303 citations · 612 across the 16 of their papers we have counts for
20 papers
Discrete gradients in short-range molecular dynamics simulations
Volker Grimm, Tobias Kliesch, G. R. W. Quispel
Discrete gradients (DG) or more exactly discrete gradient methods are time integration schemes that are custom-built to preserve first integrals or Lyapunov functions of a given or…
A geometric integration approach to nonsmooth, nonconvex optimisation
Erlend S. Riis, Matthias J. Ehrhardt, G. R. W. Quispel +1
The optimisation of nonsmooth, nonconvex functions without access to gradients is a particularly challenging problem that is frequently encountered, for example in model parameter…
Three classes of quadratic vector fields for which the Kahan discretization is the root of a generalised Manin transformation
Peter H. van der Kamp, Elena Celledoni, Robert I. McLachlan +3
We apply Kahan's discretisation method to three classes of 2-dimensional quadratic vector fields with quadratic, resp cubic, resp quartic Hamiltonians. We show that the maps obtain…
Two classes of quadratic vector fields for which the Kahan discretization is integrable
Elena Celledoni, Robert I. McLachlan, David I. McLaren +2
Applying Kahan's discretization to the reduced Nahm equations, we obtain two classes of integrable mappings.
The use of discrete gradient methods for total variation type regularization problems in image processing
V Grimm, R I McLachlan, D McLaren +2
Discrete gradient methods are well-known methods of Geometric Numerical Integration, which preserve the dissipation of gradient systems. The preservation of the dissipation of a sy…
Why geometric integration?
A. Iserles, G. R. W. Quispel
Since its emergence, GNI has become the new paradigm in numerical solution of ODEs, while making significant inroads into numerical PDEs. As often, yesterday's revolutionaries beca…
Liouville integrability and superintegrability of a generalized Lotka-Volterra system and its Kahan discretization
Theodoros E. Kouloukas, G. R. W. Quispel, Pol Vanhaecke
We prove the Liouville and superintegrability of a generalized Lotka-Volterra system and its Kahan discretization.
Travelling wave solutions of multisymplectic discretizations of semi-linear wave equations
Fleur McDonald, Robert I McLachlan, Brian E Moore +1
How well do multisymplectic discretisations preserve travelling wave solutions? To answer this question, the 5-point central difference scheme is applied to the semi-linear wave eq…
Discretization of polynomial vector fields by polarization
Elena Celledoni, Robert I. McLachlan, David I. McLaren +2
A novel integration method for quadratic vector fields was introduced by Kahan in 1993. Subsequently, it was shown that Kahan's method preserves a (modified) measure and energy whe…
Integrability properties of Kahan's method
Elena Celledoni, Robert I McLachlan, David I McLaren +2
We present several novel examples of integrable quadratic vector fields for which Kahan's discretization method preserves integrability. Our examples include generalized Suslov and…
Discrete gradient methods have an energy conservation law
Robert I McLachlan, G R W Quispel
We show for a variety of classes of conservative PDEs that discrete gradient methods designed to have a conserved quantity (here called energy) also have a time-discrete conservati…
Projection methods and discrete gradient methods for preserving first integrals of ODEs
R. A. Norton, D. I. McLaren, G. R. W. Quispel +2
In this paper we study linear projection methods for approximating the solution and simultaneously preserving first integrals of autonomous ordinary differential equations. We show…
Discrete gradient methods for preserving a first integral of an ordinary differential equation
Richard A. Norton, G. R. W. Quispel
In this paper we consider discrete gradient methods for approximating the solution and preserving a first integral (also called a constant of motion) of autonomous ordinary differe…
Geometric properties of Kahan's method
Elena Celledoni, Robert I McLachlan, Brynjulf Owren +1
We show that Kahan's discretization of quadratic vector fields is equivalent to a Runge--Kutta method. When the vector field is Hamiltonian on either a symplectic vector space or a…
Preserving energy resp. dissipation in numerical PDEs using the "Average Vector Field" method
E. Celledoni, V. Grimm, R. I. McLachlan +4
We give a systematic method for discretizing Hamiltonian partial differential equations (PDEs) with constant symplectic structure, while preserving their energy exactly. The same m…
The saddle-node-transcritical bifurcation in a population model with constant rate harvesting
K. V. I. Saputra, L. van Veen, G. R. W. Quispel
We study the interaction of saddle-node and transcritical bifurcations in a Lotka-Volterra model with a constant term representing harvesting or migration. Because some of the equi…
Construction of Integrals of Higher-Order Mappings
Ken-ichi Maruno, G. R. W. Quispel
We find that certain higher-order mappings arise as reductions of the integrable discrete A-type KP (AKP) and B-type KP (BKP) equations. We find conservation laws for the AKP and B…
Lie group foliations: Dynamical systems and integrators
Robert I. McLachlan, Matthew Perlmutter, G. Reinout W. Quispel
Foliate systems are those which preserve some (possibly singular) foliation of phase space, such as systems with integrals, systems with continuous symmetries, and skew product sys…
Numerical integrators that contract volume
Robert I McLachlan, G R W Quispel
We study numerical integrators that contract phase space volume even when the ODE does so at an arbitrarily small rate. This is done by a splitting into two-dimensional contractive…
A unified approach to Hamiltonian systems, Poisson systems, gradient systems, and systems with Lyapunov functions and/or first integrals
Robert I McLachlan, GRW Quispel, Nicolas Robidoux
Systems with a first integral (i.e., constant of motion) or a Lyapunov function can be written as ``linear-gradient systems'' for an appropriate matrix fu…