Volume 26, Issue 5
Long Time Error Analysis of Finite Difference Time Domain Methods for the Nonlinear Klein-Gordon Equation with Weak Nonlinearity

Weizhu Bao, Yue Feng & Wenfan Yi

Commun. Comput. Phys., 26 (2019), pp. 1307-1334.

Published online: 2019-08

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  • Abstract

We establish error bounds of the finite difference time domain (FDTD) methods for the long time dynamics of the nonlinear Klein-Gordon equation (NKGE) with a cubic nonlinearity, while the nonlinearity strength is characterized by ε2 with 0<ε≤1 a dimensionless parameter. When 0<ε≪1, it is in the weak nonlinearity regime and the problem is equivalent to the NKGE with small initial data, while the amplitude of the initial data (and the solution) is at O(ε). Four different FDTD methods are adapted to discretize the problem and rigorous error bounds of the FDTD methods are established for the long time dynamics, i.e. error bounds are valid up to the time at O(1/εβ) with 0 ≤ β ≤ 2, by using the energy method and the techniques of either the cut-off of the nonlinearity or the mathematical induction to bound the numerical approximate solutions. In the error bounds, we pay particular attention to how error bounds depend explicitly on the mesh size h and time step τ as well as the small parameter ε∈(0,1], especially in the weak nonlinearity regime when 0<ε≪1. Our error bounds indicate that, in order to get “correct” numerical solutions up to the time at O(1/εβ), the ε-scalability (or meshing strategy) of the FDTD methods should be taken as: h=O(εβ/2) and τ=O(εβ/2). As a by-product, our results can indicate error bounds and ε-scalability of the FDTD methods for the discretization of an oscillatory NKGE which is obtained from the case of weak nonlinearity by a rescaling in time, while its solution propagates waves with wavelength at O(1) in space and O(εβ) in time. Extensive numerical results are reported to confirm our error bounds and to demonstrate that they are sharp.

  • Keywords

Nonlinear Klein-Gordon equation, finite difference time domain methods, long time error analysis, weak nonlinearity, oscillatory nonlinear Klein-Gordon equation.

  • AMS Subject Headings

35L70, 65M06, 65M12, 65M15, 81-08

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

matbaowz@nus.edu.sg (Weizhu Bao)

fengyue@u.nus.edu (Yue Feng)

wfyi@hnu.edu.cn (Wenfan Yi)

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@Article{CiCP-26-1307, author = {Bao , Weizhu and Feng , Yue and Yi , Wenfan }, title = {Long Time Error Analysis of Finite Difference Time Domain Methods for the Nonlinear Klein-Gordon Equation with Weak Nonlinearity}, journal = {Communications in Computational Physics}, year = {2019}, volume = {26}, number = {5}, pages = {1307--1334}, abstract = {

We establish error bounds of the finite difference time domain (FDTD) methods for the long time dynamics of the nonlinear Klein-Gordon equation (NKGE) with a cubic nonlinearity, while the nonlinearity strength is characterized by ε2 with 0<ε≤1 a dimensionless parameter. When 0<ε≪1, it is in the weak nonlinearity regime and the problem is equivalent to the NKGE with small initial data, while the amplitude of the initial data (and the solution) is at O(ε). Four different FDTD methods are adapted to discretize the problem and rigorous error bounds of the FDTD methods are established for the long time dynamics, i.e. error bounds are valid up to the time at O(1/εβ) with 0 ≤ β ≤ 2, by using the energy method and the techniques of either the cut-off of the nonlinearity or the mathematical induction to bound the numerical approximate solutions. In the error bounds, we pay particular attention to how error bounds depend explicitly on the mesh size h and time step τ as well as the small parameter ε∈(0,1], especially in the weak nonlinearity regime when 0<ε≪1. Our error bounds indicate that, in order to get “correct” numerical solutions up to the time at O(1/εβ), the ε-scalability (or meshing strategy) of the FDTD methods should be taken as: h=O(εβ/2) and τ=O(εβ/2). As a by-product, our results can indicate error bounds and ε-scalability of the FDTD methods for the discretization of an oscillatory NKGE which is obtained from the case of weak nonlinearity by a rescaling in time, while its solution propagates waves with wavelength at O(1) in space and O(εβ) in time. Extensive numerical results are reported to confirm our error bounds and to demonstrate that they are sharp.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.2019.js60.03}, url = {http://global-sci.org/intro/article_detail/cicp/13266.html} }
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