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
- 2 Finished
General Linear Methods for Symplectic and Time-Reversible Problems
29/03/09 → 29/04/09
Project: Research council
Research Output
- 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
Open Access
File
5
Citations
(Scopus)
133
Downloads
(Pure)
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
File
1
Citation
(Scopus)
73
Downloads
(Pure)
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
Open Access
File
16
Citations
(Scopus)
119
Downloads
(Pure)
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
10
Citations
(Scopus)
Thesis
Structure-preserving General Linear Methods
Author: Norton, T., 29 Sep 2015Supervisor: Hill, A. (Supervisor)
Student thesis: Doctoral Thesis › PhD
File