标题： 数值模拟半直积测地线 显示英文标题

标题： Numerically modelling semidirect product geodesics

摘要： 本文数值研究了源于自半直积群结构的欧拉-庞加莱方程。从半直积作用中出现了一组非线性耦合系统，在这组系统中一个动态集合可以被认为是在另一个动态集合框架内。使用整体能量保持连续伽辽金有限元方法来研究与圆上微分同胚群自身半直积相关的测地线方程。理论预测的尖峰解作为一种涌现行为被观察到。此外，与半直积耦合相关的是复杂的非线性能量转移，在各种非线性相互作用中，我们观察到了耦合尖峰行为。使用保形（C网格）有限差分方法来研究体积保持微分同胚群自身的半直积的测地流，在涡度变量中观察到了类似的耦合行为。我们还研究了半直积群上测地线方程中的余伴随和李-泊松结构，在这种情况下，基础群首先通过中心扩张或半直积扩展。 显示英文摘要

摘要： 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.