Dynamics, numerical analysis, and some geometry
For researchers in numerical analysis and computational physics, this provides a unified perspective on geometric integration methods, though it is a review rather than a novel contribution.
This review covers structure-preserving numerical methods for differential equations, focusing on symplectic integrators for Hamiltonian systems and dynamical low-rank approximation for quantum dynamics. It synthesizes theoretical developments across these areas.
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.