arXiv subjects

Robert I McLachlan

Publications and source records attributed to Robert I McLachlan.

32 records · Page 2Linked to original sources

arXiv2017

Butcher series: A story of rooted trees and numerical methods for evolution equations

Butcher series appear when Runge-Kutta methods for ordinary differential equations are expanded in power series of the step size parameter. Each term in a Butcher series consists of a weighted elementary differential, and the set of all such differentials is isomorphic to the set of rooted trees, as noted by Cayley in the mid 19th century. A century later Butcher discovered that rooted trees can also be used to obtain the order conditions of Runge-Kutta methods, and he found a natural group structure, today known as the Butcher group. It is now known that many numerical methods also can be expanded in Butcher series; these are called B-series methods. A long-standing problem has been to characterize, in terms of qualitative features, all B-series methods. Here we tell the story of Butcher series, stretching from the early work of Cayley, to modern developments and connections to abstract algebra, and finally to the resolution of the characterization problem. This resolution introduces geometric tools and perspectives to an area traditionally explored using analysis and combinatorics.

math.NA↗

arXiv2015

Symmetry reduction for central force problems

We given an elementary illustration of symmetry reduction for central force problems, drawing phase portraits of the reduced dynamics as the intersection of Casimir and energy level sets in three dimensions. These systems form a classic example of symplectic reduction which can usefully be compared to the more-commonly seen case of the free rigid body.

math-ph↗

arXiv2015

Travelling wave solutions of multisymplectic discretizations of semi-linear wave equations

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 equation. A travelling wave ansatz leads to an ordinary difference equation, whose solutions correspond to the numerical scheme and can be compared to travelling wave solutions of the corresponding PDE. For a discontinuous nonlinearity the difference equation is solved exactly. For continuous nonlinearities the difference equation is solved using a Fourier series, and resonances that depend on the grid-size are revealed for a smooth nonlinearity. In general, the infinite dimensional functional equation, which must be solved to get the travelling wave solutions, is intractable, but backward error analysis proves to be a powerful tool, as it provides a way to study the solutions of the equation through a simple ODE that describes the behavior to arbitrarily high order. A general framework for using backward error analysis to analyze preservation of travelling waves for other equations and discretisations is presented. Then, the advantages that multisymplectic methods have over other methods are briefly highlighted.

math.NA↗

arXiv2014

Integrability properties of Kahan's method

We present several novel examples of integrable quadratic vector fields for which Kahan's discretization method preserves integrability. Our examples include generalized Suslov and Ishii systems, Nambu systems, Riccati systems, and the first Painlevé equation. We also discuss how Manin transformations arise in Kahan discretizations of certain vector fields.

nlin.SI↗

arXiv2013

Collective symplectic integrators

We construct symplectic integrators for Lie-Poisson systems. The integrators are standard symplectic (partitioned) Runge--Kutta methods. Their phase space is a symplectic vector space with a Hamiltonian action with momentum map $J$ whose range is the target Lie--Poisson manifold, and their Hamiltonian is collective, that is, it is the target Hamiltonian pulled back by $J$. The method yields, for example, a symplectic midpoint rule expressed in 4 variables for arbitrary Hamiltonians on $\mathfrak{so}(3)^*$. The method specializes in the case that a sufficiently large symmetry group acts on the fibres of $J$, and generalizes to the case that the vector space carries a bifoliation. Examples involving many classical groups are presented.

math.NA↗

arXiv2013

Symplectic integrators for index one constraints

We show that symplectic Runge-Kutta methods provide effective symplectic integrators for Hamiltonian systems with index one constraints. These include the Hamiltonian description of variational problems subject to position and velocity constraints nondegenerate in the velocities, such as those arising in sub-Riemannian geometry and control theory.

math.NA↗

arXiv2013

Discrete gradient methods have an energy conservation law

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 conservation law. The discrete conservation law has the same conserved density as the continuous conservation law, while its flux is found by replacing all derivatives of the conserved density appearing in the continuous flux by discrete gradients.

math.NA↗

arXiv2012

Geometric properties of Kahan's method

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 Poisson vector space with constant Poisson structure, the map determined by this discretization has a conserved modified Hamiltonian and an invariant measure, a combination previously unknown amongst Runge--Kutta methods applied to nonlinear vector fields. This produces large classes of integrable rational mappings in two and three dimensions, explaining some of the integrable cases that were previously known.

math.NA↗

arXiv2005

N-Particle Dynamics of the Euler Equations for Planar Diffeomorphisms

The Euler equations associated with diffeomorphism groups have received much recent study because of their links with fluid dynamics, computer vision, and mechanics. In this paper, we consider the dynamics of $N$ point particles or `blobs' moving under the action of the Euler equations associated with the group of diffeomorphisms of the plane in a variety of different metrics. The 2 body problem is always integrable, and we analyze its phase portrait under different metrics. In particular, we show that 2-body capturing orbits (in which the distances between the particles tend to 0 as $t \to \infty$) can occur when the kernel is sufficiently smooth and the relative initial velocity of the particles is sufficiently large. We compute the dynamics of these `dipoles' with respect to other test particles, and supplement the calculations with simulations for larger $N$ that illustrate the different regimes.

nlin.CD↗

arXiv2002

The algebraic entropy of classical mechanics

We describe the `Lie algebra of classical mechanics', modelled on the Lie algebra generated by kinetic and potential energy of a simple mechanical system with respect to the canonical Poisson bracket. It is a polynomially graded Lie algebra, a class we introduce. We describe these Lie algebras, give an algorithm to calculate the dimensions $c_n$ of the homogeneous subspaces of the Lie algebra of classical mechanics, and determine the value of its entropy $\lim_{n\to\infty} c_n^{1/n}$. It is $1.82542377420108...$, a fundamental constant associated to classical mechanics.

math-ph↗

arXiv1999

Area preservation in computational fluid dynamics

Incompressible two-dimensional flows such as the advection (Liouville) equation and the Euler equations have a large family of conservation laws related to conservation of area. We present two Eulerian numerical methods which preserve a discrete analog of area. The first is a fully discrete model based on a rearrangement of cells; the second is more conventional, but still preserves the area within each contour of the vorticity field. Initial tests indicate that both methods suppress the formation of spurious oscillations in the field.

math.NA↗

arXiv1998

Numerical integrators that contract volume

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 systems. We prove a sufficient condition for Runge-Kutta methods to have the appropriate contraction property for these two-dimensional systems; the midpoint rule is an example.

math.NA↗

arXiv1998

Spatial discretization of partial differential equations with integrals

We consider the problem of constructing spatial finite difference approximations on a fixed, arbitrary grid, which have analogues of any number of integrals of the partial differential equation and of some of its symmetries. A basis for the space of of such difference operators is constructed; most cases of interest involve a single such basis element. (The ``Arakawa'' Jacobian is such an element.) We show how the topology of the grid affects the complexity of the operators.

math.NA↗

arXiv1998

A unified approach to Hamiltonian systems, Poisson systems, gradient systems, and systems with Lyapunov functions and/or first integrals

Systems with a first integral (i.e., constant of motion) or a Lyapunov function can be written as ``linear-gradient systems'' $\dot x= L(x)\nabla V(x)$ for an appropriate matrix function $L$, with a generalization to several integrals or Lyapunov functions. The discrete-time analogue, $Δx/Δt = L \bar\nabla V$ where $\bar\nabla$ is a ``discrete gradient,'' preserves $V$ as an integral or Lyapunov function, respectively.

math-ph↗