Objectives and Content
The students should acquire an understanding of the process of computing solutions of differential equations, both from an analytical, geometrical and algebraic perspective. The students should get insight in the modern algebraic structures governing the computational processes.
- The core material will include:
- The exact solution of differential equations, via Taylor expansions expressed as Butcher B-series for Euclidean spaces, Lie series on manifolds and combinations of these in geometric settings.
- Runge-Kutta methods and Lie group integration
- Exponentials and flow-maps, logarithms and backward error of numerical integrators
- Invariant connections, preLie and postLie algebras and their enveloping algebras
- Hopf algebras: Connes-Kreimer Hopf algebra, tensor algebra, symmetric algebra, MKW Hopf algebra and other related Hopf algebras.
This course will be of interest to both students in computational mathematics, pure mathematics and lector students.
The student should have insight in modern algebraic structures appearing in computational algorithms for integration of differential equations. The student should obtain both analytic, geometric and algebraic understanding of this process.