finite element methods for numerical weather prediction models (Cotter and Shipton, 2012; Cotter and Thuburn, 2014; Staniforth et al, 2013; McRae and Cotter, 2014). When combined with standard limiters for the discontinuous Galerkin stage, the overall scheme remains locally bounded, addressing the previously unsolved problem of how to limit partially continuous finite element spaces that arise in the compatible finite element framework.

Finite element spaces

Bounded transport

Slope limiter for the propagator A

Flux corrected remapping

Numerical Experiments

Summary and Outlook