# ProfessorColinCotter

Faculty of Natural SciencesDepartment of Mathematics

Professor of Computational Mathematics

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### Contact

+44 (0)20 7594 3468colin.cotter

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### Location

755Huxley BuildingSouth Kensington Campus

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## Publications

Publication Type
Year
to

93 results found

Bendall TM, Cotter CJ, Shipton J, 2019, The 'recovered space' advection scheme for lowest-order compatible finite element methods, JOURNAL OF COMPUTATIONAL PHYSICS, Vol: 390, Pages: 342-358, ISSN: 0021-9991

Journal article

Cotter C, Crisan D, Holm DD, Pan W, Shevchenko Iet al., 2019, A Particle Filter for Stochastic Advection by Lie Transport (SALT): A case study for the damped and forced incompressible 2D Euler equation, Publisher: arXiv

In this work, we apply a particle filter with three additional procedures(model reduction, tempering and jittering) to a damped and forcedincompressible 2D Euler dynamics defined on a simply connected bounded domain.We show that using the combined algorithm, we are able to successfullyassimilate data from a reference system state (the truth") modelled by ahighly resolved numerical solution of the flow that has roughly $3.1\times10^6$degrees of freedom for $10$ eddy turnover times, using modest computationalhardware. The model reduction is performed through the introduction of a stochasticadvection by Lie transport (SALT) model as the signal on a coarser resolution.The SALT approach was introduced as a general theory using a geometricmechanics framework from Holm, Proc. Roy. Soc. A (2015). This work follows onthe numerical implementation for SALT presented by Cotter et al, SIAMMultiscale Model. Sim. (2019) for the flow in consideration. The modelreduction is substantial: The reduced SALT model has $4.9\times 10^4$ degreesof freedom. Forecast reliability and estimated asymptotic behaviour of the particlefilter are also presented.

Working paper

Cotter CJ, Cullen MJP, 2019, Particle relabelling symmetries and Noether's theorem for vertical slice models, Journal of Geometric Mechanics, Vol: 11, Pages: 139-151, ISSN: 1941-4889

We consider the variational formulation for vertical slice models introduced in Cotter and Holm (Proc RoySoc, 2013). These models have a Kelvin circulation theorem that holds on all materially-transported closedloops, not just those loops on isosurfaces of potential temperature. Potential vorticity conservation can bederived directly from this circulation theorem. In this paper, we show that this property is due to these modelshaving a relabelling symmetry for every single diffeomorphism of the vertical slice that preserves the density, notjust those diffeomorphisms that preserve the potential temperature. This is developed using the methodologyof Cotter and Holm (Foundations of Computational Mathematics, 2012).

Journal article

Cotter CJ, Cullen MJP, 2019, PARTICLE RELABELLING SYMMETRIES AND NOETHER'S THEOREM FOR VERTICAL SLICE MODELS, Publisher: AMER INST MATHEMATICAL SCIENCES-AIMS

Working paper

Cotter C, Cotter S, Russell P, Ensemble transport adaptive importance sampling, SIAM/ASA Journal on Uncertainty Quantification, ISSN: 2166-2525

Markov chain Monte Carlo methods are a powerful and commonly used family ofnumerical methods for sampling from complex probability distributions. As applications of thesemethods increase in size and complexity, the need for efficient methods increases. In this paper, wepresent a particle ensemble algorithm. At each iteration, an importance sampling proposal distri-bution is formed using an ensemble of particles. A stratified sample is taken from this distributionand weighted under the posterior, a state-of-the-art ensemble transport resampling method is thenused to create an evenly weighted sample ready for the next iteration. We demonstrate that thisensemble transport adaptive importance sampling (ETAIS) method outperforms MCMC methodswith equivalent proposal distributions for low dimensional problems, and in fact shows better thanlinear improvements in convergence rates with respect to the number of ensemble members. We alsointroduce a new resampling strategy, multinomial transformation (MT), which while not as accurateas the ensemble transport resampler, is substantially less costly for large ensemble sizes, and canthen be used in conjunction with ETAIS for complex problems. We also focus on how algorithmicparameters regarding the mixture proposal can be quickly tuned to optimise performance. In partic-ular, we demonstrate this methodology’s superior sampling for multimodal problems, such as thosearising from inference for mixture models, and for problems with expensive likelihoods requiring thesolution of a differential equation, for which speed-ups of orders of magnitude are demonstrated.Likelihood evaluations of the ensemble could be computed in a distributed manner, suggesting thatthis methodology is a good candidate for parallel Bayesian computations.

Journal article

Melvin T, Benacchio T, Shipway B, Wood N, Thuburn J, Cotter Cet al., 2019, A mixed finite-element, finite-volume, semi-implicit discretisation for atmospheric dynamics: Cartesian geometry, Quarterly Journal of the Royal Meteorological Society, ISSN: 0035-9009

To meet the challenges posed by future generations of massively parallel supercomputers a reformulation of the dynamical core for the Met Office’s weather and climate model is presented. This new dynamical core uses explicit finite‐volume type discretisations for the transport of scalar fields coupled with an iterated‐implicit, mixed finite‐element discretisation for all other terms. The target model aims to maintain the accuracy, stability and mimetic properties of the existing Met Office model independent of the chosen mesh while improving the conservation properties of the model. This paper details that proposed formulation and, as a first step towards complete testing, demonstrates its performance for a number of test cases in (the context of) a Cartesian domain. The new model is shown to produce similar results to both the existing semi‐implicit semi‐Lagrangian model used at the Met Office and other models in the literature on a range of bubble tests and orographically forced flows in two and three dimensions.

Journal article

Cotter CJ, Crisan D, Holm DD, Pan W, Shevchenko Iet al., Numerically Modelling Stochastic Lie Transport in Fluid Dynamics, SIAM Journal on Scientific Computing, ISSN: 1064-8275

We present a numerical investigation of stochastic transport in ideal fluids.According to Holm (Proc Roy Soc, 2015) and Cotter et al. (2017), the principlesof transformation theory and multi-time homogenisation, respectively, imply aphysically meaningful, data-driven approach for decomposing the fluid transportvelocity into its drift and stochastic parts, for a certain class of fluidflows. In the current paper, we develop new methodology to implement thisvelocity decomposition and then numerically integrate the resulting stochasticpartial differential equation using a finite element discretisation forincompressible 2D Euler fluid flows. The new methodology tested here is foundto be suitable for coarse graining in this case. Specifically, we performuncertainty quantification tests of the velocity decomposition of Cotter et al.(2017), by comparing ensembles of coarse-grid realisations of solutions of theresulting stochastic partial differential equation with the "true solutions" ofthe deterministic fluid partial differential equation, computed on a refinedgrid. The time discretization used for approximating the solution of thestochastic partial differential equation is shown to be consistent. We includecomprehensive numerical tests that confirm the non-Gaussianity of the streamfunction, velocity and vorticity fields in the case of incompressible 2D Eulerfluid flows.

Journal article

Goss ZL, Piggott MD, Kramer SC, Avdis A, Angeloudis A, Cotter CJet al., 2019, Competition effects between nearby tidal turbine arrays—optimal design for alderney race, Pages: 255-262

© 2019 Taylor & Francis Group, London. Tidal renewable energy can be described as a fledgling industry, with the world’s pilot tidal stream turbine array only recently installed. Full-sized arrays will be developed if they prove their economic, engineering and environmental viability. Reliable numerical tools are needed to optimise power yields in arrays of potentially hundreds of turbines and assess viability of new sites and designs. To demonstrate our capability to optimise the number of turbines and their spatial distribution in a region, we focus on a test case based upon the Alderney Race. The site contains the majority of the Channel Islands resource with plans from both France and Alderney to develop adjacent arrays that could impact on each other. We present a shallow-water model of the English Channel using the Thetis ocean model. Together with the hydrodynamics modelling we employ adjoint technology to optimise the micrositing of turbines for a set of scenarios.

Conference paper

Shipton J, Gibson TH, Cotter CJ, 2018, Higher-order compatible finite element schemes for the nonlinear rotating shallow water equations on the sphere, Journal of Computational Physics, Vol: 375, Pages: 1121-1137, ISSN: 0021-9991

We describe a compatible finite element discretisation for the shallow waterequations on the rotating sphere, concentrating on integrating consistentupwind stabilisation into the framework. Although the prognostic variables are velocity and layer depth, the discretisation has a diagnostic potentialvorticity that satisfies a stable upwinded advection equation through aTaylor-Galerkin scheme; this provides a mechanism for dissipating enstrophy at the gridscale whilst retaining optimal order consistency. We also use upwind discontinuous Galerkin schemes for the transport of layer depth. These transport schemes are incorporated into a semi-implicit formulation that is facilitated by a hybridisation method for solving the resulting mixed Helmholtz equation. We illustrate our discretisation with some standard rotating sphere test problems.

Journal article

Budd CJ, McRae ATT, Cotter CJ, 2018, The scaling and skewness of optimally transported meshes on the sphere, Journal of Computational Physics, Vol: 375, Pages: 540-564, ISSN: 0021-9991

In the context of numerical solution of PDEs, dynamic mesh redistribution methods (r-adaptive methods) are an important procedure for increasing the resolution in regions of interest, without modifying the connectivity of the mesh. Key to the success of these methods is that the mesh should be sufficiently refined (locally) and flexible in order to resolve evolving solution features, but at the same time not introduce errors through skewness and lack of regularity. Some state-of-the-art methods are bottom-up in that they attempt to prescribe both the local cell size and the alignment to features of the solution. However, the resulting problem is overdetermined, necessitating a compromise between these conflicting requirements. An alternative approach, described in this paper, is to prescribe only the local cell size and augment this an optimal transport condition to provide global regularity. This leads to a robust and flexible algorithm for generating meshes fitted to an evolving solution, with minimal need for tuning parameters. Of particular interest for geophysical modelling are meshes constructed on the surface of the sphere. The purpose of this paper is to demonstrate that meshes generated on the sphere using this optimal transport approach have good a-priori regularity and that the meshes produced are naturally aligned to various simple features. It is further shown that the sphere's intrinsic curvature leads to more regular meshes than the plane. In addition to these general results, we provide a wide range of examples relevant to practical applications, to showcase the behaviour of optimally transported meshes on the sphere. These range from axisymmetric cases that can be solved analytically to more general examples that are tackled numerically. Evaluation of the singular values and singular vectors of the mesh transformation provides a quantitative measure of the mesh anisotropy, and this is shown to match analytic predictions.

Journal article

Bendall TM, Cotter CJ, Statistical properties of an enstrophy conserving discretisation for the stochastic quasi-geostrophic equation, Geophysical and Astrophysical Fluid Dynamics, ISSN: 0309-1929

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.

Journal article

Bauer W, Cotter CJ, 2018, Energy-enstrophy conserving compatible finite element schemes for the rotating shallow water equations with slip boundary conditions, Journal of Computational Physics, Vol: 373, Pages: 171-187, ISSN: 0021-9991

We describe an energy-enstrophy conserving discretisation for the rotatingshallow water equations with slip boundary conditions. This relaxes theassumption of boundary-free domains (periodic solutions or the surface of asphere, for example) in the energy-enstrophy conserving formulation of McRaeand Cotter (2014). This discretisation requires extra prognostic vorticityvariables on the boundary in addition to the prognostic velocity and layerdepth variables. The energy-enstrophy conservation properties hold for anyappropriate set of compatible finite element spaces defined on arbitrary mesheswith arbitrary boundaries. We demonstrate the conservation properties of thescheme with numerical solutions on a rotating hemisphere.

Journal article

Cotter CJ, Graber PJ, Kirby RC, 2018, Mixed finite elements for global tide models with nonlinear damping, Numerische Mathematik, ISSN: 0029-599X

We study mixed finite element methods for the rotating shallow waterequations with linearized momentum terms but nonlinear drag. By means of anequivalent second-order formulation, we prove long-time stability of the systemwithout energy accumulation. We also give rates of damping in unforced systemsand various continuous dependence results on initial conditions and forcingterms. \emph{A priori} error estimates for the momentum and free surfaceelevation are given in $L^2$ as well as for the time derivative and divergenceof the momentum. Numerical results confirm the theoretical results regardingboth energy damping and convergence rates.

Journal article

Melvin T, Benacchio T, Thuburn J, Cotter Cet al., 2018, Choice of function spaces for thermodynamic variables in mixed finite-element methods, Quarterly Journal of the Royal Meteorological Society, Vol: 144, Pages: 900-916, ISSN: 0035-9009

We study the dispersion properties of three choices for the buoyancy space in a mixed finite‐element discretization of geophysical fluid flow equations. The problem is analogous to that of the staggering of the buoyancy variable in finite‐difference discretizations. Discrete dispersion relations of the two‐dimensional linear gravity wave equations are computed. By comparison with the analytical result, the best choice for the buoyancy space basis functions is found to be the horizontally discontinuous, vertically continuous option. This is also the space used for the vertical component of the velocity. At lowest polynomial order, this arrangement mirrors the Charney–Phillips vertical staggering known to have good dispersion properties in finite‐difference models. A fully discontinuous space for the buoyancy corresponding to the Lorenz finite‐difference staggering at lowest order gives zero phase velocity for high vertical wavenumber modes. A fully continuous space, the natural choice for scalar variables in a mixed finite‐element framework, with degrees of freedom of buoyancy and vertical velocity horizontally staggered at lowest order, is found to entail zero phase velocity modes at the large horizontal wavenumber end of the spectrum. Corroborating the theoretical insights, numerical results obtained on gravity wave propagation with fully continuous buoyancy highlight the presence of a computational mode in the poorly resolved part of the spectrum that fails to propagate horizontally. The spurious signal is not removed in test runs with higher‐order polynomial basis functions. Runs at higher order also highlight additional oscillations, an issue that is shown to be mitigated by partial mass‐lumping. In light of the findings and with a view to coupling the dynamical core to physical parametrizations that often force near the horizontal grid scale, the use of the fully continuous space should be avoided in favour of the horizontally discontinuous, vertically co

Journal article

Natale A, Cotter CJ, 2018, Corrigendum to: A variational H(div) finite-element discretization approach for perfect incompressible fluids, IMA Journal of Numerical Analysis, Vol: 38, Pages: 1084-1084, ISSN: 0272-4979

This is a correction to:IMA Journal of Numerical Analysis, Volume 38, Issue 3, 17 July 2018, Pages 1388–1419, https://doi.org/10.1093/imanum/drx033

Journal article

McRae ATT, Cotter CJ, Budd CJ, 2018, Optimal-transport-based mesh adaptivity on the plane and sphere using finite elements, SIAM Journal on Scientific Computing, Vol: 40, Pages: A1121-A1148

In moving mesh methods, the underlying mesh is dynamically adapted withoutchanging the connectivity of the mesh. We specifically consider the generationof meshes which are adapted to a scalar monitor function throughequidistribution. Together with an optimal transport condition, this leads to aMonge-Amp\ere equation for a scalar mesh potential. We adapt an existingfinite element scheme for the standard Monge-Amp\ere equation to this meshgeneration problem; this is a mixed finite element scheme, in which an extradiscrete variable is introduced to represent the Hessian matrix of secondderivatives. The problem we consider has additional nonlinearities over thebasic Monge-Amp\ere equation due to the implicit dependence of the monitorfunction on the resulting mesh. We also derive the equivalentMonge-Amp\ere-like equation for generating meshes on the sphere. The finiteelement scheme is extended to the sphere, and we provide numerical examples.All numerical experiments are performed using the open-source finite elementframework Firedrake.

Journal article

Gregory ACA, Cotter CJ, 2017, A seamless multilevel ensemble transform particle filter, SIAM Journal on Scientific Computing, Vol: 39, Pages: A2684-A2701, ISSN: 1095-7197

This paper presents a seamless algorithm for the application of the multilevel MonteCarlo (MLMC) method to the ensemble transform particle filter (ETPF). The algorithm uses a combi-nation of optimal coupling transformations between coarse and fine ensembles in difference estimatorswithin a multilevel framework, to minimise estimator variance. It differs from that of Gregory et al.(2016) in that strong coupling between the coarse and fine ensembles is seamlessly maintained duringall stages of the assimilation algorithm, instead of using independent transformations to equal weightsfollowed by recoupling with an assignment problem. This modification is found to lead to an increasedrate in variance decay between coarse and fine ensembles with level in the hierarchy, a key componentof MLMC. This offers the potential for greater computational cost reductions. This is shown, alongsideevidence of asymptotic consistency, in numerical examples.

Journal article

van Sebille E, Griffies SM, Abernathey R, Adams T, Berloff P, Biastoch A, Blanke B, Chassignet E, Cheng Y, Cotter C, Deleersnijder E, Doos K, Drake H, Drijfhout S, Gary S, Heemink A, Kjellsson J, Koszalka I, Lange M, Lique C, MacGilchrist G, Marsh R, Mayorga Adame G, McAdam R, Nencioli F, Paris C, Piggott M, Polton J, Ruhs S, Shah S, Thomas M, Wang J, Wolfram P, Zanna L, Zika Jet al., 2017, Lagrangian ocean analysis: fundamentals and practices, Ocean Modelling, Vol: 121, Pages: 49-75, ISSN: 1463-5003

Lagrangian analysis is a powerful way to analyse the output of ocean circulation models and other ocean velocity data such as from altimetry. In the Lagrangian approach, large sets of virtual particles are integrated within the three-dimensional, time-evolving velocity fields. Over several decades, a variety of tools and methods for this purpose have emerged. Here, we review the state of the art in the field of Lagrangian analysis of ocean velocity data, starting from a fundamental kinematic framework and with a focus on large-scale open ocean applications. Beyond the use of explicit velocity fields, we consider the influence of unresolved physics and dynamics on particle trajectories. We comprehensively list and discuss the tools currently available for tracking virtual particles. We then showcase some of the innovative applications of trajectory data, and conclude with some open questions and an outlook. The overall goal of this review paper is to reconcile some of the different techniques and methods in Lagrangian ocean analysis, while recognising the rich diversity of codes that have and continue to emerge, and the challenges of the coming age of petascale computing.

Journal article

Cotter CJ, Gottwald G, Holm DD, 2017, Stochastic partial differential fluid equations as a diffusive limit of deterministic Lagrangian multi-time dynamics, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol: 473, ISSN: 1364-5021

In Holm (Holm 2015 Proc. R. Soc. A 471, 20140963. (doi:10.1098/rspa.2014.0963)), stochastic fluid equations were derived by employing a variational principle with an assumed stochastic Lagrangian particle dynamics. Here we show that the same stochastic Lagrangian dynamics naturally arises in a multi-scale decomposition of the deterministic Lagrangian flow map into a slow large-scale mean and a rapidly fluctuating small-scale map. We employ homogenization theory to derive effective slow stochastic particle dynamics for the resolved mean part, thereby obtaining stochastic fluid partial equations in the Eulerian formulation. To justify the application of rigorous homogenization theory, we assume mildly chaotic fast small-scale dynamics, as well as a centring condition. The latter requires that the mean of the fluctuating deviations is small, when pulled back to the mean flow.

Journal article

Natale A, Cotter CJ, 2017, A variational H (div) finite-element discretization approach for perfect incompressible fluids, IMA Journal of Numerical Analysis, Vol: 38, Pages: 1388-1419, ISSN: 0272-4979

We propose a finite-element discretization approach for the incompressible Euler equations which mimicstheir geometric structure and their variational derivation. In particular, we derive a finite-element methodthat arises from a nonholonomic variational principle and an appropriately defined Lagrangian, where finite-elementH(div)vector fields are identified with advection operators; this is the first successful extensionof the structure-preserving discretization ofPavlovet al.(2009) to the finite-element setting. The resultingalgorithm coincides with the energy-conserving scheme proposed byGuzm ́anet al.(2016). Through thevariational derivation, we discover that it also satisfies a discrete analogous of Kelvin’s circulation theorem.Further, we propose an upwind-stabilized version of the scheme that dissipates enstrophy while preservingenergy conservation and the discrete Kelvin’s theorem. We prove error estimates for this version of thescheme, and we study its behaviour through numerical tests.

Journal article

Natale A, Cotter CJ, 2017, Scale-selective dissipation in energy-conserving finite element schemes for two-dimensional turbulence, Quarterly Journal of the Royal Meteorological Society, Vol: 143, Pages: 1734-1745, ISSN: 0035-9009

We analyze the multiscale properties of energy-conserving upwind-stabilized finite-element discretizations of the two-dimensional incompressible Euler equations. We focus our attention on two particular methods: the Lie derivative discretization introduced by Natale and Cotter and the Streamline Upwind/Petrov–Galerkin (SUPG) discretization of the vorticity advection equation. Such discretizations 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 characterizes two-dimensional turbulent flows.

Journal article

Yamazaki H, Shipton J, Cullen MJP, Mitchell L, Cotter CJet al., 2017, Vertical slice modelling of nonlinear Eady waves using a compatible finite element method, Journal of Computational Physics, Vol: 343, Pages: 130-149, ISSN: 1090-2716

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 idealised 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.

Journal article

Gregory ACA, Cotter CJ, 2017, On the Calibration of Multilevel Monte Carlo Ensemble Forecasts, Quarterly Journal of the Royal Meteorological Society, Vol: 143, Pages: 1929-1935, ISSN: 1477-870X

The multilevel Monte Carlo method can efficiently compute statistical estimates ofdiscretized random variables for a given error tolerance. Traditionally, only a certainstatistic is computed from a particular implementation of multilevel Monte Carlo. Thisarticle considers the multilevel case in which one wants to verify and evaluate a singleensemble that forms an empirical approximation to many different statistics, namely anensemble forecast. We propose a simple algorithm that, in the univariate case, allows oneto derive a statistically consistent single ensemble forecast from the hierarchy of ensemblesthat are formed during an implementation of multilevel Monte Carlo. This ensembleforecast then allows the entire multilevel hierarchy of ensembles to be evaluated usingstandard ensemble forecast verification techniques. We demonstrate the case of evaluatingthe calibration of the forecast.

Journal article

Natale A, Shipton J, Cotter CJ, 2016, Compatible finite element spaces for geophysical fluid dynamics, Dynamics and Statistics of the Climate System, Vol: 1, ISSN: 2059-6987

Compatible finite elements provide a framework for preserving important structures in equations of geophysical uid 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 uid 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.

Journal article

McRae ATT, Mitchell L, Bercea, Ham DA, Cotteret 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.

Journal article

Abolghasemi M, Piggott MD, Spinneken J, Vire A, Cotter CJ, Crammond Set 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.

Journal article

Cotter CJ, Eldering J, Holm DD, Jacobs HO, Meier DMet 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.

Journal article

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.

Journal article

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.

Journal article

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

Journal article

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