Projects per year
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.
Error estimates for semi-Lagrangian finite difference methods applied to Burgers' equation in one dimensionCook, 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 › peer-reviewOpen AccessFile9 Downloads (Pure)
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 › peer-reviewOpen AccessFile7 Citations (Scopus)157 Downloads (Pure)
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 › peer-reviewFile1 Citation (Scopus)95 Downloads (Pure)
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 › peer-reviewOpen AccessFile19 Citations (Scopus)152 Downloads (Pure)
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