A Continuous-Stage Modified Leap-Frog Scheme for High-Dimensional Semi-Linear Hamiltonian Wave Equations
Year: 2020
Author: Xinyuan Wu, Bin Wang, Yonglei Fang, Xinyuan Wu, Yonglei Fang
Numerical Mathematics: Theory, Methods and Applications, Vol. 13 (2020), Iss. 3 : pp. 814–844
Abstract
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 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 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/nmtma.OA-2019-0115
Numerical Mathematics: Theory, Methods and Applications, Vol. 13 (2020), Iss. 3 : pp. 814–844
Published online: 2020-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 31
Keywords: Leap-frog scheme Hamiltonian wave equations modified Leap-frog scheme continuous-stage methods.