Adrian Hill


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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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  • 3 Similar Profiles
General Linear Methods Mathematics
Differential equations Engineering & Materials Science
Linear multistep Methods Mathematics
Hamiltonians Engineering & Materials Science
Symplectic Methods Mathematics
Generalized Eigenproblem Mathematics
Linear Time-varying Systems Mathematics
Algebraic Riccati Equation Mathematics

Network Recent external collaboration on country level. Dive into details by clicking on the dots.

Projects 2009 2012

Research Output 2002 2019

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 journalArticle

5 Citations (Scopus)
104 Downloads (Pure)

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 journalArticle

Open Access
General Linear Methods
Runge Kutta methods
Differential equations
One-step Method

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 journalArticle

Computational Results
Euler equations
Time Scales
14 Citations (Scopus)
87 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 journalArticle

Open Access
Symplectic Methods
General Linear Methods
Small Perturbations
10 Citations (Scopus)

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 journalArticle

Linear Time-varying Systems
Exponential Stability
Asymptotic stability
Linear systems
Stability Estimates


Structure-preserving General Linear Methods

Author: Norton, T., 29 Sep 2015

Supervisor: Hill, A. (Supervisor)

Student thesis: Doctoral ThesisPhD