# DrColinCotter

Faculty of Natural SciencesDepartment of Mathematics

Reader in Numerical Analysis and Scientific Computing

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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
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86 results found

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

© 2018 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

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

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

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

JOURNAL ARTICLE

Natale A, Cotter CJ, 2018, A variational $\boldsymbol{H}({\rm div})$ finite-element discretization approach for perfect incompressible fluids, IMA Journal of Numerical Analysis, Vol: 38, Pages: 1388-1419, ISSN: 0272-4979

JOURNAL ARTICLE

Natale A, Cotter CJ, 2018, 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

JOURNAL ARTICLE

van Sebille E, Griffies SM, Abernathey R, Adams TP, Berloff P, Biastoch A, Blanke B, Chassignet EP, Cheng Y, Cotter CJ, Deleersnijder E, Doos K, Drake HF, Drijfhout S, Gary SF, Heemink AW, Kjellsson J, Koszalka IM, Lange M, Lique C, MacGilchrist GA, Marsh R, Adame CGM, McAdam R, Nencioli F, Paris CB, Piggott MD, Polton JA, Ruehs S, Shah SHAM, Thomas MD, Wang J, Wolfram PJ, Zanna L, Zika JDet al., 2018, Lagrangian ocean analysis: Fundamentals and practices, OCEAN MODELLING, Vol: 121, Pages: 49-75, ISSN: 1463-5003

JOURNAL ARTICLE

Cotter CJ, Gottwald GA, 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

JOURNAL ARTICLE

Gregory A, 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: 0035-9009

JOURNAL ARTICLE

Gregory A, Cotter CJ, 2017, A SEAMLESS MULTILEVEL ENSEMBLE TRANSFORM PARTICLE FILTER, SIAM JOURNAL ON SCIENTIFIC COMPUTING, Vol: 39, Pages: A2684-A2701, ISSN: 1064-8275

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

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: 0021-9991

JOURNAL ARTICLE

Abolghasemi MA, 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: 0889-9746

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

Cotter CJ, Kirby RC, 2016, Mixed finite elements for global tide models, NUMERISCHE MATHEMATIK, Vol: 133, Pages: 255-277, ISSN: 0029-599X

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

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: 1064-8275

JOURNAL ARTICLE

Mcrae ATT, Bercea G-T, Mitchell L, Ham DA, Cotter CJet al., 2016, AUTOMATED GENERATION AND SYMBOLIC MANIPULATION OF TENSOR PRODUCT FINITE ELEMENTS, SIAM JOURNAL ON SCIENTIFIC COMPUTING, Vol: 38, Pages: S25-S47, ISSN: 1064-8275

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

JOURNAL ARTICLE

Jordi BE, Cotter CJ, Sherwin SJ, 2015, An adaptive selective frequency damping method, PHYSICS OF FLUIDS, Vol: 27, ISSN: 1070-6631

JOURNAL ARTICLE

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

BOOK

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

JOURNAL ARTICLE

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

JOURNAL ARTICLE

Cotter CJ, Holm DD, Jacobs HO, Meier DMet al., 2014, A jetlet hierarchy for ideal fluid dynamics, JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, Vol: 47, ISSN: 1751-8113

JOURNAL ARTICLE

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

CONFERENCE PAPER

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, ISSN: 0021-9991

JOURNAL ARTICLE

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

JOURNAL ARTICLE

Jordi BE, Cotter CJ, Sherwin SJ, 2014, Encapsulated formulation of the selective frequency damping method, PHYSICS OF FLUIDS, Vol: 26, ISSN: 1070-6631

JOURNAL ARTICLE

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.

JOURNAL ARTICLE

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

JOURNAL ARTICLE

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