@Article{NMTMA-13-3,
author = {Xinyuan, Wu and Wang, Bin and Yonglei, Fang and Xinyuan, Wu and Yonglei, Fang},
title = {A Continuous-Stage Modified Leap-Frog Scheme for High-Dimensional Semi-Linear Hamiltonian Wave Equations},
journal = {Numerical Mathematics: Theory, Methods and Applications},
year = {2020},
volume = {13},
number = {3},
pages = {814--844},
abstract = {<p style="text-align: justify;">Among the typical time integrations for PDEs, Leap-frog scheme is the well-known method which can easily be used. A most welcome feature of the Leap-frog scheme is that it has very simple scheme and is easy to be implemented. The main purpose of this paper is to propose and analyze an improved Leap-frog scheme, the so-called continuous-stage modified&nbsp; Leap-frog scheme for high-dimensional semi-linear Hamiltonian wave equations. To this end, under the assumption of periodic boundary conditions, we begin with the formulation of the nonlinear Hamiltonian equation as an abstract second-order ordinary differential equation (ODE) and its operator-variation-of-constants formula (the Duhamel Principle). Then the continuous-stage modified Leap-frog scheme is formulated. Accordingly, the convergence, energy preservation, symplecticity conservation and&nbsp; long-time behaviour of explicit schemes are rigorously analysed. Numerical results demonstrate the remarkable advantage and efficiency of the improved Leap-frog scheme compared with the existing mostly used numerical schemes in the literature.</p>},
issn = {2079-7338},
doi = {https://doi.org/10.4208/nmtma.OA-2019-0115},
url = {https://global-sci.com/article/90369/a-continuous-stage-modified-leap-frog-scheme-for-high-dimensional-semi-linear-hamiltonian-wave-equations}
}