What makes nonholonomic integrators work?

A nonholonomic system is a mechanical system with velocity constraints not originating from position constraints; rolling without slipping is the typical example. A nonholonomic integrator is a numerical method specifically designed for nonholonomic systems. It has been observed numerically that many nonholonomic integrators exhibit excellent long-time behaviour when applied to various test problems. The excellent performance is often attributed to some underlying discrete version of the Lagrange--d'Alembert principle. Instead, in this paper, we give evidence that reversibility is behind the observed behaviour. Indeed, we show that many standard nonholonomic test problems have the structure of being foliated over reversible integrable systems. As most nonholonomic integrators preserve the foliation and the reversible structure, near conservation of the first integrals is a consequence of reversible KAM theory. Therefore, to fully evaluate nonholonomic integrators one has to consider also non-reversible nonholonomic systems. To this end we construct perturbed test problems that are integrable but no longer reversible (with respect to the standard reversibility map). Applying various nonholonomic integrators from the literature to these problems we observe that no method performs well on all problems. This further indicates that reversibility is the main mechanism behind near conservation of first integrals for nonholonomic integrators. A list of relevant open problems is given.Comment: 27 pages, 9 figure

Time-Periodic Solutions of the Burgers Equation

We investigate the time periodic solutions to the viscous Burgers equation $u_t -\mu u_{xx} + uu_x = f$ for irregular forcing terms. We prove that the corresponding Burgers operator is a diffeomorphism between appropriate function spaces

Multi-symplectic discretisation of wave map equations

We present a new multi-symplectic formulation of constrained Hamiltonian partial differential equations, and we study the associated local conservation laws. A multi-symplectic discretisation based on this new formulation is exemplified by means of the Euler box scheme. When applied to the wave map equation, this numerical scheme is explicit, preserves the constraint and can be seen as a generalisation of the Shake algorithm for constrained mechanical systems. Furthermore, numerical experiments show excellent conservation properties of the numerical solutions

Integrators on homogeneous spaces: Isotropy choice and connections

We consider numerical integrators of ODEs on homogeneous spaces (spheres, affine spaces, hyperbolic spaces). Homogeneous spaces are equipped with a built-in symmetry. A numerical integrator respects this symmetry if it is equivariant. One obtains homogeneous space integrators by combining a Lie group integrator with an isotropy choice. We show that equivariant isotropy choices combined with equivariant Lie group integrators produce equivariant homogeneous space integrators. Moreover, we show that the RKMK, Crouch--Grossman or commutator-free methods are equivariant. To show this, we give a novel description of Lie group integrators in terms of stage trees and motion maps, which unifies the known Lie group integrators. We then proceed to study the equivariant isotropy maps of order zero, which we call connections, and show that they can be identified with reductive structures and invariant principal connections. We give concrete formulas for connections in standard homogeneous spaces of interest, such as Stiefel, Grassmannian, isospectral, and polar decomposition manifolds. Finally, we show that the space of matrices of fixed rank possesses no connection

Numerical Study of Nonlinear Dispersive Wave Models with SpecTraVVave

In nonlinear dispersive evolution equations, the competing effects of nonlinearity and dispersion make a number of interesting phenomena possible. In the current work, the focus is on the numerical approximation of traveling-wave solutions of such equations. We describe our efforts to write a dedicated Python code which is able to compute traveling-wave solutions of nonlinear dispersive equations of the general form \begin{equation*} u_t + [f(u)]_{x} + \mathcal{L} u_x = 0, \end{equation*} where $\mathcal{L}$ is a self-adjoint operator, and $f$ is a real-valued function with $f(0) = 0$. The SpectraVVave code uses a continuation method coupled with a spectral projection to compute approximations of steady symmetric solutions of this equation. The code is used in a number of situations to gain an understanding of traveling-wave solutions. The first case is the Whitham equation, where numerical evidence points to the conclusion that the main bifurcation branch features three distinct points of interest, namely a turning point, a point of stability inversion, and a terminal point which corresponds to a cusped wave. The second case is the so-called modified Benjamin-Ono equation where the interaction of two solitary waves is investigated. It is found that is possible for two solitary waves to interact in such a way that the smaller wave is annihilated. The third case concerns the Benjamin equation which features two competing dispersive operators. In this case, it is found that bifurcation curves of periodic traveling-wave solutions may cross and connect high up on the branch in the nonlinear regime

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

A minimal-variable symplectic integrator on spheres

We construct a symplectic, globally defined, minimal-coordinate, equivariant integrator on products of 2-spheres. Examples of corresponding Hamiltonian systems, called spin systems, include the reduced free rigid body, the motion of point vortices on a sphere, and the classical Heisenberg spin chain, a spatial discretisation of the Landau-Lifschitz equation. The existence of such an integrator is remarkable, as the sphere is neither a vector space, nor a cotangent bundle, has no global coordinate chart, and its symplectic form is not even exact. Moreover, the formulation of the integrator is very simple, and resembles the geodesic midpoint method, although the latter is not symplectic