Numerically modelling semidirect product geodesics

TL;DR

This paper numerically explores geodesic equations on semidirect product groups, revealing emergent peakon solutions and complex energy transfer behaviors using finite element and finite difference methods.

Contribution

It introduces a numerical framework for studying semidirect product geodesics, including energy-preserving and mimetic discretizations, and investigates their coadjoint and Lie-Poisson structures.

Findings

Emergence of peakon solutions in semidirect product geodesics

Observation of nonlinear energy transfer and coupling behaviors

Validation of numerical methods for complex group structures

Abstract

This paper numerically investigates Euler-Poincar\'e equations arising from a self-semidirect product group structure. Nonlinearly coupled systems of equations emerge from the semidirect product action where one set of dynamics can be considered in the frame of another. A monolithic energy-preserving continuous Galerkin finite element method is used to study geodesic equations associated with the semidirect product of the diffeomorphism group on a circle with itself. Theoretically predicted peakon solutions are observed as an emergent behaviour. In addition, complicated nonlinear transfers of energy are associated with the semidirect product coupling, where amongst various nonlinear interactions, we observe coupled peakon behaviour. A mimetic (C-grid) finite difference method is used to study the geodesic flow of the semidirect product of the volume preserving diffeomorphism group with itself, where similar coupling behaviour is observed in the vorticity variables. We also investigate coadjoint and Lie-Poisson structures in the context of geodesic equations on semidirect product groups, where the underlying group is first extended by central extension or semidirect product.