American Institute of Mathematical Sciences

Advanced
2005, 2005(Special): 792-797. doi: 10.3934/proc.2005.2005.792

An adaptive splitting algorithm for the sine-Gordon equation

Qin Sheng 1, , David A. Voss 2, and  Q. M. Khaliq 3,

1. 

Department of Mathematics, Baylor University, Waco, TX76798-7328, United States
2. 

Department of Mathematics, Western Illinois University, Macomb, IL 61455, United States
3. 

Department of Mathematical Sciences, Middle Tennessee State University, Murfreesboro, Tennessee 37132, United States

Received  September 2004 Revised  March 2005 Published  September 2005

This preliminary investigation concerns an adaptive splitting scheme for the numerical solution of two dimensional sine-Gordon equation. The dispersive wave equation allows for soliton-alike solutions, an ubiquitous phenomenon in a large variety of physical problems. The system of nonlinear differential equations obtained via the method of lines is then attached by a recurrence procedure whose solution yields second order accuracy. The numerical solution of the system is designed using the Peaceman-Rachford splitting to avoid solving a nonlinear system of equations at each step and allows more efficient implementations of the difference scheme.
Keywords: numerical stability., sequential splitting, finite difference method, Solitary waves, Adaptation.
Mathematics Subject Classification: Primary: 65M06, 65M20; Secondary: 35Q5.
Citation: Qin Sheng, David A. Voss, Q. M. Khaliq. An adaptive splitting algorithm for the sine-Gordon equation. Conference Publications, 2005, 2005 (Special) : 792-797. doi: 10.3934/proc.2005.2005.792
[1]

Yones Esmaeelzade Aghdam, Hamid Safdari, Yaqub Azari, Hossein Jafari, Dumitru Baleanu. Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference scheme. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020402
[2]

H. Kalisch. Stability of solitary waves for a nonlinearly dispersive equation. Discrete & Continuous Dynamical Systems, 2004, 10 (3) : 709-717. doi: 10.3934/dcds.2004.10.709
[3]

Pavlos Xanthopoulos, Georgios E. Zouraris. A linearly implicit finite difference method for a Klein-Gordon-Schrödinger system modeling electron-ion plasma waves. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 239-263. doi: 10.3934/dcdsb.2008.10.239
[4]

Santosh Bhattarai. Stability of normalized solitary waves for three coupled nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 1789-1811. doi: 10.3934/dcds.2016.36.1789
[5]

Steve Levandosky, Yue Liu. Stability and weak rotation limit of solitary waves of the Ostrovsky equation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 793-806. doi: 10.3934/dcdsb.2007.7.793
[6]

Amjad Khan, Dmitry E. Pelinovsky. Long-time stability of small FPU solitary waves. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 2065-2075. doi: 10.3934/dcds.2017088
[7]

Khaled El Dika. Asymptotic stability of solitary waves for the Benjamin-Bona-Mahony equation. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 583-622. doi: 10.3934/dcds.2005.13.583
[8]

Sevdzhan Hakkaev. Orbital stability of solitary waves of the Schrödinger-Boussinesq equation. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1043-1050. doi: 10.3934/cpaa.2007.6.1043
[9]

Berat Karaagac. Numerical treatment of Gray-Scott model with operator splitting method. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020143
[10]

Hongsong Feng, Shan Zhao. A multigrid based finite difference method for solving parabolic interface problem. Electronic Research Archive, , () : -. doi: 10.3934/era.2021031
[11]

Junxiang Li, Yan Gao, Tao Dai, Chunming Ye, Qiang Su, Jiazhen Huo. Substitution secant/finite difference method to large sparse minimax problems. Journal of Industrial & Management Optimization, 2014, 10 (2) : 637-663. doi: 10.3934/jimo.2014.10.637
[12]

Gonzalo Galiano, Julián Velasco. Finite element approximation of a population spatial adaptation model. Mathematical Biosciences & Engineering, 2013, 10 (3) : 637-647. doi: 10.3934/mbe.2013.10.637
[13]

Nghiem V. Nguyen, Zhi-Qiang Wang. Existence and stability of a two-parameter family of solitary waves for a 2-coupled nonlinear Schrödinger system. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 1005-1021. doi: 10.3934/dcds.2016.36.1005
[14]

Nabile Boussïd, Andrew Comech. Spectral stability of bi-frequency solitary waves in Soler and Dirac-Klein-Gordon models. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1331-1347. doi: 10.3934/cpaa.2018065
[15]

José R. Quintero. Nonlinear stability of solitary waves for a 2-d Benney--Luke equation. Discrete & Continuous Dynamical Systems, 2005, 13 (1) : 203-218. doi: 10.3934/dcds.2005.13.203
[16]

Andrew Comech, Elena Kopylova. Orbital stability and spectral properties of solitary waves of Klein–Gordon equation with concentrated nonlinearity. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021063
[17]

Guoliang Ju, Can Chen, Rongliang Chen, Jingzhi Li, Kaitai Li, Shaohui Zhang. Numerical simulation for 3D flow in flow channel of aeroengine turbine fan based on dimension splitting method. Electronic Research Archive, 2020, 28 (2) : 837-851. doi: 10.3934/era.2020043
[18]

Xiaozhong Yang, Xinlong Liu. Numerical analysis of two new finite difference methods for time-fractional telegraph equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3921-3942. doi: 10.3934/dcdsb.2020269
[19]

Z. Jackiewicz, B. Zubik-Kowal, B. Basse. Finite-difference and pseudo-spectral methods for the numerical simulations of in vitro human tumor cell population kinetics. Mathematical Biosciences & Engineering, 2009, 6 (3) : 561-572. doi: 10.3934/mbe.2009.6.561
[20]

José Raúl Quintero, Juan Carlos Muñoz Grajales. Solitary waves for an internal wave model. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5721-5741. doi: 10.3934/dcds.2016051

 Impact Factor: 

Tools

Metrics

  • PDF downloads (37)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences