A composite scheme for solving a differential equation driven by a system of time-varying vector fields

Thesis

Many problems in physics and chemistry are based around the interactions of large numbers of particles. Motivated by this, we consider numerical methods aimed towards the accurate modelling of a 'particle' under the influence of a large cloud of other particles. The trajectory of this 'particle' can be modelled by the solution to an ordinary differential equation driven by a high-dimensional system where the vector field defining the ordinary differential equation is given by the sum of a large number of time-varying vector fields. In this thesis, we shall describe and develop a algorithm for solving such equations.

To make progress, we need to find a way to summarise the large amount of information provided by the vector fields. We do this by performing a three-step simplification; first we localise the problem and expand the vector fields in space using Taylor series, thus projecting the trajectory onto the finite-dimensional subspace of polynomial vector fields of the infinite-dimensional space of vector fields that initially control the evolution of the `particle' over time. We then separate the system into a series of space-dependent vector fields and a time-varying driving path. Finally, we approximate the system's behaviour in time by finding the log-signatures of the driving path and contracting them with the appropriate vector fields, thus producing a flow map.

We consider three existing models and also construct and investigate two new test problems, the Interacting Harmonic Oscillators and Interacting Simple Pendula systems, which are two-dimensional dynamical systems driven by a large number of vector fields, in order to expose the weaknesses in our initial implementations. We then discuss how our implementation could be made faster and more accurate, and how it might be extended for use on higher-dimensional systems.

Authors: Campbell, L

• Publication date: 2010
• Type of award: MSc by Research
• Level of award: Masters
• Awarding institution: Oxford University, UK
• Language: English
• Keywords: high dimensional dynamics; rough paths; mathematical methods; Taylor series
• Subjects: Numerical analysis; Dynamical systems and ergodic theory (mathematics); Ordinary differential equations; Approximations and expansions; Mathematics; Probability theory and stochastic processes