arXiv subjects

Chris J. Budd

Publications and source records attributed to Chris J. Budd.

3 recordsLinked to original sources

arXiv2020

An analysis of the periodically forced PP04 climate model, using the theory of non-smooth dynamical systems

In this paper we perform a careful analysis of the forced PP04 model for climate change, in particular the behaviour of the ice-ages. This system models the transition from a glacial to an inter-glacial state through a sudden release of oceanic Carbon Dioxide into the atmosphere. This process can be cast in terms of a Filippov dynamical system, with a discontinuous change in its dynamics related to the Carbon Dioxide release. By using techniques from the theory of non-smooth dynamical systems, we give an analysis of this model in the cases of both no insolation forcing and also periodic insolation forcing. This reveals a rich, and novel, dynamical structure to the solutions of the PP04 model. In particular we see synchronised periodic solutions with subtle regions of existence which depend on the amplitude and frequency of the forcing. The orbits can be created/destroyed in both smooth and discontinuity induced bifurcations. We study both the orbits and the transitions between them and make comparisons with actual climate dynamics.

nlin.CD↗

arXiv2018

The scaling and skewness of optimally transported meshes on the sphere

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 aniso...

math.NA↗

arXiv2017

Optimal-transport-based mesh adaptivity on the plane and sphere using finite elements

In moving mesh methods, the underlying mesh is dynamically adapted without changing the connectivity of the mesh. We specifically consider the generation of meshes which are adapted to a scalar monitor function through equidistribution. Together with an optimal transport condition, this leads to a Monge-Ampère equation for a scalar mesh potential. We adapt an existing finite element scheme for the standard Monge-Ampère equation to this mesh generation problem; this is a mixed finite element scheme, in which an extra discrete variable is introduced to represent the Hessian matrix of second derivatives. The problem we consider has additional nonlinearities over the basic Monge-Ampère equation due to the implicit dependence of the monitor function on the resulting mesh. We also derive the equivalent Monge-Ampère-like equation for generating meshes on the sphere. The finite element scheme is extended to the sphere, and we provide numerical examples. All numerical experiments are performed using the open-source finite element framework Firedrake.

math.NA↗