Olivier Verdier

B-series methods are exactly the affine equivariant methods

Butcher series, also called B-series, are a type of expansion, fundamental in the analysis of numerical integration. Numerical methods that can be expanded in B-series are defined in all dimensions, so they correspond to sequences of maps—one map for each dimension. A long-standing problem has been to characterise those sequences of maps that arise from B-series. This problem is solved here: we prove that a sequence of smooth maps between vector fields on affine spaces has a B-series expansion if and only if it is affine equivariant, meaning it respects all affine maps between affine spaces.

Aromatic Butcher Series

We show that without other further assumption than affine equivariance and locality, a numerical integrator has an expansion in a generalized form of Butcher series (B-series), which we call aromatic B-series. We obtain an explicit description of aromatic B-series in terms of elementary differentials associated to aromatic trees, which are directed graphs generalizing trees. We also define a new class of integrators, the class of aromatic Runge–Kutta methods, that extends the class of Runge–Kutta methods and have aromatic B-series expansion but are not B-series methods. Finally, those results are partially extended to the case of more general affine group equivariance.

Geometric Generalisations of SHAKE and RATTLE

A geometric analysis of the shake and rattle methods for constrained Hamiltonian problems is carried out. The study reveals the underlying differential geometric foundation of the two methods, and the exact relation between them. In addition, the geometric insight naturally generalises shake and rattle to allow for a strictly larger class of constrained Hamiltonian systems than in the classical setting.In order for shake and rattle to be well defined, two basic assumptions are needed. First, a nondegeneracy assumption, which is a condition on the Hamiltonian, i.e., on the dynamics of the system. Second, a coisotropy assumption, which is a condition on the geometry of the constrained phase space. Non-trivial examples of systems fulfilling, and failing to fulfill, these assumptions are given.

High Order Semi-Lagrangian Methods for the Incompressible Navier---Stokes Equations

We propose a class of semi-Lagrangian methods of high approximation order in space and time, based on spectral element space discretizations and exponential integrators of Runge–Kutta type. The methods were presented in Celledoni and Kometa (J Sci Comput 41(1):139–164, 2009) for simpler convection–diffusion equations. We discuss the extension of these methods to the Navier–Stokes equations, and their implementation using projections. Semi-Lagrangian methods up to order three are implemented and tested on various examples. The good performance of the methods for convection-dominated problems is demonstrated with numerical experiments.

Symplectic integrators for spin systems

We present a symplectic integrator, based on the implicit midpoint method, for classical spin systems where each spin is a unit vector in R{3}. Unlike splitting methods, it is defined for all Hamiltonians and is O(3)-equivariant, i.e., coordinate-independent. It is a rare example of a generating function for symplectic maps of a noncanonical phase space. It yields a new integrable discretization of the spinning top.

Collective Lie-Poisson integrators on R3

We develop Lie-Poisson integrators for general Hamiltonian systems on

Integrability of Nonholonomically Coupled Oscillators

\mathbf{R}^{3}

Collective symplectic integrators

equipped with the rigid body bracket. The method uses symplectic realisation of

Reductions of operator pencils

\mathbf{R}^{3}

Collective Lie-Poisson integrators on

on