Publications
87 results found
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.
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
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.
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.
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
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- Citations: 1
Melvin T, Benacchio T, Thuburn J, et 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
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
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.
van Sebille E, Griffies SM, Abernathey R, et al., 2018, Lagrangian ocean analysis: Fundamentals and practices, OCEAN MODELLING, Vol: 121, Pages: 49-75, ISSN: 1463-5003
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- Citations: 10
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
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- Citations: 6
Yamazaki H, Shipton J, Cullen MJP, et 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
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
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
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- Citations: 1
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
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- Citations: 1
Abolghasemi MA, 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: 0889-9746
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- Citations: 6
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.
Cotter CJ, Kirby RC, 2016, Mixed finite elements for global tide models, NUMERISCHE MATHEMATIK, Vol: 133, Pages: 255-277, ISSN: 0029-599X
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- Citations: 1
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
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- Citations: 2
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
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- Citations: 7
Mcrae ATT, Bercea G-T, Mitchell L, et 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
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- Citations: 6
Natale A, Shipton J, Cotter CJ, 2016, Compatible finite element spaces for geophysical fluid dynamics, Dynamics and Statistics of the Climate System, Vol: 1
Jordi BE, Cotter CJ, Sherwin SJ, 2015, An adaptive selective frequency damping method, PHYSICS OF FLUIDS, Vol: 27, ISSN: 1070-6631
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- Citations: 4
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
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, Holm DD, Jacobs HO, et al., 2014, A jetlet hierarchy for ideal fluid dynamics, JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, Vol: 47, ISSN: 1751-8113
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- Citations: 4
Cotter CJ, McRae ATT, 2014, Compatible finite elements for numerical weather prediction, ECMWF Annual Seminar
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: 1070-6631
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- Citations: 12
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
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- Citations: 24
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
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