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Dive into the research topics where Olivier Verdier is active.

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Featured researches published by Olivier Verdier.


Numerische Mathematik | 2016

B-series methods are exactly the affine equivariant methods

Robert I. McLachlan; Klas Modin; Hans Z. Munthe-Kaas; Olivier Verdier

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.


Foundations of Computational Mathematics | 2016

Aromatic Butcher Series

Hans Z. Munthe-Kaas; Olivier Verdier

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.


Foundations of Computational Mathematics | 2014

Geometric Generalisations of SHAKE and RATTLE

Robert I. McLachlan; Klas Modin; Olivier Verdier; Matt Wilkins

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.


Journal of Scientific Computing | 2016

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

Elena Celledoni; Bawfeh Kingsley Kometa; Olivier Verdier

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.


Physical Review E | 2014

Symplectic integrators for spin systems

Robert I. McLachlan; Klas Modin; Olivier Verdier

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.


Ima Journal of Numerical Analysis | 2015

Collective Lie-Poisson integrators on R3

Robert I. McLachlan; Klas Modin; Olivier Verdier

We develop Lie-Poisson integrators for general Hamiltonian systems on


Discrete and Continuous Dynamical Systems | 2013

Integrability of Nonholonomically Coupled Oscillators

Klas Modin; Olivier Verdier

\mathbf{R}^{3}


Nonlinearity | 2014

Collective symplectic integrators

Robert I. McLachlan; Klas Modin; Olivier Verdier

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


Mathematics of Computation | 2013

Reductions of operator pencils

Olivier Verdier

\mathbf{R}^{3}


Ima Journal of Numerical Analysis | 2013

Collective Lie-Poisson integrators on

Robert I. McLachlan; Klas Modin; Olivier Verdier

on

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Klas Modin

Chalmers University of Technology

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Bawfeh Kingsley Kometa

Norwegian University of Science and Technology

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Elena Celledoni

Norwegian University of Science and Technology

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