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High Order Energy-Preserving Method of the “Good” Boussinesq Equation

Part of: Numerical approximation and computational geometry

Published online by Cambridge University Press:  15 February 2016

Chaolong Jiang ,
Jianqiang Sun ,
Xunfeng He  and
Lanlan Zhou
Chaolong Jiang
Affiliation:
Department of Mathematics, College of Information Science and Technology, Hainan University, Haikou, 570228, China
Jianqiang Sun*
Affiliation:
Department of Mathematics, College of Information Science and Technology, Hainan University, Haikou, 570228, China
Xunfeng He
Affiliation:
Department of Mathematics, College of Information Science and Technology, Hainan University, Haikou, 570228, China
Lanlan Zhou
Affiliation:
Department of Mathematics, College of Information Science and Technology, Hainan University, Haikou, 570228, China
*
*Corresponding author. Email addresses: [email protected] (C.-L. Jiang), [email protected] (J.-Q. Sun), [email protected] (X.-F. He), [email protected] (L.-L. Zhou)
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Abstract

The fourth order average vector field (AVF) method is applied to solve the “Good” Boussinesq equation. The semi-discrete system of the “good” Boussinesq equation obtained by the pseudo-spectral method in spatial variable, which is a classical finite dimensional Hamiltonian system, is discretizated by the fourth order average vector field method. Thus, a new high order energy conservation scheme of the “good” Boussinesq equation is obtained. Numerical experiments confirm that the new high order scheme can preserve the discrete energy of the “good” Boussinesq equation exactly and simulate evolution of different solitary waves well.

Keywords

“Good” Boussinesq equationaverage vector field methodsolitary wavesconservation law

MSC classification

Secondary: 65D17: Computer aided design (modeling of curves and surfaces)
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 9 , Issue 1 , February 2016 , pp. 111 - 122
Copyright
Copyright © Global-Science Press 2016 

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References

[1]Ablowitz, M.J and Segur, H., Solitons and the Inverse Scattering Transform, SIAM Studies in Applied Mathematics, Philadelphia, USA, 1981.CrossRefGoogle Scholar
[2]Aydin, A., and Karasözen, B., Symplectic and multisymplectic Lobatto methods for the “good” Boussinesq equation, J. Math. Phys., Vol. 49 (2008), pp. 083509.CrossRefGoogle Scholar
[3]Bridges, T.J., Reich, S., Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity, Phys. Lett. A., Vol. 284 (2001), pp. 184–193.CrossRefGoogle Scholar
[4]Chen, J.B., Multisymplectic geometry, local conservation laws and Fourier pseudo-spectral discretization for the “good” Boussinesq equation, Appl. Math. Comput., Vol. 161 (2005), pp. 55–67.Google Scholar
[5]Celledoni, E., Grimm, V., McLachlan, R.I., et al., Preserving energy resp. dissipation in numerical PDEs using the “average vector field” method, J. Comput. Phys, Vol. 231 (2012), pp. 6770–6789.CrossRefGoogle Scholar
[6]Frutos, J.D., Ortega, T. and Sanz-Serna, J.M., Pseudo-spectral method for the “Good” Boussinesq equation, Math. Comp, Vol. 57 (1991), pp. 109–122.Google Scholar
[7]Huang, L.Y., Zeng, W.P. and Qin, M.Z., A new muliti-symplectic scheme for nonlinear “good” Boussinesq equation., J. Comput. Math, Vol. 21 (2003), pp. 703–714.Google Scholar
[8]Hu, W.P., and Deng, Z.C., Multi-symplectic method for generalized Boussinesq equation., Appl. Math. Mech.-Engl. ed., Vol. 29 (2008), pp. 927–932.Google Scholar
[9]Hairer, E., Lubich, C. and Wanner, G., Geometric Numerical Integration: Structure-Preserving Algorithm for Ordinary Differential Equatuins, Springer, Berlin, 2nd ed., 2006.Google Scholar
[10]Ismial, M.S. and Farida, M., A fourth order finite difference method for the “good” Boussinesq equation., Abstract and Analysis, Article ID 323260, 2014.CrossRefGoogle Scholar
[11]Kang, F., Qin, M.Z., Symplectic Geometric Algorithms for Hamiltonian Systems, Springer and Zhejiang Science and Technology Publishing House, Heidelberg and Hangzhou, 2010.Google Scholar
[12]Manoranjan, V.S., Mitchell, A.R. and Morris, J.Ll., Numerical solutions of the “Good” Boussinesq equation., SIAM J. Sci. Comput., Vol. 5 (1984), pp. 946–957.CrossRefGoogle Scholar
[13]Manoranjan, V.S., Ortega, T. and Sanz-Serna, J.M, Soliton and antisoliton interaction in the “Good” Boussinesq equation, J. Math. Phys., Vol. 29 (1988), pp. 1964–1968.CrossRefGoogle Scholar
[14]McLachlan, R.I., Quispel, G.R.W. and Robidoux, N., Geometric integration using discrete gradients., Phil. Trans. R. Soc. A., Vol. 357 (1999), pp. 1021–1045.CrossRefGoogle Scholar
[15]Ortega, T. and Sanz-Serna, J.M., Nonlinear stability and convergence of finite-difference methods for the “good” Boussinesq equation., Numer. Math., Vol. 58 (1990), pp. 215–229.CrossRefGoogle Scholar
[16]Qin, M.Z., Wang, Y.S., Structure-Preserving Algorithms for Partial Differential Equation, Zhejiang Science and Technology Publishing House, Hangzhou, (in Chinese), 2012.Google Scholar
[17]Quispel, G.R., McLaren, G.R.W. and McLaren, D.I., A new class of energy-preserving numerical integration methods., J. Phys. A: Math. Theor., Vol. 41 (2008), pp. 045206.CrossRefGoogle Scholar
[18]Sanz-Serna, J.M., Symplectic Runge-Kutta and related methods: Recent result, Phys. D., Vol. 60 (1992), pp. 293–302.CrossRefGoogle Scholar
[19]Whitham, G.B., Linear and Nonlinear Waves,Wiley-Interscience, New York, NY, USA, 1974.Google Scholar
[20]Zoheiry, H.E-, Numerical investigation for the solitary waves interaction of the “good” Boussinesq equation., Appl. Numer. Math, Vol. 45 (2003), pp. 161–173.CrossRefGoogle Scholar
[21]Zeng, W.P., The multi-symplectic algorithm for “Good” Boussinesq equation., Appl. Math. Mech.-Engl.Ed., Vol. 23 (2002), pp. 835–841.Google Scholar