arXiv subjects

C. J. Cotter

Publications and source records attributed to C. J. Cotter.

At least 19 recordsLinked to original sources

arXiv2023

The r-Camassa-Holm equation: smooth and singular solutions

This paper introduces the r-Camassa-Holm (r-CH) equation, which describes a geodesic flow on the manifold of diffeomorphisms acting on the real line induced by the W1,r metric. The conserved energy is for the problem is given by the full W1,r norm and the for r = 2, we recover the Camassa-Holm equation. We compute the Lie symmetries for r-CH and study various symmetry reductions. We introduce singular weak solutions of the r-CH equation for r >= 2 and demonstrates their robustness in numerical simulations of their nonlinear interactions in both overtaking and head-on collisions. Several open questions are formulated about the unexplored properties of the r-CH weak singular solutions, including the question of whether they would emerge from smooth initial conditions.

math.AP↗

arXiv2023

A compatible finite element discretisation for the nonhydrostatic vertical slice equations

We present a compatible finite element discretisation for the vertical slice compressible Euler equations, at next-to-lowest order (i.e., the pressure space is bilinear discontinuous functions). The equations are numerically integrated in time using a fully implicit timestepping scheme which is solved using monolithic GMRES preconditioned by a linesmoother. The linesmoother only involves local operations and is thus suitable for domain decomposition in parallel. It allows for arbitrarily large timesteps but with iteration counts scaling linearly with Courant number in the limit of large Courant number. This solver approach is implemented using Firedrake, and the additive Schwarz preconditioner framework of PETSc. We demonstrate the robustness of the scheme using a standard set of testcases that may be compared with other approaches.

math.NA↗

arXiv2018

Higher-order compatible finite element schemes for the nonlinear rotating shallow water equations on the sphere

We describe a compatible finite element discretisation for the shallow water equations on the rotating sphere, concentrating on integrating consistent upwind stabilisation into the framework. Although the prognostic variables are velocity and layer depth, the discretisation has a diagnostic potential vorticity that satisfies a stable upwinded advection equation through a Taylor-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.

math.NA↗

arXiv2016

Weak dual pairs and jetlet methods for ideal incompressible fluid models in $n\geq 2$ dimensions

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 $n \geq 2$ 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.

physics.flu-dyn↗

arXiv2016

Embedded discontinuous Galerkin transport schemes with localised limiters

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

math.NA↗

arXiv2014

On the shallow atmosphere approximation in finite element dynamical cores

We provide an approach to implementing the shallow atmosphere approximation in three dimensional finite element discretisations for dynamical cores. The approach makes use of the fact that the shallow atmosphere approximation metric can be obtained by writing equations on a three-dimensional manifold embedded in $\mathbb{R}^4$ with a restriction of the Euclidean metric. We show that finite element discretisations constructed this way are equivalent to the use of a modified three dimensional mesh for the construction of metric terms. We demonstrate our approach via a convergence test for a prototypical elliptic problem.

math.NA↗

arXiv2014

A jetlet hierarchy for ideal fluid dynamics

Truncated Taylor expansions of smooth flow maps are used in Hamilton's principle to derive a multiscale Lagrangian particle representation of ideal fluid dynamics. Numerical simulations for scattering of solutions at one level of truncation are found to produce solutions at higher levels. These scattering events to higher levels in the Taylor expansion are interpreted as modeling a cascade to smaller scales.

physics.flu-dyn↗

arXiv2014

Compatible finite element methods for numerical weather prediction

This article takes the form of a tutorial on the use of a particular class of mixed finite element methods, which can be thought of as the finite element extension of the C-grid staggered finite difference method. The class is often referred to as compatible finite elements, mimetic finite elements, discrete differential forms or finite element exterior calculus. We provide an elementary introduction in the case of the one-dimensional wave equation, before summarising recent results in applications to the rotating shallow water equations on the sphere, before taking an outlook towards applications in three-dimensional compressible dynamical cores.

math.NA↗

arXiv2013

Encapsulated formulation of the Selective Frequency Damping method

We present an alternative "encapsulated" formulation of the Selective Frequency Damping method for finding unstable equilibria of dynamical systems, which is particularly useful when analysing the stability of fluid flows. The formulation makes use of splitting methods, which means that it can be wrapped around an existing time-stepping code as a "black box". The method is first applied to a scalar problem in order to analyse its stability and highlight the roles of the control coefficient $χ$ and the filter width $Δ$ in the convergence (or not) towards the steady-state. Then the steady-state of the incompressible flow past a two-dimensional cylinder at $Re=100$, obtained with a code which implements the spectral/hp element method, is presented.

physics.flu-dyn↗

arXiv2013

A finite element exterior calculus framework for the rotating shallow-water equations

We describe discretisations of the shallow water equations on the sphere using the framework of finite element exterior calculus, which are extensions of the mimetic finite difference framework presented in Ringler, Thuburn, Klemp, and Skamarock (Journal of Computational Physics, 2010). The exterior calculus notation provides a guide to which finite element spaces should be used for which physical variables, and unifies a number of desirable properties. We present two formulations: a ``primal'' formulation in which the finite element spaces are defined on a single mesh, and a ``primal-dual'' formulation in which finite element spaces on a dual mesh are also used. Both formulations have velocity and layer depth as prognostic variables, but the exterior calculus framework leads to a conserved diagnostic potential vorticity. In both formulations we show how to construct discretisations that have mass-consistent (constant potential vorticity stays constant), stable and oscillation-free potential vorticity advection.

math.NA↗

arXiv2013

Variational formulations of sound-proof models

We derive a family of ideal (nondissipative) 3D sound-proof fluid models that includes both the Lipps-Hemler anelastic approximation (AA) and the Durran pseudo-incompressible approximation (PIA). This family of models arises in the Euler-Poincaré framework involving a constrained Hamilton's principle expressed in the Eulerian fluid description. The derivation in this framework establishes the following properties of each member of the entire family: the Kelvin-Noether circulation theorem, conservation of potential vorticity on fluid parcels, a Lie-Poisson Hamiltonian formulation possessing conserved Casimirs, a conserved domain integrated energy and an associated variational principle satisfied by the equilibrium solutions. \smallskip Having set the stage with the derivations of 3D models using the constrained Hamilton's principle, we then derive the corresponding 2D vertical slice models for these sound-proof theories.

physics.flu-dyn↗

arXiv2013

A variational formulation of vertical slice models

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

math.DS↗

arXiv2012

Bayesian data assimilation in shape registration

In this paper we apply a Bayesian framework to the problem of geodesic curve matching. Given a template curve, the geodesic equations provide a mapping from initial conditions for the conjugate momentum onto topologically equivalent shapes. Here, we aim to recover the well-defined posterior distribution on the initial momentum which gives rise to observed points on the target curve; this is achieved by explicitly including a reparameterisation in the formulation. Appropriate priors are chosen for the functions which together determine this field and the positions of the observation points, the initial momentum $p_0$ and the reparameterisation vector field $ν$, informed by regularity results about the forward model. Having done this, we illustrate how Maximum Likelihood Estimators (MLEs) can be used to find regions of high posterior density, but also how we can apply recently developed \SLC{Markov chain Monte Carlo (MCMC)} methods on function spaces to characterise the whole of the posterior density. These illustrative examples also include scenarios where the posterior distribution is multimodal and irregular, leading us to the conclusion that knowledge of a state of global maximal posterior density does not always give us the whole picture, and full posterior sampling can give better quantification of likely states and the overall uncertainty inherent in the problem.

math.PR↗

arXiv2012

Mixed finite elements for numerical weather prediction

We show how two-dimensional mixed finite element methods that satisfy the conditions of finite element exterior calculus can be used for the horizontal discretisation of dynamical cores for numerical weather prediction on pseudo-uniform grids. This family of mixed finite element methods can be thought of in the numerical weather prediction context as a generalisation of the popular polygonal C-grid finite difference methods. There are a few major advantages: the mixed finite element methods do not require an orthogonal grid, and they allow a degree of flexibility that can be exploited to ensure an appropriate ratio between the velocity and pressure degrees of freedom so as to avoid spurious mode branches in the numerical dispersion relation. These methods preserve several properties of the C-grid method when applied to linear barotropic wave propagation, namely: a) energy conservation, b) mass conservation, c) no spurious pressure modes, and d) steady geostrophic modes on the $f$-plane. We explain how these properties are preserved, and describe two examples that can be used on pseudo-uniform grids: the recently-developed modified RT0-Q0 element pair on quadrilaterals and the BDFM1-\pdg element pair on triangles. All of these mixed finite element methods have an exact 2:1 ratio of velocity degrees of freedom to pressure degrees of freedom. Finally we illustrate the properties with some numerical examples.

math.NA↗

arXiv2010

Numerical wave propagation for the triangular $P1_{DG}$-$P2$ finite element pair

Inertia-gravity mode and Rossby mode dispersion properties are examined for discretisations of the linearized rotating shallow-water equations using the $P1_{DG}$-$P2$ finite element pair on arbitrary triangulations in planar geometry. A discrete Helmholtz decomposition of the functions in the velocity space based on potentials taken from the pressure space is used to provide a complete description of the numerical wave propagation for the discretised equations. In the $f$-plane case, this decomposition is used to obtain decoupled equations for the geostrophic modes, the inertia-gravity modes, and the inertial oscillations. As has been noticed previously, the geostrophic modes are steady. The Helmholtz decomposition is used to show that the resulting inertia-gravity wave equation is third-order accurate in space. In general the \pdgp finite element pair is second-order accurate, so this leads to very accurate wave propagation. It is further shown that the only spurious modes supported by this discretisation are spurious inertial oscillations which have frequency $f$, and which do not propagate. The Helmholtz decomposition also allows a simple derivation of the quasi-geostrophic limit of the discretised $P1_{DG}$-$P2$ equations in the $β$-plane case, resulting in a Rossby wave equation which is also third-order accurate.

math.NA↗

arXiv2009

Geodesic boundary value problems with symmetry

This paper shows how left and right actions of Lie groups on a manifold may be used to complement one another in a variational reformulation of optimal control problems equivalently as geodesic boundary value problems with symmetry. We prove an equivalence theorem to this effect and illustrate it with several examples. In finite-dimensions, we discuss geodesic flows on the Lie groups SO(3) and SE(3) under the left and right actions of their respective Lie algebras. In an infinite-dimensional example, we discuss optimal large-deformation matching of one closed curve to another embedded in the same plane. In the curve-matching example, the manifold $\Emb(S^1, \mathbb{R}^2)$ comprises the space of closed curves $S^1$ embedded in the plane $\mathbb{R}^2$. The diffeomorphic left action $\Diff(\mathbb{R}^2)$ deforms the curve by a smooth invertible time-dependent transformation of the coordinate system in which it is embedded, while leaving the parameterisation of the curve invariant. The diffeomorphic right action $\Diff(S^1)$ corresponds to a smooth invertible reparameterisation of the $S^1$ domain coordinates of the curve. As we show, this right action unlocks an important degree of freedom for geodesically matching the curve shapes using an equivalent fixed boundary value problem, without being constrained to match corresponding points along the template and target curves at the endpoint in time.

math.OC↗

arXiv2009

Solving the Poisson equation on small aspect ratio domains using unstructured meshes

We discuss the ill conditioning of the matrix for the discretised Poisson equation in the small aspect ratio limit, and motivate this problem in the context of nonhydrostatic ocean modelling. Efficient iterative solvers for the Poisson equation in small aspect ratio domains are crucial for the successful development of nonhydrostatic ocean models on unstructured meshes. We introduce a new multigrid preconditioner for the Poisson problem which can be used with finite element discretisations on general unstructured meshes; this preconditioner is motivated by the fact that the Poisson problem has a condition number which is independent of aspect ratio when Dirichlet boundary conditions are imposed on the top surface of the domain. This leads to the first level in an algebraic multigrid solver (which can be extended by further conventional algebraic multigrid stages), and an additive smoother. We illustrate the method with numerical tests on unstructured meshes, which show that the preconditioner makes a dramatic improvement on a more standard multigrid preconditioner approach, and also show that the additive smoother produces better results than standard SOR smoothing. This new solver method makes it feasible to run nonhydrostatic unstructured mesh ocean models in small aspect ratio domains.

math.NA↗

arXiv2009

The variational particle-mesh method for matching curves

Diffeomorphic matching (only one of several names for this technique) is a technique for non-rigid registration of curves and surfaces in which the curve or surface is embedded in the flow of a time-series of vector fields. One seeks the flow between two topologically-equivalent curves or surfaces which minimises some metric defined on the vector fields, \emph{i.e.} the flow closest to the identity in some sense. In this paper, we describe a new particle-mesh discretisation for the evolution of the geodesic flow and the embedded shape. Particle-mesh algorithms are very natural for this problem because Lagrangian particles (particles moving with the flow) can represent the movement of the shape whereas the vector field is Eulerian and hence best represented on a static mesh. We explain the derivation of the method, and prove conservation properties: the discrete method has a set of conserved momenta corresponding to the particle-relabelling symmetry which converge to conserved quantities in the continuous problem. We also introduce a new discretisation for the geometric current matching condition of (Vaillant and Glaunes, 2005). We illustrate the method and the derived properties with numerical examples.

math.NA↗