##### Speaker

Dr. Vu Thai Luan, Assistant Professor, Department of Mathematics & Statistics, MSU

##### Title

Mathematics Seminar Series

##### Subtitle

Rooted trees, B-series and their applications to numerical methods

##### Physical Location

Allen 14

**Abstract: ** In graph theory, rooted trees are defined as undirected and connected graphs without cycles, in which a vertex has been distinguished as the root. Surprisingly, they can be used to represent certain differential operators, which are related to elementary differentials, and derive order conditions for Runge--Kutta methods and other time integration methods in a convenient way. This leads to the introduction of the concept of B-series, following the terminology of E. Hairer, G. Wanner, and C. Lubich in their popular book “*Geometric Numerical Integration*” (2006; cited 6771 times), which is one of the essential tools in the study of properties of numerical methods for evolution equations.

The aim of this talk is to provide an introduction to rooted trees and B-series and to illustrate how to apply them to derive order conditions for classical Runge--Kutta methods. Additionally, we will also briefly show a recent result on their application to obtain order conditions for exponential Runge--Kutta methods up to order 6 [V.T. Luan & T. Alshmy, *Applied Mathematics Letters* (to appear), 2024].

The talk is designed for a general audience and is accessible to both undergraduate and graduate students.