Some algebraic techniques in the numerical analysis of ordinary differential equations

Applied and Computational Mathematics Seminar
Thursday, October 29, 2015 - 11:00
1 hour (actually 50 minutes)
Skiles 006
INRIA Rennes, Université de Rennes I, ENS Rennes

Joint with School of Math Colloquium. Special time (colloquium time).

In this talk, I will introduce B-series, which are formal series indexed by trees, and briefly expose the two laws operating on them. The presentation of algebraic aspects will here be focused on applications to numerical analysis. I will then show how B-series can be used on two examples: modified vector fields techniques, which allow for the construction of arbitrarly high-order schemes, and averaging methods, which lie at the core of many numerical schemes highly-oscillatory evolution equations. Ultimately and if time permits, I will illustrate how these concepts lead to the accelerated simulation of the rigid body and the (nonlinear) Schrödinger equations. A significant part of the talk will remain expository and aimed at a general mathematical audience.