arXiv subjects

Colin J. Cotter

Publications and source records attributed to Colin J. Cotter.

43 records · Page 3Linked to original sources

arXiv2013

A framework for the automation of generalised stability theory

The traditional approach to investigating the stability of a physical system is to linearise the equations about a steady base solution, and to examine the eigenvalues of the linearised operator. Over the past several decades, it has been recognised that this approach only determines the asymptotic stability of the system, and neglects the possibility of transient perturbation growth arising due to the nonnormality of the system. This observation motivated the development of a more powerful generalised stability theory (GST), which focusses instead on the singular value decomposition of the linearised propagator of the system. While GST has had significant successes in understanding the stability of phenomena in geophysical fluid dynamics, its more widespread applicability has been hampered by the fact that computing the SVD requires both the tangent linear operator and its adjoint: deriving the tangent linear and adjoint models is usually a considerable challenge, and manually embedding them inside an eigensolver is laborious. In this paper, we present a framework for the automation of generalised stability theory, which overcomes these difficulties. Given a compact high-level symbolic representation of a finite element discretisation implemented in the FEniCS system, efficient C++ code is automatically generated to assemble the forward, tangent linear and adjoint models; these models are then used to calculate the optimally growing perturbations to the forward model, and their growth rates. By automating the stability computations, we hope to make these powerful tools a more routine part of computational analysis. The efficiency and generality of the framework is demonstrated with applications drawn from geophysical fluid dynamics, phase separation and quantum mechanics.

cs.MS↗

arXiv2013

Energy- and enstrophy-conserving schemes for the shallow-water equations, based on mimetic finite elements

This paper presents a family of spatial discretisations of the nonlinear rotating shallow-water equations that conserve both energy and potential enstrophy. These are based on two-dimensional mixed finite element methods and hence, unlike some finite difference methods, do not require an orthogonal grid. Numerical verification of the aforementioned properties is also provided.

math.NA↗

arXiv2012

Ensemble filter techniques for intermittent data assimilation - a survey

This survey paper is written with the intention of giving a mathematical introduction to filtering techniques for intermittent data assimilation, and to survey some recent advances in the field. The paper is divided into three parts. The first part introduces Bayesian statistics and its application to statistical inference and estimation. Basic aspects of Markov processes, as they typically arise from scientific models in the form of stochastic differential and/or difference equations, are covered in the second part. The third and final part describes the filtering approach to estimation of model states by assimilation of observational data into scientific models. While most of the material is of survey type, very recent advances in the field of nonlinear data assimilation covered in this paper include a discussion of Bayesian inference in the context of optimal transportation and coupling of random variables, as well as a discussion of recent advances in ensemble transform filters. References and sources for further reading material will be listed at the end of each section.

math.NA↗

arXiv2010

Geostrophic balance preserving interpolation in mesh adaptive shallow-water ocean modelling

The accurate representation of geostrophic balance is an essential requirement for numerical modelling of geophysical flows. Significant effort is often put into the selection of accurate or optimal balance representation by the discretisation of the fundamental equations. The issue of accurate balance representation is particularly challenging when applying dynamic mesh adaptivity, where there is potential for additional imbalance injection when interpolating to new, optimised meshes. In the context of shallow-water modelling, we present a new method for preservation of geostrophic balance when applying dynamic mesh adaptivity. This approach is based upon interpolation of the Helmholtz decomposition of the Coriolis acceleration. We apply this in combination with a discretisation for which states in geostrophic balance are exactly steady solutions of the linearised equations on an f-plane; this method guarantees that a balanced and steady flow on a donor mesh remains balanced and steady after interpolation onto an arbitrary target mesh, to within machine precision. We further demonstrate the utility of this interpolant for states close to geostrophic balance, and show that it prevents pollution of the resulting solutions by imbalanced perturbations introduced by the interpolation.

physics.ao-ph↗

arXiv2008

An Euler Poincaré framework for the multilayer Green Nagdhi equations

The Green Nagdhi equations are frequently used as a model of the wave-like behaviour of the free surface of a fluid, or the interface between two homogeneous fluids of differing densities. Here we show that their multilayer extension arises naturally from a framework based on the Euler Poincare theory under an ansatz of columnar motion. The framework also extends to the travelling wave solutions of the equations. We present numerical solutions of the travelling wave problem in a number of flow regimes. We find that the free surface and multilayer waves can exhibit intriguing differences compared to the results of single layer or rigid lid models.

physics.flu-dyn↗

arXiv2006

The remapped particle-mesh advection scheme

We describe the remapped particle-mesh method, a new mass-conserving method for solving the density equation which is suitable for combining with semi-Lagrangian methods for compressible flow applied to numerical weather prediction. In addition to the conservation property, the remapped particle-mesh method is computationally efficient and at least as accurate as current semi-Lagrangian methods based on cubic interpolation. We provide results of tests of the method in the plane, results from incorporating the advection method into a semi-Lagrangian method for the rotating shallow-water equations in planar geometry, and results from extending the method to the surface of a sphere.

physics.ao-ph↗

arXiv2006

Singular solutions, momentum maps and computational anatomy

This paper describes the variational formulation of template matching problems of computational anatomy (CA); introduces the EPDiff evolution equation in the context of an analogy between CA and fluid dynamics; discusses the singular solutions for the EPDiff equation and explains why these singular solutions exist (singular momentum map). Then it draws the consequences of EPDiff for outline matching problem in CA and gives numerical examples.

nlin.SI↗