Colin J. Cotter

Andrew T. T. McRae, Colin J. Cotter, Chris J. Budd

In moving mesh methods, the underlying mesh is dynamically adapted without changing the connectivity of the mesh. We specifically consider the generation of meshes which are adapted to a scalar monitor function through equidistribution. Together with an optimal transport condition, this leads to a Monge-Ampère equation for a scalar mesh potential. We adapt an existing finite element scheme for the standard Monge-Ampère equation to this mesh generation problem; this is a mixed finite element scheme, in which an extra discrete variable is introduced to represent the Hessian matrix of second derivatives. The problem we consider has additional nonlinearities over the basic Monge-Ampère equation due to the implicit dependence of the monitor function on the resulting mesh. We also derive the equivalent Monge-Ampère-like equation for generating meshes on the sphere. The finite element scheme is extended to the sphere, and we provide numerical examples. All numerical experiments are performed using the open-source finite element framework Firedrake.

John Thuburn, Colin J. Cotter

A new numerical method is presented for solving the rotating shallow water equations on a rotating sphere using quasi-uniform polygonal meshes. The method uses special families of finite element function spaces to mimic key mathematical properties of the continuous equations and thereby capture several desirable physical properties related to balance and conservation. The method relies on two novel features. The first is the use of compound finite elements to provide suitable finite element spaces on general polygonal meshes. The second is the use of dual finite element spaces on the dual of the original mesh, along with suitably defined discrete Hodge star operators to map between the primal and dual meshes, enabling the use of a finite volume scheme on the dual mesh to compute potential vorticity fluxes. The resulting method has the same mimetic properties as a finite volume method presented previously, but is more accurate on a number of standard test cases.

Andrea Natale, Jemma Shipton, Colin J. Cotter

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 nonlinear rotating shallow water equations, and the three-dimensional compressible Euler equations. We summarise 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.

Colin J. Cotter, Sebastian Reich

This survey paper is written with the intention of giving a mathematical introduction to filtering techniques for intermittent data assimilation, and to survey some recent advances in the field. The paper is divided into three parts. The first part introduces Bayesian statistics and its application to statistical inference and estimation. Basic aspects of Markov processes, as they typically arise from scientific models in the form of stochastic differential and/or difference equations, are covered in the second part. The third and final part describes the filtering approach to estimation of model states by assimilation of observational data into scientific models. While most of the material is of survey type, very recent advances in the field of nonlinear data assimilation covered in this paper include a discussion of Bayesian inference in the context of optimal transportation and coupling of random variables, as well as a discussion of recent advances in ensemble transform filters. References and sources for further reading material will be listed at the end of each section.

Andrea Natale, Colin J. Cotter

We propose a finite element discretisation approach for the incompressible Euler equations which mimics their geometric structure and their variational derivation. In particular, we derive a finite element method that arises from a nonholonomic variational principle and an appropriately defined Lagrangian, where finite element H(div) vector fields are identified with advection operators; this is the first successful extension of the structure-preserving discretisation of Pavlov et al. (2009) to the finite element setting. The resulting algorithm coincides with the energy-conserving scheme presented in Guzmán et al. (2016). Through the variational derivation, we discover that it also satisfies a discrete analogous of Kelvin's circulation theorem. Further, we propose an upwind-stabilised version of the scheme which dissipates enstrophy whilst preserving energy conservation and the discrete Kelvin's theorem. We prove error estimates for this version of the scheme, and we study its behaviour through numerical tests.

Hiroe Yamazaki, Jemma Shipton, Michael J. P. Cullen et al.

A vertical slice model is developed for the Euler-Boussinesq equations with a constant temperature gradient in the direction normal to the slice (the Eady-Boussinesq model). The model is a solution of the full three-dimensional equations with no variation normal to the slice, which is an idealized problem used to study the formation and subsequent evolution of weather fronts. A compatible finite element method is used to discretise the governing equations. To extend the Charney-Phillips grid staggering in the compatible finite element framework, we use the same node locations for buoyancy as the vertical part of velocity and apply a transport scheme for a partially continuous finite element space. For the time discretisation, we solve the semi-implicit equations together with an explicit strong-stability-preserving Runge-Kutta scheme to all of the advection terms. The model reproduces several quasi-periodic lifecycles of fronts despite the presence of strong discontinuities. An asymptotic limit analysis based on the semi-geostrophic theory shows that the model solutions are converging to a solution in cross-front geostrophic balance. The results are consistent with the previous results using finite difference methods, indicating that the compatible finite element method is performing as well as finite difference methods for this test problem. We observe dissipation of kinetic energy of the cross-front velocity in the model due to the lack of resolution at the fronts, even though the energy loss is not likely to account for the large gap on the strength of the fronts between the model result and the semi-geostrophic limit solution.

Andrea Natale, Colin J. Cotter

We analyse the multiscale properties of energy-conserving upwind-stabilised finite element discretisations of the two-dimensional incompressible Euler equations. We focus our attention on two particular methods: the Lie derivative discretisation introduced in Natale and Cotter (2016a) and the Streamline Upwind/Petrov-Galerkin (SUPG) discretisation of the vorticity advection equation. Such discretisations provide control on enstrophy by modelling different types of scale interactions. We quantify the performance of the schemes in reproducing the non-local energy backscatter that characterises two-dimensional turbulent flows.

Colin J. Cotter, Robert C. Kirby

We study mixed finite element methods for the linearized rotating shallow water equations with linear drag and forcing terms. By means of a strong energy estimate for an equivalent second-order formulation for the linearized momentum, we prove long-time stability of the system without energy accumulation -- the geotryptic state. A priori error estimates for the linearized momentum and free surface elevation are given in $L^2$ as well as for the time derivative and divergence of the linearized momentum. Numerical results confirm the theoretical results regarding both energy damping and convergence rates.

Andreas Bock, Colin J. Cotter, Robert C. Kirby

We study parameterisation-independent closed planar curve matching as a Bayesian inverse problem. The motion of the curve is modelled via a curve on the diffeomorphism group acting on the ambient space, leading to a large deformation diffeomorphic metric mapping (LDDMM) functional penalising the kinetic energy of the deformation. We solve Hamilton's equations for the curve matching problem using the Wu-Xu element [S. Wu, J. Xu, Nonconforming finite element spaces for $2m^\text{th}$ order partial differential equations on $\mathbb{R}^n$ simplicial grids when $m=n+1$, Mathematics of Computation 88 (316) (2019) 531-551] which provides mesh-independent Lipschitz constants for the forward motion of the curve, and solve the inverse problem for the momentum using Bayesian inversion. Since this element is not affine-equivalent we provide a pullback theory which expedites the implementation and efficiency of the forward map. We adopt ensemble Kalman inversion using a negative Sobolev norm mismatch penalty to measure the discrepancy between the target and the ensemble mean shape. We provide several numerical examples to validate the approach.

Thomas M. Bendall, Colin J. Cotter

A framework of variational principles for stochastic fluid dynamics was presented by Holm (2015), and these stochastic equations were also derived by Cotter et al. (2017). We present a conforming finite element discretisation for the stochastic quasi-geostrophic equation that was derived from this framework. The discretisation preserves the first two moments of potential vorticity, i.e. the mean potential vorticity and the enstrophy. Following the work of Dubinkina and Frank (2007), who investigated the statistical mechanics of discretisations of the deterministic quasi-geostrophic equation, we investigate the statistical mechanics of our discretisation of the stochastic quasi-geostrophic equation. We compare the statistical properties of our discretisation with the Gibbs distribution under assumption of these conserved quantities, finding that there is agreement between the statistics under a wide range of set-ups.

Andreas Bock, Colin J. Cotter

We study the problem of diffeomorphometric geodesic landmark matching where the objective is to find a diffeomorphism that via its group action maps between two sets of landmarks. It is well-known that the motion of the landmarks, and thereby the diffeomorphism, can be encoded by an initial momentum leading to a formulation where the landmark matching problem can be solved as an optimisation problem over such momenta. The novelty of our work lies in the application of a derivative-free Bayesian inverse method for learning the optimal momentum encoding the diffeomorphic mapping between the template and the target. The method we apply is the ensemble Kalman filter, an extension of the Kalman filter to nonlinear observation operators. We describe an efficient implementation of the algorithm and show several numerical results for various target shapes.

Thomas M. Bendall, Colin J. Cotter, Jemma Shipton

We present a new compatible finite element advection scheme for the compressible Euler equations. Unlike the discretisations described in Cotter and Kuzmin (2016) and Shipton et al (2018), the discretisation uses the lowest-order family of compatible finite element spaces, but still retains second-order numerical accuracy. This scheme obtains this second-order accuracy by first `recovering' the function in higher-order spaces, before using the discontinuous Galerkin advection schemes of Cotter and Kuzmin (2016). As well as describing the scheme, we also present its stability properties and a strategy for ensuring boundedness. We then demonstrate its properties through some numerical tests, before presenting its use within a model solving the compressible Euler equations.

Colin J. Cotter, P. Jameson Graber, Robert C. Kirby

We study mixed finite element methods for the rotating shallow water equations with linearized momentum terms but nonlinear drag. By means of an equivalent second-order formulation, we prove long-time stability of the system without energy accumulation. We also give rates of damping in unforced systems and various continuous dependence results on initial conditions and forcing terms. \emph{A priori} error estimates for the momentum and free surface elevation are given in $L^2$ as well as for the time derivative and divergence of the momentum. Numerical results confirm the theoretical results regarding both energy damping and convergence rates.