Boundary element solution of the two-dimensional sine-Gordon equation using continuous linear elements. (English) Zbl 1166.65390
Summary: This article studies the boundary element solution of a two-dimensional sine-Gordon (SG) equation by using continuous linear element approximation. Non-linear and in-homogeneous terms are converted to the boundary by the dual reciprocity method and a predictor-corrector scheme is employed to eliminate the non-linearity. The procedure developed in this paper, is applied to various problems involving line and ring solitons considered references [J. Argyris, M. Haase and J. C. Heinrich, Comput. Methods Appl. Mech. Eng. 86, No. 1, 1–26 (1991; Zbl 0762.65073); A. G. Bratsos, ANACM, Appl. Numer. Anal. Comput. Math. 2, No. 2, 189–211 (2005; Zbl 1075.65111); erratum ibid. 2, No. 3, 365 (2005); Numer. Algorithms 43, No. 4, 295–308 (2006; Zbl 1112.65077); J. Comput. Appl. Math. 206, No. 1, 251–277 (2007; Zbl 1117.65126); Math. Comput. Simul. 76, No. 4, 271–282 (2007; Zbl 1135.65358); P. L. Christiansen and P. S. Lomdahl, Physica D: Nonlinear Phenom 2; No. 3, 482–494 (1981); K. Djidjeli, W. G. Price and E. H. Twizell, J. Eng. Math. 29, No. 4, 347–369 (1995; Zbl 0841.65083); M. Dehghan and D. Mirzaei, Comput. Methods Appl. Mech. Eng. 197, No. 6–8, 476–486 (2008)]. Using continuous linear elements approximation produces more accurate results than constant ones. By using this approach all cases associated to SG equation, which exist in literature, are investigated.
MSC:
| 65M99 | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
Keywords:
sine-Gordon equation; soliton; boundary element method; dual reciprocity method; continuous linear elementsCitations:
Zbl 0762.65073; Zbl 1075.65111; Zbl 1112.65077; Zbl 1117.65126; Zbl 1135.65358; Zbl 0841.65083Software:
SERBAReferences:
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