Dynamics, numerical analysis, and some geometry

TL;DR

This paper reviews the development of geometric numerical methods, focusing on symplectic integrators, low-rank approximations, and their applications in Hamiltonian systems and quantum dynamics.

Contribution

It provides a comprehensive overview of geometric aspects in structure-preserving numerical methods for differential equations, highlighting recent advances in symplectic integrators and low-rank tensor techniques.

Findings

Symplectic integrators effectively preserve Hamiltonian structures.

Low-rank approximations improve computational efficiency for large systems.

Applications in quantum dynamics demonstrate the practical relevance of these methods.

Abstract

Geometric aspects play an important role in the construction and analysis of structure-preserving numerical methods for a wide variety of ordinary and partial differential equations. Here we review the development and theory of symplectic integrators for Hamiltonian ordinary and partial differential equations, of dynamical low-rank approximation of time-dependent large matrices and tensors, and its use in numerical integrators for Hamiltonian tensor network approximations in quantum dynamics.