## Publications

99 results found

McRae ATT, Mitchell L, Bercea,
et al., 2016, Automated Generation and Symbolic Manipulation of Tensor Product Finite Elements, *SIAM Journal on Scientific Computing*, Vol: 38, Pages: S25-S47, ISSN: 1095-7197

We describe and implement a symbolic algebra for scalar and vector-valued finite elements, enabling the computer generation of elements with tensor product structure on quadrilateral, hexahedral, and triangular prismatic cells. The algebra is implemented as an extension to the domain-specific language UFL, the Unified Form Language. This allows users to construct many finite element spaces beyond those supported by existing software packages. We have made corresponding extensions to FIAT, the FInite element Automatic Tabulator, to enable numerical tabulation of such spaces. This tabulation is consequently used during the automatic generation of low-level code that carries out local assembly operations, within the wider context of solving finite element problems posed over such function spaces. We have done this work within the code-generation pipeline of the software package Firedrake; we make use of the full Firedrake package to present numerical examples.

Abolghasemi M, Piggott MD, Spinneken J,
et al., 2016, Simulating tidal turbines with multi-scale mesh optimisation techniques, *Journal of Fluids and Structures*, Vol: 66, Pages: 69-90, ISSN: 1095-8622

Embedding tidal turbines within simulations of realistic large-scale tidal flows is a highly multi-scale problem that poses significant computational challenges. Here this problem is tackled using actuator disc momentum (ADM) theory and Reynolds-averaged Navier-Stokes (RANS) with, for the first time, dynamically adaptive mesh optimisation techniques. Both k-ω and k-ω SST RANS models have been developed within the Fluidity framework, an adaptive mesh CFD solver, and the model is validated against two sets of experimental flume test results. A brief comparison against a similar OpenFOAM model is presented to portray the benefits of the finite element discretisation scheme employed in the Fluidity ADM model. This model has been developed with the aim that it will be seamlessly combined with larger numerical models simulating tidal flows in realistic domains. This is where the mesh optimisation capability is a major advantage as it enables the mesh to be refined dynamically in time and only in the locations required, thus making optimal use of limited computational resources.

Cotter CJ, Eldering J, Holm DD,
et al., 2016, Weak dual pairs and jetlet methods for ideal incompressible fluid models in n >= 2 dimensions, *Journal of Nonlinear Science*, Vol: 26, Pages: 1723-1765, ISSN: 1432-1467

We review the role of dual pairs in mechanics and use them to derive particle-like solutions to regularized incompressible fluid systems. In our case we have a dual pair resulting from the action of diffeomorphisms on point particles (essentially by moving the points). We then augment our dual pair by considering the action of diffeomorphisms on Taylor series, also known as jets. The augmented weak dual pairs induce a hierarchy of particle-like solutions and conservation laws with particles carrying a copy of a jet group. We call these augmented particles jetlets. The jet groups serve as finite-dimensional models of the diffeomorphism group itself, and so the jetlet particles serve as a finite-dimensional model of the self-similarity exhibited by ideal incompressible fluids. The conservation law associated to jetlet solutions is shown to be a shadow of Kelvin’s circulation theorem. Finally, we study the dynamics of infinite time particle mergers. We prove that two merging particles at the zeroth level in the hierarchy yield dynamics which asymptotically approach that of a single particle in the first level in the hierarchy. This merging behavior is then verified numerically as well as the exchange of angular momentum which must occur during a near collision of two particles. The resulting particle-like solutions suggest a new class of meshless methods which work in dimensions n≥2n≥2 and which exhibit a shadow of Kelvin’s circulation theorem. More broadly, this provides one of the first finite-dimensional models of self-similarity in ideal fluids.

Gregory A, Cotter CJ, Reich S, 2016, Multilevel Ensemble Transform Particle Filtering, *SIAM Journal on Scientific Computing*, Vol: 38, Pages: A1317-A1338, ISSN: 1095-7197

This paper extends the multilevel Monte Carlo variance reduction technique tononlinear filtering. In particular, multilevel Monte Carlo is applied to a certain variant of the particlefilter, the ensemble transform particle filter (EPTF). A key aspect is the use of optimal transportmethods to re-establish correlation between coarse and fine ensembles after resampling; this controlsthe variance of the estimator. Numerical examples present a proof of concept of the effectivenessof the proposed method, demonstrating significant computational cost reductions (relative to thesingle-level ETPF counterpart) in the propagation of ensembles.

Cotter CJ, Kuzmin D, 2016, Embedded discontinuous Galerkin transport schemes with localised limiters, *Journal of Computational Physics*, Vol: 311, Pages: 363-373, ISSN: 0021-9991

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

Jordi BE, Cotter CJ, Sherwin SJ, 2015, An adaptive selective frequency damping method, *Physics of Fluids*, Vol: 27, ISSN: 1089-7666

The selective frequency damping (SFD) method is an alternative to classical Newton’smethod to obtain unstable steady-state solutions of dynamical systems. However, thismethod has two main limitations: it does not converge for arbitrary control parameters,and when it does converge, the time necessary to reach a steady-state solution may bevery long. In this paper, we present an adaptive algorithm to address these two issues.We show that by evaluating the dominant eigenvalue of a “partially converged” steadyflow, we can select a control coefficient and a filter width that ensure an optimumconvergence of the SFD method. We apply this adaptive method to several classicaltest cases of computational fluid dynamics and we show that a steady-state solution canbe obtained with a very limited (or without any) a priori knowledge of the flow stabilityproperties

Cotter CJ, Kirby RC, 2015, Mixed finite elements for global tide models, *Numerische Mathematik*, Vol: 133, Pages: 255-277, ISSN: 0945-3245

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 L2L2 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.

Reich S, Cotter CJ, 2015, Probabilistic Forecasting and Bayesian Data Assimilation, http://www.cambridge.org/us/academic/subjects/mathematics/computational-science/probabilistic-forecasting-and-bayesian-data-assimilation, Publisher: Cambridge University Press, ISBN: 9781107663916

Abolghasemi M, Piggott MD, Spinneken J, et al., Simulating tidal turbines with mesh optimisation and RANS turbulence models, 2015 European Wave and Tidal Energy Conference

Thuburn J, Cotter CJ, 2015, A primal-dual mimetic finite element scheme for the rotating shallow water equations on polygonal spherical meshes, *JOURNAL OF COMPUTATIONAL PHYSICS*, Vol: 290, Pages: 274-297, ISSN: 0021-9991

Cotter CJ, McRae ATT, Compatible finite element methods for numerical weather prediction, *Proceedings of the EMCWF Annual Seminar*

This article takes the form of a tutorial on the use of a particular class ofmixed finite element methods, which can be thought of as the finite elementextension of the C-grid staggered finite difference method. The class is oftenreferred to as compatible finite elements, mimetic finite elements, discretedifferential forms or finite element exterior calculus. We provide anelementary introduction in the case of the one-dimensional wave equation,before summarising recent results in applications to the rotating shallow waterequations on the sphere, before taking an outlook towards applications inthree-dimensional compressible dynamical cores.

Cotter CJ, McRae ATT, 2014, Compatible finite elements for numerical weather prediction, ECMWF Annual Seminar

Thuburn J, Cotter CJ, Dubos T, 2014, A mimetic, semi-implicit, forward-in-time, finite volume shallow water model: comparison of hexagonal–icosahedral and cubed-sphere grids, *Geoscientific Model Development*, Vol: 7, Pages: 909-929

Melvin T, Staniforth A, Cotter C, 2014, A two-dimensional mixed finite-element pair on rectangles, *Quarterly Journal of the Royal Meteorological Society*, Vol: 140, Pages: 930-942, ISSN: 0035-9009

Jordi BE, Cotter CJ, Sherwin SJ, 2014, Encapsulated formulation of the selective frequency damping method, *Physics of Fluids*, Vol: 26, ISSN: 1089-7666

We present an alternative “encapsulated” formulation of the selective frequencydamping method for finding unstable equilibria of dynamical systems, which isparticularly useful when analysing the stability of fluid flows. The formulation makesuse of splitting methods, which means that it can be wrapped around an existingtime-stepping code as a “black box.” The method is first applied to a scalar problemin order to analyse its stability and highlight the roles of the control coefficient χand the filter width in the convergence (or not) towards the steady-state. Then thesteady-state of the incompressible flow past a two-dimensional cylinder at Re = 100,obtained with a code which implements the spectral/hp element method, is presented.

Farrell PE, Cotter CJ, Funke SW, 2014, A framework for the automation of generalised stability theory, *SIAM Journal on Scientific Computing*, Vol: 36, Pages: C25-C48

Cotter CJ, Thuburn J, 2014, A finite element exterior calculus framework for the rotating shallow-water equations, *Journal of Computational Physics*, Vol: 257, Pages: 1506-1526

McRae ATT, Cotter CJ, 2014, Energy- and enstrophy-conserving schemes for the shallow-water equations, based on mimetic finite elements, *Quarterly Journal of the Royal Meteorological Society*

This paper presents a family of spatial discretisations of the nonlinearrotating shallow-water equations that conserve both energy and potentialenstrophy. These are based on two-dimensional mixed finite element methods, andhence, unlike some finite difference methods, do not require an orthogonalgrid. Numerical verification of the aforementioned properties is also provided.

Visram AR, Cotter CJ, Cullen MJP, 2014, A framework for evaluating model error using asymptotic convergence in the Eady model, *Quarterly Journal of the Royal Meteorological Society*, Pages: n/a-n/a, ISSN: 0035-9009

Cotter CJ, Holm DD, 2014, Variational formulations of sound-proof models, *Quarterly Journal of the Royal Meteorological Society*

Vire A, Jiang J, Piggott MD, et al., 2014, Towards the Numerical Modelling of Floating Offshore Renewables, Fluid-Structure-Sound Interactions and Control, Editors: Zhou, Yang, Huang, Hodges, Publisher: Springer Berlin Heidelberg, Pages: 413-417, ISBN: 978-3-642-40370-5

Cotter CJ, Holm DD, Jacobs HJ,
et al., 2014, A jetlet hierarchy for ideal fluid dynamics, *J Phys A*, Vol: 47

Maddison JR, Cotter CJ, Farrell PE, 2013, Geostrophic balance preserving interpolation in mesh adaptive linearised shallow-water ocean modelling (vol 37, pg 35, 2011), *OCEAN MODELLING*, Vol: 68, Pages: 106-106, ISSN: 1463-5003

Cotter CJ, Holm DD, 2013, A variational formulation of vertical slice models, *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 469, ISSN: 1364-5021

A variational framework is defined for vertical slice models with three-dimensional velocity depending only on x and z. The models that result from this framework are Hamiltonian, and have a Kelvin–Noether circulation theorem that results in a conserved potential vorticity in the slice geometry. These results are demonstrated for the incompressible Euler–Boussinesq equations with a constant temperature gradient in the y-direction (the Eady–Boussinesq model), which is an idealized problem used to study the formation and subsequent evolution of weather fronts. We then introduce a new compressible extension of this model. Unlike the incompressible model, the compressible model does not produce solutions that are also solutions of the three-dimensional equations, but it does reduce to the Eady–Boussinesq model in the low Mach number limit. Hence, the new model could be used in asymptotic limit error testing for compressible weather models running in a vertical slice configuration.

Staniforth A, Melvin T, Cotter C, 2013, Analysis of a mixed finite-element pair proposed for an atmospheric dynamical core, *QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY*, Vol: 139, Pages: 1239-1254, ISSN: 0035-9009

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- Citations: 15

Cotter CJ, 2013, Data assimilation on the exponentially accurate slow manifold, *Journal: Philosophical Transactions A: Mathematical, Physical and Engineering Sciences*, Vol: 371

I describe an approach to data assimilation making use of an explicit map that defines a coordinate system on the slow manifold in the semi-geostrophic scaling in Lagrangian coordinates, and apply the approach to a simple toy system that has previously been proposed as a low-dimensional model for the semi-geostrophic scaling. The method can be extended to Lagrangian particle methods such as Hamiltonian particle–mesh and smooth-particle hydrodynamics applied to the rotating shallow-water equations, and many of the properties will remain for more general Eulerian methods. Making use of Hamiltonian normal-form theory, it has previously been shown that, if initial conditions for the system are chosen as image points of the map, then the fast components of the system have exponentially small magnitude for exponentially long times as ϵ→0, and this property is preserved if one uses a symplectic integrator for the numerical time stepping. The map may then be used to parametrize initial conditions near the slow manifold, allowing data assimilation to be performed without introducing any fast degrees of motion (more generally, the precise amount of fast motion can be selected).

Cotter CJ, Cotter SL, Vialard FX, 2013, Bayesian data assimilation in shape registration, *Inverse Problems in Science and Engineering*, Vol: 29

In this paper we apply a Bayesian framework to the problem of geodesic curve matching. Given a template curve, the geodesic equations provide a mapping from initial conditions for the conjugate momentum onto topologically equivalent shapes. Here, we aim to recover the well-defined posterior distribution on the initial momentum which gives rise to observed points on the target curve; this is achieved by explicitly including a reparameterization in the formulation. Appropriate priors are chosen for the functions which together determine this field and the positions of the observation points, the initial momentum p0 and the reparameterization vector field ν, informed by regularity results about the forward model. Having done this, we illustrate how maximum likelihood estimators can be used to find regions of high posterior density, but also how we can apply recently developed Markov chain Monte Carlo methods on function spaces to characterize the whole of the posterior density. These illustrative examples also include scenarios where the posterior distribution is multimodal and irregular, leading us to the conclusion that knowledge of a state of global maximal posterior density does not always give us the whole picture, and full posterior sampling can give better quantification of likely states and the overall uncertainty inherent in the problem.

Fogell NA, Sherwin S, Cotter CJ, et al., 2013, Fluid-structure iInteraction simulation of theinflated shape of ram-air parachutes, 22nd AIAA Aerodynamic Decelerator Systems Technology Conference, Daytona Beach, Florida [Best Student Paper Award]

Rognes ME, Ham DA, Cotter CJ,
et al., 2013, Automating the solution of PDEs on the sphere and other manifolds in FEniCS 1.2, *GEOSCIENTIFIC MODEL DEVELOPMENT*, Vol: 6, Pages: 2099-2119, ISSN: 1991-959X

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- Citations: 21

Vire A, Xiang J, Piggott M, et al., 2013, Towards the fully-coupled numerical modelling of floating wind turbines, 10th Deep Sea Offshore Wind R and D Conference (DeepWind), Publisher: ELSEVIER SCIENCE BV, Pages: 43-51, ISSN: 1876-6102

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- Citations: 11

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