Projects per year

## Personal profile

### Research interests

Numerical time-stepping methods which preserve geometric properties of time-symmetric and Hamiltonian problems over very long times. In particular, the analysis and construction of general linear methods (GLMs), which are a generalization of Runge-Kutta and linear multistep methods.

Recent work with Prof John Butcher (University of Auckland) and Terence Norton (PhD student, Bath) has shown that G-symplectic time-symmetric GLMs can solve non-separable Hamiltonian systems more efficiently than symplectic Runge- Kutta methods. Methods up to order 6 have been constructed. The underlying one-step method of such methods has been shown to be both symmetric and symplectic.

A study of starting and finishing methods has led to a re-evaluation of the way that classical numerical methods, such as Leapfrog and Crank-Nicolson, should be implemented on symplectic and near symplectic systems, such as the equations modelling weather.

Other interests include nonautonomous stability for both continuous and discrete dynamical systems.

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## Projects 2009 2012

- 2 Finished

### General Linear Methods for Symplectic and Time-Reversible Problems

29/03/09 → 29/04/09

Project: Research council

## Research Output 2002 2019

- 16 Article

### Error estimates for semi-Lagrangian finite difference methods applied to Burgers' equation in one dimension

Cook, S., Budd, C., Hill, A. & Melvin, T., 1 Nov 2019, In : Applied Numerical Mathematics. 145, p. 261-282 22 p.Research output: Contribution to journal › Article

### Symmetric general linear methods

Butcher, J. C., Hill, A. T. & Norton, T. J. T., 1 Dec 2016, In : BIT Numerical Mathematics. 56, 4, p. 1189-1212 24 p.Research output: Contribution to journal › Article

### An iterative starting method to control parasitism for the Leapfrog method

Hill, A. & Norton, T., 1 Jan 2015, In : Applied Numerical Mathematics. 87, 1, p. 145 156 p., Volume 87.Research output: Contribution to journal › Article

### The control of parasitism in $G$-symplectic methods

Butcher, J. C., Habib, Y., Hill, A. T. & Norton, T. J. T., 14 Oct 2014, In : SIAM Journal on Numerical Analysis (SINUM). 52, 5, p. 2440-2465 26 p.Research output: Contribution to journal › Article

### Exponential stability of time-varying linear systems

Hill, A. T. & Ilchmann, A., Jul 2011, In : IMA Journal of Numerical Analysis. 31, 3, p. 865-885 21 p.Research output: Contribution to journal › Article

## Thesis

## Structure-preserving General Linear Methods

Author: Norton, T., 29 Sep 2015Supervisor: Hill, A. (Supervisor)

Student thesis: Doctoral Thesis › PhD