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Split Local Artificial Boundary Conditions for the Two-Dimensional Sine-Gordon Equation on R2

Published online by Cambridge University Press:  20 August 2015

Houde Han  and
Zhiwen Zhang
Houde Han*
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
Zhiwen Zhang*
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
*
Corresponding author.Email:hhan@math.tsinghua.edu.cn
Email:zhangzhiwen02@mails.tsinghua.edu.cn
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Abstract

In this paper the numerical solution of the two-dimensional sine-Gordon equation is studied. Split local artificial boundary conditions are obtained by the operator splitting method. Then the original problem is reduced to an initial boundary value problem on a bounded computational domain, which can be solved by the finite difference method. Several numerical examples are provided to demonstrate the effectiveness and accuracy of the proposed method, and some interesting propagation and collision behaviors of the solitary wave solutions are observed.

Keywords

52B1065D1868U0568U07Sine-Gordon equationoperator splitting methodartificial boundary conditionsoli-tonunbounded domain
Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 5 , November 2011 , pp. 1161 - 1183
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Josephson, J. D., Supercurrents through barriers, Adv. Phys., 14 (1965), 419–451.Google Scholar
[2]Christiansen, P. L. and Lomdahl, P. S., Numerical solution of 2+1 dimensional sine-Gordon solitons, Phys. D, 2 (1981), 482–494.Google Scholar
[3]Bogolyubski, I. L. and Makhankov, V. G., Lifetime of pulsating solitons in certain classical models, JETP Lett., 24(1) (1976), 12–14.Google Scholar
[4]Han, H., The Artifial Boundary Method-Numerical Solutions of Partial Differential Equations on Unbounded Domains, Frontier and Prospects of Comtemporary Applied Mathematics, 2005, Higher Education Press, World Scientific.Google Scholar
[5]Guo, B.-Y., Pascual, P. J., Rodriguez, M. J. and Vazquez, L., Numerical solution of the sine-Gordon equation, Appl. Math. Comput., 18 (1986), 1–14.Google Scholar
[6]Xin, J. X., Modeling light bullets with the two-dimensional sine-Gordon equation, Phys. D, 135 (2000), 345–368.Google Scholar
[7]Christiansen, P. L. and Lomdahl, P. S., Numerical solutions of 2+1 dimensional sine-Gordon solitons, Phys. D, 2 (1981), 482–494.CrossRefGoogle Scholar
[8]Argyris, J., Haase, M. and Heinrich, J. C., Finite element approximation to two-dimensional sine-Gordon solitons, Comput. Methods Appl. Mech. Eng., 86 (1991), 1–26.Google Scholar
[9]Sheng, Q., Khaliq, A. Q. M. and Voss, D. A., Numerical simulation of two-dimensional sine-Gordon solitons via a split cosine scheme, Math. Comput. Sim., 68 (2005), 355–373.Google Scholar
[10]Djidjeli, K., Price, W. G. and Twizell, E. H., Numerical solutions of a damped sine-Gordon equation in two space variables, J. Eng. Math., 29 (1995), 347–369.Google Scholar
[11]Bratsos, A.G., An explicit numerical scheme for the sine-Gordon equation in 2+1 dimensions, Appl. Numer. Anal. Comput. Math., 2(2) (2005), 189–211.CrossRefGoogle Scholar
[12]Bratsos, A. G., The solution of the two-dimensional sine-Gordon equation using the method of lines, J. Comput. Appl. Math., 206 (2007), 251–277.Google Scholar
[13]Dehghan, M. and Shokri, A., A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions, Math. Comput. Sim., 79 (2008), 700–715.Google Scholar
[14]Mirzaei, D. and Dehghan, M., Boundary element solution of the two-dimensional sine-Gordonequation using continuous linear elements, Eng. Anal. Bound. Elem., 33 (2009), 12–24.Google Scholar
[15]Engquist, B. and Majda, A., Absorbing boundary conditions for the numerical simulation of waves, Math. Comput., 31 (1986), 629–651.Google Scholar
[16]Givoli, D., High-order local non-reflecting boundary conditions: a review, Wave Motion, 39 (2004), 319–326.Google Scholar
[17]Hagstrom, T., Radiation boundary conditions for the numerical simulation of waves, Acta. Number., 8 (1999), 47–106.Google Scholar
[18]Grote, M. J. and Keller, J. B., Exact non-reflecting boundary conditions for the time dependent wave equation, SIAM J. Appl. Math., 55(2) (1995), 280–297.Google Scholar
[19]Han, H. and Zheng, C., Exact non-reflecting boundary conditions for an acoustic problem in three dimensions, J. Comput. Math., 21(1) (2003), 15–24.Google Scholar
[20]Han, H., Wu, X. and Xu, Z., Artificial boundary method for Burger’s equation using nonlinear boundary conditions, J. Comput. Math., 24 (2006), 295–304.Google Scholar
[21]Xu, Z., Han, H. and Wu, X., Numerical method for the determimistic Kardar-Parisi-Zhang equation in unbounded domians, Commun. Comput. Phys., 1 (2006), 479–493.Google Scholar
[22]Han, H. and Zhang, Z. W., Split local absorbing conditions for one-dimensional nonlinear Klein-Gordon equation on unbounded domain, J. Comput. Phys., 227 (2008), 8992–9004.CrossRefGoogle Scholar
[23]Zheng, C., Exact nonreflection boundary conditions for one-dimensional cubic nonlinear Schrädinger equations, J. Comput. Phys., 215 (2006), 552–565.Google Scholar
[24]Zheng, C., Numerical solution to the sine-Gordon equation defined on the whole real axis, SIAM J. Sci. Comput., 29(6) (2007), 2494–2506.Google Scholar
[25]Engquist, B. and Majda, A., Radiation boundary conditions for acoustic and elastic wave calculations, Commun. Pure Appl. Math., 32 (1979), 314–358.Google Scholar
[26]Higdon, R. L., Absorbing boundary conditions for difference approximations to the multidimensional wave equation, Math. Comput., 47 (1986), 437–459.Google Scholar
[27]Higdon, R. L., Numerical absorbing boundary conditions for the wave equation, Math. Comput., 49 (1987), 65–90.Google Scholar
[28]Higdon, R. L., Radiation boundary conditions for dispersive waves, SIAM J. Numer. Anal., 31 (1994), 64–100.Google Scholar
[29]Strang, G., On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5 (1968), 506–517.Google Scholar
[30]Yu, S., Zhao, S. and Wei, G. W., Local spectral time splitting method for first- and second-order partial differential equations, J. Comput. Phys., 206 (2005), 727–780.Google Scholar