TY - JOUR
T1 - A Continuous-Stage Modified Leap-Frog Scheme for High-Dimensional Semi-Linear Hamiltonian Wave Equations
AU - Wang , Bin
AU - Wu , Xinyuan
AU - Fang , Yonglei
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 3
SP - 814
EP - 844
PY - 2020
DA - 2020/03
SN - 13
DO - http://doi.org/10.4208/nmtma.OA-2019-0115
UR - https://global-sci.org/intro/article_detail/nmtma/15786.html
KW - Leap-frog scheme, Hamiltonian wave equations, modified Leap-frog scheme, continuous-stage methods.
AB - <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>