An efficient finite difference scheme for the 2D sine-Gordon equation

Volume 10, Issue 6, pp 2998--3012 http://dx.doi.org/10.22436/jnsa.010.06.14

Publication Date: June 25, 2017 Submission Date: January 12, 2017

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 2339 Downloads
• 3248 Views

Authors

Xiaorong Kang - School of Science, Southwest University of Science and Technology, Mianyang, Sichuan 621010, China. Wenqiang Feng - Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996, USA. Kelong Cheng - School of Science, Southwest University of Science and Technology, Mianyang, Sichuan 621010, China. Chunxiang Guo - School of Business, Sichuan University, Chengdu, Sichuan 610064, China.

Abstract

We present an efficient second-order finite difference scheme for solving the 2D sine-Gordon equation, which can inherit the discrete energy conservation for the undamped model theoretically. Due to the semi-implicit treatment for the nonlinear term, it leads to a sequence of nonlinear coupled equations. We use a linear iteration algorithm, which can solve them efficiently, and the contraction mapping property is also proven. Based on truncation errors of the numerical scheme, the convergence analysis in the discrete \(l^2\)-norm is investigated in detail. Moreover, we carry out various numerical simulations, such as verifications of the second order accuracy, tests of energy conservation and circular ring solitons, to demonstrate the efficiency and the robustness of the proposed scheme.

Share and Cite

Keywords

• invex set
• conservative
• difference scheme
• linear iteration
• convergence.

MSC

•  65M06
•  65M12

References

• [1] J. Argyris, M. Haase, J. C. Heinrich, Finite element approximation to two-dimensional sine-Gordon solitons, Comput. Methods Appl. Mech. Engrg. , 86 (1991), 1–26.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] Z. Asgari, S. M. Hosseini, Numerical solution of two-dimensional sine-Gordon and MBE models using Fourier spectral and high order explicit time stepping methods, Comput. Phys. Commun., 184 (2013), 565–572.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] A. G. Bratsos, A modified predictor-corrector scheme for the two-dimensional sine-Gordon equation, Numer. Algorithms, 43 (2006), 295–308.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] A. G. Bratsos, The solution of the two-dimensional sine-Gordon equation using the method of lines, J. Comput. Appl. Math., 206 (2007), 251–277.
  • View Article
  • MathSciNet
  • Google Scholar


• [5] W. Chen, W. Feng, C. Wang, S. Wise, A second order energy stable scheme for the Cahn-Hilliard-Hele-Shaw equations, ArXiv, 2016 (2016), 34 pages.
  • Google Scholar


• [6] K.-L. Cheng, W.-Q. Feng, S. Gottlieb, C. Wang, A Fourier pseudospectral method for the ”good” Boussinesq equation with second-order temporal accuracy, Numer. Methods Partial Differential Equations, 31 (2015), 202–224.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] R. J. Cheng, K. M. Liew, Analyzing two-dimensional sine-Gordon equation with the mesh-free reproducing kernel particle Ritz method, Comput. Methods Appl. Mech. Engrg., 245/246 (2012), 132–143.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [8] K.-L. Cheng, C. Wang, S. M. Wise, X.-Y. Yue, A second-order, weakly energy-stable pseudo-spectral scheme for the Cahn- Hilliard equation and its solution by the homogeneous linear iteration method, J. Sci. Comput., 69 (2016), 1083–1114.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] M.-R. Cui, High order compact alternating direction implicit method for the generalized sine-Gordon equation, J. Comput. Appl. Math., 235 (2010), 837–849.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] M. Dehghan, A. Ghesmati, Numerical simulation of two-dimensional sine-Gordon solitons via a local weak meshless technique based on the radial point interpolation method (RPIM), Comput. Phys. Comm., 181 (2010), 772–786.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] M. Dehghan, D. Mirzaei, The dual reciprocity boundary element method (DRBEM) for two-dimensional sine-Gordon equation, Comput. Methods Appl. Mech. Engrg., 197 (2008), 476–486.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] M. Dehghan, A. Shokri, A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions, Math. Comput. Simulation, 79 (2008), 700–715.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [13] J.-S. Hu, K.-L. Zheng, M.-B. Zheng, Numerical simulation and convergence analysis of a high-order conservative difference scheme for SRLW equation, Appl. Math. Model., 38 (2014), 5573–5581.
  • View Article
  • MathSciNet
  • Google Scholar


• [14] R. Jiwari, S. Pandit, R. C. Mittal, Numerical simulation of two-dimensional sine-Gordon solitons by differential quadrature method, Comput. Phys. Commun., 183 (2012), 600–616.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] S. Li, L. Vu-Quoc, Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation, SIAM J. Numer. Anal., 32 (1995), 1839–1875.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] J. K. Perring, T. H. R. Skyrme, A model unified field equation, Nuclear Phys., 31 (1962), 550–555.
  • MathSciNet


• [17] W. Strauss, L. Vazquez, Numerical solution of a nonlinear Klein-Gordon equation, J. Comput. Phys., 28 (1978), 271–278.
  • View Article
  • Google Scholar


• [18] L.-D.Wang,W.-B. Chen, C.Wang, An energy-conserving second order numerical scheme for nonlinear hyperbolic equation with an exponential nonlinear term, J. Comput. Appl. Math., 280 (2015), 347–366.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [19] C. Wang, S. M. Wise, An energy stable and convergent finite-difference scheme for the modified phase field crystal equation, SIAM J. Numer. Anal., 49 (2011), 945–969.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] G. B. Whitham, Linear and nonlinear waves, Reprint of the 1974 original, Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York (1999)
  • Google Scholar


• [21] J. X. Xin, Modeling light bullets with the two-dimensional sine-Gordon equation, Phys. D, 135 (2000), 345–368.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] X.-J. Yang, A new integral transform method for solving steady heat-transfer problem, Therm. Sci., 20 (2016), S639–S642.
  • Google Scholar


• [23] X.-J. Yang, A new integral transform with an application in heat-transfer problem, Therm. Sci., 20 (2016), S677–S681.
  • View Article
  • Google Scholar


• [24] X.-J. Yang, A new integral transform operator for solving the heat-diffusion problem, Appl. Math. Lett., 64 (2017), 193– 197.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [25] X.-J. Yang, F. Gao, H. M. Srivastava, Exact travelling wave solutions for the local fractional two-dimensional Burgers-type equations, Comput. Math. Appl., 73 (2017), 203–210.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [26] X.-J. Yang, J. A. Tenreiro Machado, D. Baleanu, C. Cattani, On exact traveling-wave solutions for local fractional Korteweg-de Vries equation, Chaos, 26 (2016), 5 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [27] J. Zagrodziński, Particular solutions of the sine-Gordon equation in 2+1 dimensions, Phys. Lett. A, 72 (1979), 284–286.
  • View Article
  • MathSciNet
  • Google Scholar


• [28] K.-L. Zheng, J.-S. Hu, High-order conservative Crank-Nicolson scheme for regularized long wave equation, Adv. Difference Equ., 2013 (2013), 12 pages.
  • View Article
  • MathSciNet
  • Google Scholar