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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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

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Projects 2009 2012

General Linear Methods

Hill, A.


Project: UK charity

Research Output 2002 2016

5 Citations (Scopus)

Symmetric general linear methods

Butcher, J. C., Hill, A. T. & Norton, T. J. T., 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)

The control of parasitism in $G$-symplectic methods

Butcher, J. C., Habib, Y., Hill, A. T. & Norton, T. J. T., 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
9 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
8 Citations (Scopus)

Algebraically stable diagonally implicit general linear methods

Hewitt, L. L. & Hill, A. T., Jun 2010, In : Applied Numerical Mathematics. 60, 6, p. 629-636 8 p.

Research output: Contribution to journalArticle

General Linear Methods
Implicit Method
Differential equations
Stiff Differential Equations