Imperial College London


Faculty of Natural SciencesDepartment of Mathematics

Professor of Computational Mathematics



+44 (0)20 7594 3468colin.cotter




755Huxley BuildingSouth Kensington Campus





Publication Type

112 results found

Clare MCA, Wallwork JG, Kramer SC, Weller H, Cotter CJ, Piggott MDet al., 2022, Multi-scale hydro-morphodynamic modelling using mesh movement methods, GEM: International Journal on Geomathematics, Vol: 13, ISSN: 1869-2672

Hydro-morphodynamic modelling is an important tool that can be used in the protection of coastal zones. The models can be required to resolve spatial scales ranging from sub-metre to hundreds of kilometres and are computationally expensive. In this work, we apply mesh movement methods to a depth-averaged hydro-morphodynamic model for the first time, in order to tackle both these issues. Mesh movement methods are particularly well-suited to coastal problems as they allow the mesh to move in response to evolving flow and morphology structures. This new capability is demonstrated using test cases that exhibit complex evolving bathymetries and have moving wet-dry interfaces. In order to be able to simulate sediment transport in wet-dry domains, a new conservative discretisation approach has been developed as part of this work, as well as a sediment slide mechanism. For all test cases, we demonstrate how mesh movement methods can be used to reduce discretisation error and computational cost. We also show that the optimum parameter choices in the mesh movement monitor functions are fairly predictable based upon the physical characteristics of the test case, facilitating the use of mesh movement methods on further problems.

Journal article

Clare MCA, Piggott MD, Cotter CJ, 2022, Assessing erosion and flood risk in the coastal zone through the application of multilevel Monte Carlo methods, Coastal Engineering, Vol: 174, ISSN: 0378-3839

Coastal zones are vulnerable to both erosion and flood risk, which can be assessed using coupled hydro-morphodynamic models. However, the use of such models as decision support tools suffers from a high degreeof uncertainty, due to both incomplete knowledge and natural variability in the system. In this work, we showfor the first time how the multilevel Monte Carlo method (MLMC) can be applied in hydro-morphodynamiccoastal ocean modelling, here using the popular model XBeach, to quantify uncertainty by computing statisticsof key output variables given uncertain input parameters. MLMC accelerates the Monte Carlo approach throughthe use of a hierarchy of models with different levels of resolution. Several theoretical and real-world coastalzone case studies are considered here, for which output variables that are key to the assessment of flood anderosion risk, such as wave run-up height and total eroded volume, are estimated. We show that MLMC cansignificantly reduce computational cost, resulting in speed up factors of 40 or greater compared to a standardMonte Carlo approach, whilst keeping the same level of accuracy. Furthermore, a sophisticated ensemblegenerating technique is used to estimate the cumulative distribution of output variables from the MLMC output.This allows for the probability of a variable exceeding a certain value to be estimated, such as the probabilityof a wave run-up height exceeding the height of a seawall. This is a valuable capability that can be used toinform decision-making under uncertainty

Journal article

Clare MCA, Kramer SC, Cotter CJ, Piggott MDet al., 2022, Calibration, inversion and sensitivity analysis for hydro-morphodynamic models through the application of adjoint methods, Computers and Geosciences, Vol: 163, Pages: 1-13, ISSN: 0098-3004

The development of reliable, sophisticated hydro-morphodynamic models is essential for protecting the coastal environment against hazards such as flooding and erosion. There exists a high degree of uncertainty associated with the application of these models, in part due to incomplete knowledge of various physical, empirical and numerical closure related parameters in both the hydrodynamic and morphodynamic solvers. This uncertainty can be addressed through the application of adjoint methods. These have the notable advantage that the number and/or dimension of the uncertain parameters has almost no effect on the computational cost associated with calculating the model sensitivities. Here, we develop the first freely available and fully flexible adjoint hydro-morphodynamic model framework. This flexibility is achieved through using the pyadjoint library, which allows us to assess the uncertainty of any parameter with respect to any model functional, without further code implementation. The model is developed within the coastal ocean model Thetis constructed using the finite element code-generation library Firedrake. We present examples of how this framework can perform sensitivity analysis, inversion and calibration for a range of uncertain parameters based on the final bedlevel. These results are verified using so-called dual-twin experiments, where the ‘correct’ parameter value is used in the generation of synthetic model test data, but is unknown to the model in subsequent testing. Moreover, we show that inversion and calibration with experimental data using our framework produces physically sensible optimum parameters and that these parameters always lead to more accurate results. In particular, we demonstrate how our adjoint framework can be applied to a tsunami-like event to invert for the tsunami wave from sediment deposits.

Journal article

Wimmer GA, Cotter CJ, Bauer W, 2022, Energy conserving SUPG methods for compatible finite element schemes in numerical weather prediction, The SMAI journal of computational mathematics, Vol: 7, Pages: 267-300

Journal article

Bock A, Cotter CJ, 2021, Learning landmark geodesics using the ensemble Kalman filter, Foundations of Data Science, Vol: 3, Pages: 701-727, ISSN: 2639-8001

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 operators. We describe an efficient implementation of the algorithm and show several numerical results for various target shapes.

Journal article

Bendall TM, Cotter CJ, Holm DD, 2021, Perspectives on the formation of peakons in the stochastic Camassa-Holm equation, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol: 477, ISSN: 1364-5021

A famous feature of the Camassa–Holm equation is its admission of peaked soliton solutions known as peakons. We investigate this equation under the influence of stochastic transport. Noting that peakons are weak solutions of the equation, we present a finite-element discretization for it, which we use to explore the formation of peakons. Our simulations using this discretization reveal that peakons can still form in the presence of stochastic perturbations. Peakons can emerge both through wave breaking, as the slope turns vertical, and without wave breaking as the inflection points of the velocity profile rise to reach the summit.

Journal article

Bauer W, Behrens J, Cotter C, 2021, A structure-preserving approximation of the discrete split rotating shallow water equations, European Numerical Mathematics and Advanced Applications Conference 2019, Publisher: Springer Verlag, Pages: 103-113, ISSN: 1439-7358

We introduce an efficient split finite element (FE) discretization of a y-independent (slice) model of the rotating shallow water equations. The study of this slice model provides insight towards developing schemes for the full 2D case. Using the split Hamiltonian FE framework (Bauer, Behrens and Cotter, 2019), we result in structure-preserving discretizations that are split into topological prognostic and metric-dependent closure equations. This splitting also accounts for the schemes' properties: the Poisson bracket is responsible for conserving energy (Hamiltonian) as well as mass, potential vorticity and enstrophy (Casimirs), independently from the realizations of the metric closure equations. The latter, in turn, determine accuracy, stability, convergence and discrete dispersion properties. We exploit this splitting to introduce structure-preserving approximations of the mass matrices in the metric equations avoiding to solve linear systems. We obtain a fully structure-preserving scheme with increased efficiency by a factor of two.

Conference paper

Clare MCA, Percival JR, Angeloudis A, Cotter CJ, Piggott MDet al., 2021, Hydro-morphodynamics 2D modelling using a discontinuous Galerkin discretisation, Computers and Geosciences, Vol: 146, Pages: 1-13, ISSN: 0098-3004

The development of morphodynamic models to simulate sediment transport accurately is a challenging process that is becoming ever more important because of our increasing exploitation of the coastal zone, as well as sea-level rise and the potential increase in strength and frequency of storms due to a changing climate. Morphodynamic models are highly complex given the non-linear and coupled nature of the sediment transport problem. Here we implement a new depth-averaged coupled hydrodynamic and sediment transport model within the coastal ocean model Thetis, built using the code generating framework Firedrake which facilitates code flexibility and optimisation benefits. To the best of our knowledge, this represents the first full morphodynamic model including both bedload and suspended sediment transport which uses a discontinuous Galerkin based finite element discretisation. We implement new functionalities within Thetis extending its existing capacity to model scalar transport to modelling suspended sediment transport, incorporating within Thetis options to model bedload transport and bedlevel changes. We apply our model to problems with non-cohesive sediment and account for effects of gravity and helical flow by adding slope gradient terms and parametrising secondary currents. For validation purposes and in demonstrating model capability, we present results from test cases of a migrating trench and a meandering channel comparing against experimental data and the widely-used model Telemac-Mascaret.

Journal article

Cotter CJ, Deasy J, Pryer T, 2020, The r-Hunter-Saxton equation, smooth and singular solutions and their approximation, Nonlinearity, Vol: 33, Pages: 7016-7039, ISSN: 0951-7715

In this work we introduce the r-Hunter–Saxton equation, a generalisation of the Hunter–Saxton equation arising as extremals of an action principle posed in Lr. We characterise solutions to the Cauchy problem, quantifying the blow-up time and studying various symmetry reductions. We construct piecewise linear functions and show that they are weak solutions to the r-Hunter–Saxton equation.

Journal article

Bendall TM, Gibson TH, Shipton J, Cotter CJ, Shipway Bet al., 2020, A compatible finite-element discretisation for the moist compressible Euler equations, Quarterly Journal of the Royal Meteorological Society, Vol: 146, Pages: 3187-3205, ISSN: 0035-9009

A promising development of the last decade in the numerical modelling of geophysical fluids has been the compatible finite‐element framework. Indeed, this will form the basis for the next‐generation dynamical core of the Met Office. For this framework to be useful for numerical weather prediction models, it must be able to handle descriptions of unresolved and diabatic processes. These processes offer a challenging test for any numerical discretisation, and have not yet been described within the compatible finite‐element framework. The main contribution of this article is to extend a discretisation using this new framework to include moist thermodynamics. Our results demonstrate that discretisations within the compatible finite‐element framework can be robust enough also to describe moist atmospheric processes.We describe our discretisation strategy, including treatment of moist processes, and present two configurations of the model using different sets of function spaces with different degrees of finite element. The performance of the model is demonstrated through several test cases. Two of these test cases are new cloudy‐atmosphere variants of existing test cases: inertia–gravity waves in a two‐dimensional vertical slice and a three‐dimensional rising thermal.

Journal article

Kramer S, Wilson C, Davies R, Funke SW, Greaves T, Avdis A, Lange M, Candy A, Cotter CJ, Pain C, Percival J, Mouradian S, Bhutani G, Gorman G, Gibson A, Duvernay T, Guo X, Maddison JR, Rathgeber F, Farrell P, Weiland M, Robinson D, Ham DA, Goffin M, Piggott M, Gomes J, Dargaville S, Everett A, Jacobs CT, Cavendish ABet al., 2020, FluidityProject/fluidity: New test cases "Analytical solutions for mantle flow in cylindrical and spherical shells"

This release adds new test cases described in the GMD paper "Analytical solutions for mantle flow in cylindrical and spherical shells"


Cotter C, Crisan D, Holm DD, Pan W, Shevchenko Iet al., 2020, Modelling uncertainty using stochastic transport noise in a 2-layer quasi-geostrophic model, Publisher: arXiv

The stochastic variational approach for geophysical fluid dynamics wasintroduced by Holm (Proc Roy Soc A, 2015) as a framework for derivingstochastic parameterisations for unresolved scales. This paper applies thevariational stochastic parameterisation in a two-layer quasi-geostrophic modelfor a beta-plane channel flow configuration. We present a new method forestimating the stochastic forcing (used in the parameterisation) to approximateunresolved components using data from the high resolution deterministicsimulation, and describe a procedure for computing physically-consistentinitial conditions for the stochastic model. We also quantify uncertainty ofcoarse grid simulations relative to the fine grid ones in homogeneous (teamedwith small-scale vortices) and heterogeneous (featuring horizontally elongatedlarge-scale jets) flows, and analyse how the spread of stochastic solutionsdepends on different parameters of the model. The parameterisation is tested bycomparing it with the true eddy-resolving solution that has reached somestatistical equilibrium and the deterministic solution modelled on alow-resolution grid. The results show that the proposed parameterisationsignificantly depends on the resolution of the stochastic model and gives goodensemble performance for both homogeneous and heterogeneous flows, and theparameterisation lays solid foundations for data assimilation.

Working paper

Bauer W, Behrens J, Cotter CJ, 2020, A structure-preserving approximation of the discrete split rotating shallow water equations, EGU General Assembly 2020, Publisher: Copernicus GmbH, ISSN: 0090-8312

We introduce an efficient split finite element (FE) discretization of a y-independent (slice) model of the rotating shallow water equations. The study of this slice model provides insight towards developing schemes for the full 2D case. Using the split Hamiltonian FE framework [1,2], we result in structure-preserving discretizations that are split into topological prognostic and metric-dependent closure equations. This splitting also accounts for the schemes' properties: the Poisson bracket is responsible for conserving energy (Hamiltonian) as well as mass, potential vorticity and enstrophy (Casimirs), independently from the realizations of the metric closure equations. The latter, in turn, determine accuracy, stability, convergence and discrete dispersion properties. We exploit this splitting to introduce structure-preserving approximations of the mass matrices in the metric equations avoiding to solve linear systems. We obtain a fully structure-preserving scheme with increased efficiency by a factor of two.

Conference paper

Cotter C, Crisan D, Holm D, Pan W, Shevchenko Iet al., 2020, Data Assimilation for a Quasi-Geostrophic Model with Circulation-Preserving Stochastic Transport Noise, Publisher: SPRINGER

Working paper

Gibson T, Mitchell L, Ham D, Cotter Cet al., 2020, Slate: extending Firedrake's domain-specific abstraction to hybridized solvers for geoscience and beyond, Geoscientific Model Development, Vol: 13, Pages: 735-761, ISSN: 1991-959X

Within the finite element community, discontinuous Galerkin (DG) and mixed finite element methods have becomeincreasingly popular in simulating geophysical flows. However, robust and efficient solvers for the resulting saddle-point andelliptic systems arising from these discretizations continue to be an on-going challenge. One possible approach for addressingthis issue is to employ a method known as hybridization, where the discrete equations are transformed such that classic staticcondensation and local post-processing methods can be employed. However, it is challenging to implement hybridization as performant parallel code within complex models, whilst maintaining separation of concerns between applications scientistsand software experts. In this paper, we introduce a domain-specific abstraction within the Firedrake finite element library thatpermits the rapid execution of these hybridization techniques within a code-generating framework. The resulting frameworkcomposes naturally with Firedrake’s solver environment, allowing for the implementation of hybridization and static condensa-tion as runtime-configurable preconditioners via the Python interface to PETSc, petsc4py. We provide examples derived from second order elliptic problems and geophysical fluid dynamics. In addition, we demonstrate that hybridization shows greatpromise for improving the performance of solvers for mixed finite element discretizations of equations related to large-scalegeophysical flows.

Journal article

Wimmer GA, Cotter CJ, Bauer W, 2020, Energy conserving upwinded compatible finite element schemes for the rotating shallow water equations, Journal of Computational Physics, Vol: 401, Pages: 1-18, ISSN: 0021-9991

We present an energy conserving space discretisation of the rotating shallow water equations using compatible finite elements. It is based on an energy and enstrophy conserving Hamiltonian formulation as described in McRae and Cotter (2014), and extends it to include upwinding in the velocity and depth advection to increase stability. Upwinding for velocity in an energy conserving context was introduced for the incompressible Euler equations in Natale and Cotter (2017), while upwinding in the depth field in a Hamiltonian finite element context is newly described here. The energy conserving property is validated by coupling the spatial discretisation to an energy conserving time discretisation. Further, the discretisation is demonstrated to lead to an improved field development with respect to stability when upwinding in the depth field is included.

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, Vol: 145, Pages: 2835-2853, 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

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

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.

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


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 C, Cotter S, Russell P, 2019, Ensemble transport adaptive importance sampling, SIAM/ASA Journal on Uncertainty Quantification, Vol: 7, Pages: 444-471, 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

Cotter CJ, Crisan D, Holm DD, Pan W, Shevchenko Iet al., 2019, Numerically modelling stochastic lie transport in fluid dynamics, SIAM Journal on Scientific Computing, Vol: 17, Pages: 192-232, 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

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

Bock A, Arnaudon A, Cotter C, 2019, Selective Metamorphosis for Growth Modelling with Applications to Landmarks, Publisher: SPRINGER INTERNATIONAL PUBLISHING AG

Working paper

Gibson T, McRae ATT, Cotter C, Mitchell L, Ham Det al., 2019, Compatible finite element methods for geophysical flows: Automation and implementation using Firedrake, Publisher: Springer International Publishing, ISBN: 9783030239565

This book introduces recently developed mixed finite element methods for large-scale geophysical flows that preserve essential numerical properties for accurate simulations. The methods are presented using standard models of atmospheric flows and are implemented using the Firedrake finite element library. Examples guide the reader through problem formulation, discretisation, and automated implementation.The so-called “compatible” finite element methods possess key numerical properties which are crucial for real-world operational weather and climate prediction. The authors summarise the theory and practical implications of these methods for model problems, introducing the reader to the Firedrake package and providing open-source implementations for all the examples covered.Students and researchers with engineering, physics, mathematics, or computer science backgrounds will benefit from this book. Those readers who are less familiar with the topic are provided with an overview of geophysical fluid dynamics.


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

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

Cotter CJ, Graber PJ, Kirby RC, 2018, Mixed finite elements for global tide models with nonlinear damping, Numerische Mathematik, Vol: 140, Pages: 963-991, 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

Bendall TM, Cotter CJ, 2018, Statistical properties of an enstrophy conserving finite element discretisation for the stochastic quasi-geostrophic equation, Geophysical and Astrophysical Fluid Dynamics, Vol: 113, Pages: 491-504, 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

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