arXiv subjects

Colin J Cotter

Publications and source records attributed to Colin J Cotter.

7 recordsLinked to original sources

arXiv2026

Shifted HSS solvers for the indefinite Helmholtz equation

We provide an iterative solution approach for the indefinite Helmholtz equation discretised using finite elements, based upon a Hermitian Skew-Hermitian Splitting (HSS) iteration applied to the shifted operator, and prove that the iteration is k- and mesh-robust when O(k) HSS iterations are performed. The HSS iterations involve solving a shifted operator that is suitable for approximation by multigrid using standard smoothers and transfer operators, and hence we can use O(N) parallel processors in a high performance computing implementation, where N is the total number of degrees of freedom. We argue that the algorithm converges in O(k) wallclock time when within the range of scalability of the multigrid. We provide numerical results in both 2D and 3D verifying our proofs and demonstrating this claim, establishing a method that can make use of large scale high performance computing systems.

math.NA↗

arXiv2024

New limiter regions for multidimensional flows

Accurate transport algorithms are crucial for computational fluid dynamics and more accurate and efficient schemes are always in development. One dimensional limiting is commonly employed to suppress nonphysical oscillations. However, the application of such limiters can reduce accuracy. It is important to identify the weakest set of sufficient conditions required on the limiter as to allow the development of successful numerical algorithms. The main goal of this paper is to identify new less restrictive sufficient conditions for flux form in-compressible advection to remain monotonic. We identify additional necessary conditions for incompressible flux form advection to be monotonic, demonstrating that the Spekreijse limiter region is not sufficient for incompressible flux form advection to remain monotonic. Then a convex combination argument is used to derive new sufficient conditions that are less restrictive than the Sweby region for a discrete maximum principle. This allows the introduction of two new more general limiter regions suitable for flux form incompressible advection.

math.NA↗

arXiv2023

Time parallel integration and phase averaging for the nonlinear shallow water equations on the sphere

We describe a proof-of-concept development and application of a phase averaging technique to the nonlinear rotating shallow water equations on the sphere, discretised using compatible finite element methods. Phase averaging consists of averaging the nonlinearity over phase shifts in the exponential of the linear wave operator. Phase averaging aims to capture the slow dynamics in a solution that is smoother in time (in transformed variables) so that larger timesteps may be taken. We overcome the two key technical challenges that stand in the way of studying the phase averaging and advancing its implementation: 1) we have developed a stable matrix exponential specific to finite elements and 2) we have developed a parallel finite averaging proceedure. Following Peddle et al (2019), we consider finite width phase averaging windows, since the equations have a finite timescale separation. In our numerical implementation, the averaging integral is replaced by a Riemann sum, where each term can be evaluated in parallel. This creates an opportunity for parallelism in the timestepping method, which we use here to compute our solutions. Here, we focus on the stability and accuracy of the numerical solution. We confirm there is an optimal averaging window, in agreement with theory. Critically, we observe that the combined time discretisation and averaging error is much smaller than the time discretisation error in a semi-implicit method applied to the same spatial discretisation. An evaluation of the parallel aspects will follow in later work.

math.NA↗

arXiv2018

Energy-enstrophy conserving compatible finite element schemes for the rotating shallow water equations with slip boundary conditions

We describe an energy-enstrophy conserving discretisation for the rotating shallow water equations with slip boundary conditions. This relaxes the assumption of boundary-free domains (periodic solutions or the surface of a sphere, for example) in the energy-enstrophy conserving formulation of McRae and Cotter (2014). This discretisation requires extra prognostic vorticity variables on the boundary in addition to the prognostic velocity and layer depth variables. The energy-enstrophy conservation properties hold for any appropriate set of compatible finite element spaces defined on arbitrary meshes with arbitrary boundaries. We demonstrate the conservation properties of the scheme with numerical solutions on a rotating hemisphere.

math.NA↗

arXiv2017

Stochastic partial differential fluid equations as a diffusive limit of deterministic Lagrangian multi-time dynamics

In {\em{Holm}, Proc. Roy. Soc. A 471 (2015)} 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 justifying 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 centering condition. The latter requires that the mean of the fluctuating deviations is small, when pulled back to the mean flow.

math.AP↗

arXiv2006

Semi-geostrophic particle motion and exponentially accurate normal forms

We give an exponentially-accurate normal form for a Lagrangian particle moving in a rotating shallow-water system in the semi-geostrophic limit, which describes the motion in the region of an exponentially-accurate slow manifold (a region of phase space for which dynamics on the fast scale are exponentially small in the Rossby number). The result extends to numerical solutions of this problem via backward error analysis, and extends to the Hamiltonian Particle-Mesh (HPM) method for the shallow-water equations where the result shows that HPM stays close to balance for exponentially-long times in the semi-geostrophic limit. We show how this result is related to the variational asymptotics approach of [Oliver, 2005]; the difference being that on the Hamiltonian side it is possible to obtain strong bounds on the growth of fast motion away from (but near to) the slow manifold.

math.DS↗

arXiv2006

Discrete momentum maps for lattice EPDiff

We focus on the spatial discretization produced by the Variational Particle-Mesh (VPM) method for a prototype fluid equation the known as the EPDiff equation}, which is short for Euler-Poincaré equation associated with the diffeomorphism group (of $\mathbb{R}^d$, or of a $d$-dimensional manifold $Ω$). The EPDiff equation admits measure valued solutions, whose dynamics are determined by the momentum maps for the left and right actions of the diffeomorphisms on embedded subspaces of $\mathbb{R}^d$. The discrete VPM analogs of those dynamics are studied here. Our main results are: (i) a variational formulation for the VPM method, expressed in terms of a constrained variational principle principle for the Lagrangian particles, whose velocities are restricted to a distribution $D_{\VPM}$ which is a finite-dimensional subspace of the Lie algebra of vector fields on $Ω$; (ii) a corresponding constrained variational principle on the fixed Eulerian grid which gives a discrete version of the Euler-Poincaré equation; and (iii) discrete versions of the momentum maps for the left and right actions of diffeomorphisms on the space of solutions.

math.NA↗