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- Philippe Chartier, Ernst Hairer, Gilles Vilmart
- Math. Comput.
- 2007

Inspired by the theory of modified equations (backward error analysis), a new approach to high-order, structure-preserving numerical integrators for ordinary differential equations is developed. Thisâ€¦ (More)

- Assyr Abdulle, Gilles Vilmart, Konstantinos C. Zygalakis
- SIAM J. Numerical Analysis
- 2014

We introduce new sufficient conditions for a numerical method to approximate with high order of accuracy the invariant measure of an ergodic system of stochastic differential equations, independentlyâ€¦ (More)

- Assyr Abdulle, David Cohen, Gilles Vilmart, Konstantinos C. Zygalakis
- SIAM J. Scientific Computing
- 2012

Inspired by recent advances in the theory of modified differential equations, we propose a new methodology for constructing numerical integrators with high weak order for the time integration ofâ€¦ (More)

- E. Hairer, Gilles Vilmart
- 2006

The Discrete Moserâ€“Veselov algorithm is an integrable discretisation of the equations of motion for the free rigid body. It is symplectic and time-reversible, and it conserves all first integrals ofâ€¦ (More)

Using composition procedures, we build up high order splitting methods to solve evolution equations posed in finite or infinite dimensional spaces. Since high-order splitting methods with real timeâ€¦ (More)

- Philippe Chartier, Ernst Hairer, Gilles Vilmart
- Foundations of Computational Mathematics
- 2010

B-series are a fundamental tool in practical and theoretical aspects of numerical integrators for ordinary differential equations. A composition law for B-series permits an elegant derivation ofâ€¦ (More)

- Assyr Abdulle, Gilles Vilmart, Konstantinos C. Zygalakis
- SIAM J. Scientific Computing
- 2013

We introduce a new family of explicit integrators for stiff ItÃ´ stochastic differential equations (SDEs) of weak order two. These numerical methods belong to the class of one-step stabilized methodsâ€¦ (More)

- Assyr Abdulle, Gilles Vilmart
- Math. Comput.
- 2014

An analysis of the finite element heterogeneous multiscale method for a class of quasilinear elliptic homogenization problems of nonmonotone type is proposed. We obtain optimal convergence resultsâ€¦ (More)

- Gilles Vilmart
- SIAM J. Scientific Computing
- 2015

The concept of effective order is a popular methodology in the deterministic literature for the construction of efficient and accurate integrators for differential equations over long times. The ideaâ€¦ (More)

- Assyr Abdulle, Gilles Vilmart, Konstantinos C. Zygalakis
- SIAM J. Numerical Analysis
- 2015

A new characterization of sufficient conditions for the Lie-Trotter splitting to capture the numerical invariant measure of nonlinear ergodic Langevin dynamics up to an arbitrary order is discussed.â€¦ (More)