Trees, Algebras, and Differential Equations: Extending the B-series.jl package for numerical analysis of initial value problems

The bseries.jl software package (https://ranocha.de/BSeries.jl/stable/) is designed to facilitate analysis of numerical methods for initial value problems, by utilizing the relationship between Taylor series, rooted trees, and Hopf algebras. It implements a range of graph-based algorithms that enable the study of errors in numerical methods, for instance revealing how the energy of the approximated system will evolve. It also allows for the design of novel methods. Its current functionality is primarily focused on Runge-Kutta methods.In this project we seek to extend the capabilities of bseries.jl to new classes of methods and/or new kinds of analysis. There are a number of possible directions and the specific one chosen will depend on the interests and knowledge of the student. Possibilities include extensions to:- Multi-derivative methods - Partitioned methods (e.g. for Hamiltonian systems) - General linear (multistep, multistage) methods - Exponential methods - Alternative bases for order conditions - Application of simplifying assumptions in method design - Generalized additive Runge-Kutta methods - Characterization of energy-preserving B-series - Extensions of B-series, such as aromatic B-series, exotic B-series, and S-series.Additional topics and references for some of these topics can be found at https://github.com/ranocha/BSeries.jl/issues/8.

To be discussed in interview with applicant