David I. McLaren

A new class of energy-preserving numerical integration methods

The first ever energy-preserving B-series numerical integration method for (ordinary) differential equations is presented and applied to several Hamiltonian systems. Related novel Lie algebraic results are also discussed.

Preserving energy resp. dissipation in numerical PDEs using the Average Vector Field method

We give a systematic method for discretizing Hamiltonian partial differential equations (PDEs) with constant symplectic structure, while preserving their energy exactly. The same method, applied to PDEs with constant dissipative structure, also preserves the correct monotonic decrease of energy. The method is illustrated by many examples. In the Hamiltonian case these include: the sine-Gordon, Korteweg-de Vries, nonlinear Schrodinger, (linear) time-dependent Schrodinger, and Maxwell equations. In the dissipative case the examples are: the Allen-Cahn, Cahn-Hilliard, Ginzburg-Landau, and heat equations.

Integral-preserving integrators

Ordinary differential equations having a first integral may be solved numerically using one of several methods, with the integral preserved to machine accuracy. One such method is the discrete gradient method. It is shown here that the order of the method can be bootstrapped repeatedly to higher orders of accuracy. The method is illustrated using the Henon–Heiles system.

Studies to assess the suitability of Uromyces pencanus as a biological control agent for Nassella neesiana (Poaceae) in Australia and New Zealand.

Nassella neesiana (Chilean needle grass), a South American species, is an intractable weed invading managed pastures and natural grasslands that has become a target for biological control in Australia and New Zealand. Studies have been carried out to assess the potential of three rusts naturally infecting this grass species in Argentina: Uromyces pencanus, Puccinia graminella and P. nassellae, as biocontrol agents. Of the three, U. pencanus was recognised as the most promising candidate. It causes significant damage to its host in the field and there is an isolate that can infect most Australian populations of the weed tested. Herein are described methods for: maintaining the rust in the glasshouse; storing urediniospores over 12 months; and, for inoculating urediniospores in order to test the host-specificity of selected isolates of the rust. Evidence from the literature, and a preliminary host range test indicates that U. pencanus is sufficiently host-specific for use as a classical biocontrol agent. Attempts at elucidating the life cycle of U. pencanus were unsuccessful as teliospores did not germinate. It appears that these have become redundant and the rust cycles as urediniospores on its grass host.

Explicit volume-preserving and symplectic integrators for trigonometric polynomial flows

We introduce explicit volume-preserving and symplectic integrators for the case of generalized trigonometric polynomial flows. The method is demonstrated using the Arter flow, and computational trials are conducted using a 4-dimensional vector field.

Projection methods and discrete gradient methods for preserving first integrals of ODEs

In this paper we study linear projection methods for approximating the solution and simultaneously preserving first integrals of autonomous ordinary differential equations. We show that each (linear) projection method is equivalent to a class of discrete gradient methods, in both single and multiple first integral cases, and known results for discrete gradient methods also apply to projection methods. Thus we prove that in the single first integral case, under certain mild conditions, the numerical solution for a projection method exists and is locally unique, and preserves the order of accuracy of the underlying method. Our results allow considerable freedom for the choice of projection direction and do not have a time step restriction close to critical points.

Discretization of polynomial vector fields by polarization

A novel integration method for quadratic vector fields was introduced by Kahan in 1993. Subsequently, it was shown that Kahans method preserves a (modified) measure and energy when applied to quadratic Hamiltonian vector fields. Here we generalize Kahans method to cubic resp. higher degree polynomial vector fields and show that the resulting discretization also preserves modified versions of the measure and energy when applied to cubic resp. higher degree polynomial Hamiltonian vector fields.

Plant/pathogen interactions observed during host range testing of the rust fungus Uromyces pencanus, a classical biological control agent for Chilean needle grass (Nassella neesiana) in Australia and New Zealand

ABSTRACT Nassella neesiana (Chilean needle grass) is a South American grass species that is a serious weed in Australia and New Zealand. The rust fungus Uromyces pencanus is a promising biocontrol agent that could be used to control the weed in both countries. Extensive host range testing has been conducted to explore the specificity of the rust. In this paper we discuss the different degrees of invasion by the rust of the tissues of target and non-target species; the plant defences elicited by such invasion at the cellular level; and their relevance to the biological control of Chilean needle grass.