Gilles Vilmart

Algebraic Structures of B-series

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 of order conditions, and a substitution law gives much insight into modified differential equations of backward error analysis. These two laws give rise to algebraic structures (groups and Hopf algebras of trees) that have recently received much attention also in the non-numerical literature. This article emphasizes these algebraic structures and presents interesting relationships among them.

Numerical integrators based on modified differential equations

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. This approach is illustrated with the implicit midpoint rule applied to the full dynamics of the free rigid body. Special attention is paid to methods represented as B-series, for which explicit formulae for the modified differential equation are given. A new composition law on B-series, called substitution law, is presented.

Preprocessed discrete Moser?Veselov algorithm for the full dynamics of a rigid body

The discrete Moser?Veselov algorithm is an integrable discretization of the equations of motion for a free rigid body. It is symplectic and time reversible, and it conserves all first integrals of the system. The only drawback is its low order. We present a modification of this algorithm to arbitrarily high order which has negligible overhead but considerably improves the accuracy.

Analysis of the finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems

A fully discrete analysis of the finite element heterogeneous multiscale method for a class of nonlinear elliptic homogenization problems of nonmonotone type is proposed. In contrast to previous results obtained for such problems in dimension

High Order Numerical Approximation of the Invariant Measure of Ergodic SDEs

dleq2

Coupling heterogeneous multiscale FEM with Runge-Kutta methods for parabolic homogenization problems: a fully discrete space-time analysis

for the

A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems

H^1

Weak second order explicit stabilized methods for stiff stochastic differential equations

norm and for a semi-discrete formulation [W.E, P. Ming and P. Zhang, J. Amer. Math. Soc. 18 (2005), no. 1, 121–156], we obtain optimal convergence results for dimension

HIGH WEAK ORDER METHODS FOR STOCHASTIC DIFFERENTIAL EQUATIONS BASED ON MODIFIED EQUATIONS

dleq3

Reducing round-off errors in rigid body dynamics

and for a fully discrete method, which takes into account the microscale discretization. In addition, our results are also valid for quadrilateral finite elements, optimal a-priori error estimates are obtained for the