Geometric Numerical Methods with Lie Groups

Abstract

Due to the increasing demands for modeling large-scale and complex systems, designing optimal controls, and conducting optimization tasks, many real-world applications require sophisticated models. Geometric methods are designed to capture the underlying structure of the system at hand and to preserve the global qualitative or geometric properties of the flow, such as symplecticity, volume preservation and symmetry. A survey on three of such structure preserving numerical methods is proposed in the present article. Testing the validity of such simulations is achieved by exhibiting analytically solvable models and comparing the result of simulations with their exact behavior.