Embedded discontinuous Galerkin transport schemes with localised limiters

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

Compatible finite element methods for geophysical fluid dynamics

This article surveys research on the application of compatible finite element methods to large scale atmosphere and ocean simulation. Compatible finite element methods extend Arakawa's C-grid finite difference scheme to the finite element world. They are constructed from a discrete de Rham complex, which is a sequence of finite element spaces which are linked by the operators of differential calculus. The use of discrete de Rham complexes to solve partial differential equations is well established, but in this article we focus on the specifics of dynamical cores for simulating weather, oceans and climate. The most important consequence of the discrete de Rham complex is the Hodge-Helmholtz decomposition, which has been used to exclude the possibility of several types of spurious oscillations from linear equations of geophysical flow. This means that compatible finite element spaces provide a useful framework for building dynamical cores. In this article we introduce the main concepts of compatible finite element spaces, and discuss their wave propagation properties. We survey some methods for discretising the transport terms that arise in dynamical core equation systems, and provide some example discretisations, briefly discussing their iterative solution. Then we focus on the recent use of compatible finite element spaces in designing structure preserving methods, surveying variational discretisations, Poisson bracket discretisations, and consistent vorticity transport.Comment: correction of some typo

Compatible finite element spaces for geophysical fluid dynamics

This is the final version. Available from Oxford University Press via the DOI in this record.Compatible finite elements provide a framework for preserving important structures in equations of geophysical fluid dynamics and are becoming important in their use for building atmosphere and ocean models. We survey the application of compatible finite element spaces to geophysical fluid dynamics, including the application to the non-linear rotating shallow water equations, and the three-dimensional compressible Euler equations. We summarize analytic results about dispersion relations and conservation properties and present new results on approximation properties in three dimensions on the sphere and on hydrostatic balance properties

On coupling resolved and unresolved physical processes in finite element discretisations of geophysical fluids

At the heart of modern numerical weather forecasting and climate modelling lie simulations of two geophysical fluids: the atmosphere and the ocean. These endeavours rely on numerically solving the equations that describe these fluids. A key challenge is that the fluids contain motions spanning a range of scales. As the small-scale processes (unresolved by the numerical model) affect the resolved motions, they need to be described in the model, which is known as parametrisation. One major class of methods for numerically solving such partial differential equations is the finite element method. This thesis focuses on the coupling of such parametrised processes to the resolved flow within finite element discretisations. Four sets of research are presented, falling under two main categories. The first is the development of a compatible finite element discretisation for use in numerical weather prediction models, so as to avoid the bottleneck in computational scalability associated with the convergence at the poles of latitude-longitude grids. We present a transport scheme for use with the lowest-order function spaces in such a compatible finite element method, which is motivated by the coupling of the resolved and unresolved processes within the model. This then facilitates the use of the lower-order spaces within Gusto, a toolkit for studying such compatible finite element discretisations. Then, we present a compatible finite element discretisation of the moist compressible Euler equations, parametrising the unresolved moist processes. This is a major step in the development of Gusto, extending it to describe its first unresolved processes. The second category with which this thesis is concerned is the stochastic variational framework presented by Holm [Variational principles for stochastic fluid dynamics, P. Roy. Soc. A-Math. Phy. 471 (2176), (2015)]. In this framework, the effect of the unresolved processes and their uncertainty is expressed through a stochastic component to the advecting velocity. This framework ensures the circulation theorem is preserved by the stochastic equations. We consider the application of this formulation to two simple geophysical fluid models. First, we discuss the statistical properties of an enstrophy-preserving finite element discretisation of the stochastic quasi-geostrophic equation. We find that the choice of discretisation and the properties that it preserves affects the statistics of the solution. The final research presented is a finite element discretisation of the stochastic Camassa-Holm equation, which is used to numerically investigate the formation of ‘peakons’ within this set-up, finding that they do still always form despite the noise’s presence.Open Acces

Two-dimensional discontinuous Galerkin shallow water model for practical flood modelling applications

Finite volume (FV) numerical solvers to the two-dimensional shallow water equations are the foundation of the current state-of-the-practice, industry-standard flood models. The second-order Discontinuous Galerkin (DG2) alternative show a promising way to improve current FV-based flood model formulations, but is yet under-studied and rarely utilised to support flood modelling applications. This is contributed by the mathematical complexity constructed within the DG2 formulation that could lead to large computational costs and compromise its stability and robustness when used for practical modelling. Therefore, this PhD research aims to develop a new flood model based on simplified DG2 solver that is improved for flood modelling practices. To achieve this aim, three objectives have been formed and addressed through analyses involving academic and experimental test cases, as well as test cases that are recommended by the UK Environment Agency to validate 2D flood model capabilities, whilst benchmarking the simplified DG2 solver against four FV-based industrial models. Key research findings indicate that the simplified DG2 solver can equally retain conservative properties and provide second-order accurate predictions as the standard DG2 solver whilst offering around 2.6 times runtime speed up. Additionally, the simplified DG2 solver can be reliably efficient to provide predictions close to the outputs of the industrial models, in simulating flood scenarios covering large catchment-scale areas and at a grid resolution greater or equal than 5 m, particularly when the local limiting is disabled. However, the local limiting is still needed by the simplified DG2 solver when modelling detailed velocity fields at sub-metre grid resolutions, particularly in regions of highly active wave-structure interactions as commonly encountered in urban flooding around steep-sloped building structures

Proceedings of the FEniCS Conference 2017

Proceedings of the FEniCS Conference 2017 that took place 12-14 June 2017 at the University of Luxembourg, Luxembourg

Compatible finite element methods for atmospheric dynamical cores

A key part of numerical weather prediction is the simulation of the partial differential equations governing atmospheric flow over the Earth's surface. This is typically performed on supercomputers at national and international centres around the world. In the last decade, there has been a relative plateau in single-core computing performance. Running ever-finer forecasting models has necessitated the use of ever-larger numbers of CPU cores. Several current forecasting models, including those favoured by the Met Office, use an underlying latitude--longitude grid. This facilitates the development of finite difference discretisations with favourable numerical properties. However, such models are inherently unable to make efficient use of large numbers of processors, as a result of the excessive concentration of gridpoints in the vicinity of the poles. A certain class of mixed finite element methods have recently been proposed in order to obtain favourable numerical properties on an arbitrary -- in particular, quasi-uniform -- mesh. This thesis supports the proposition that such finite element methods, which we label ``compatible'', or ``mimetic'', are suitable for discretising the equations used in an atmospheric dynamical core. We firstly show promising results applying these methods to the nonlinear rotating shallow-water equations. We then develop sophisticated tensor product finite elements for use in 3D. Finally, we give a discretisation for the fully-compressible 3D equations.Open Acces