39 citations · 312 across the 32 of their papers we have counts for
17 papers · 1 filter
The aromatic bicomplex for the description of divergence-free aromatic forms and volume-preserving integrators
Adrien Laurent, Robert I. McLachlan, Hans Z. Munthe-Kaas +1
Aromatic B-series were introduced as an extension of standard Butcher-series for the study of volume-preserving integrators. It was proven with their help that the only volume-pres…
What makes nonholonomic integrators work?
Klas Modin, Olivier Verdier
A nonholonomic system is a mechanical system with velocity constraints not originating from position constraints; rolling without slipping is the typical example. A nonholonomic in…
A Numerical Algorithm for C2-splines on Symmetric Spaces
Geir Bogfjellmo, Klas Modin, Olivier Verdier
Cubic spline interpolation on Euclidean space is a standard topic in numerical analysis, with countless applications in science and technology. In several emerging fields, for exam…
Currents and finite elements as tools for shape space
James Benn, Stephen Marsland, Robert I McLachlan +2
The nonlinear spaces of shapes (unparameterized immersed curves or submanifolds) are of interest for many applications in image analysis, such as the identification of shapes that…
Numerical Study of Nonlinear Dispersive Wave Models with SpecTraVVave
Henrik Kalisch, Daulet Moldabayev, Olivier Verdier
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…
Full affine equivariance and weak natural transformations in numerical analysis - the case of B-Series
Olivier Verdier
Many algorithms in numerical analysis are affine equivariant: they are immune to changes of affine coordinates. This is because those algorithms are defined using affine invariant…
Butcher series: A story of rooted trees and numerical methods for evolution equations
Robert I McLachlan, Klas Modin, Hans Munthe-Kaas +1
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 o…
A classification of volume preserving generating forms in R^3
Olivier Verdier, Huiyan Xue, Antonella Zanna
In earlier work, Lomeli and Meiss used a generalization of the symplectic approach to study volume preserving generating differential forms. In particular, for the c…
Multi-symplectic discretisation of wave map equations
David Cohen, Olivier Verdier
We present a new multi-symplectic formulation of constrained Hamiltonian partial differential equations, and we study the associated local conservation laws. A multi-symplectic dis…
B-series methods are exactly the affine equivariant methods
Robert I. McLachlan, Klas Modin, Hans Munthe-Kaas +1
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…
Integrators on homogeneous spaces: Isotropy choice and connections
Hans Munthe-Kaas, Olivier Verdier
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 i…
Collective symplectic integrators
Robert I McLachlan, Klas Modin, Olivier Verdier
We construct symplectic integrators for Lie-Poisson systems. The integrators are standard symplectic (partitioned) Runge--Kutta methods. Their phase space is a symplectic vector sp…
Aromatic Butcher Series
Hans 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) whic…
Collective Lie-Poisson integrators on
Robert McLachlan, Klas Modin, Olivier Verdier
We develop Lie-Poisson integrators for general Hamiltonian systems on equipped with the rigid body bracket. The method uses symplectic realisation of $\mathbf{R}^{…
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-Ku…
Geometric Generalisations of SHAKE and RATTLE
Robert I. McLachlan, Klas Modin, Olivier Verdier +1
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…
Reduction and Normal Forms of Matrix Pencils
Olivier Verdier
Matrix pencils, or pairs of matrices, may be used in a variety of applications. In particular, a pair of matrices (E,A) may be interpreted as the differential equation E x' + A x =…