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.
https://eta.impa.br/dl/PL006.pdf
https://arxiv.org/abs/1710.03946
本文首先介绍常微分方程Hamilton系统的近似求解的数值方法，引入流的辛性和辛积分子（integrators），然后讨论具有多重时间尺度的有限维Hamilton系统的数值方法，引入调制Fourier展开。作者研究了Hamilton偏微分方程组（例如非线性波动方程和非线性Schrödinger方程）的适当数值离散化的长时间结果。最后，作者讨论了依赖于时间的大型矩阵和张量的动态低秩近似，及其在Hamiltonian张量网络近似中的应用。
相关附件
18-PL006 Lubich
