Volume 13, Issue 3
A Continuous-Stage Modified Leap-Frog Scheme for High-Dimensional Semi-Linear Hamiltonian Wave Equations

Bin Wang, Xinyuan Wu & Yonglei Fang

Numer. Math. Theor. Meth. Appl., 13 (2020), pp. 814-844.

Published online: 2020-03

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  • 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.

  • Keywords

Leap-frog scheme, Hamiltonian wave equations, modified Leap-frog scheme, continuous-stage methods.

  • AMS Subject Headings

65M12, 65M20, 65M70, 65P10

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

wangbinmaths@xjtu.edu.cn (Bin Wang)

xywu@nju.edu.cn (Xinyuan Wu)

ylfangmath@163.com (Yonglei Fang)

  • BibTex
  • RIS
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@Article{NMTMA-13-814, author = {Wang , Bin and Wu , Xinyuan and Fang , Yonglei}, 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 = {

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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2019-0115}, url = {http://global-sci.org/intro/article_detail/nmtma/15786.html} }
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 -

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.

Bin Wang, Xinyuan Wu & Yonglei Fang. (2020). A Continuous-Stage Modified Leap-Frog Scheme for High-Dimensional Semi-Linear Hamiltonian Wave Equations. Numerical Mathematics: Theory, Methods and Applications. 13 (3). 814-844. doi:10.4208/nmtma.OA-2019-0115
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