Numerical Analysis and Simulation of Solutions to a Class of Boussinesq Systems With Source Terms

HE Yubo; DONG Xiaoliang; LIN Xiaoyan

doi:10.21656/1000-0887.380126

Volume 39 Issue 8

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HE Yubo, DONG Xiaoliang, LIN Xiaoyan. Numerical Analysis and Simulation of Solutions to a Class of Boussinesq Systems With Source Terms[J]. Applied Mathematics and Mechanics, 2018, 39(8): 961-978. doi: 10.21656/1000-0887.380126

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Numerical Analysis and Simulation of Solutions to a Class of Boussinesq Systems With Source Terms

doi: 10.21656/1000-0887.380126

¹School of Mathematics and Computation Science, Huaihua University, Huaihua, Hunan 418008, P.R.China;²School of Mathematics and Information Science, North Minzu University, Yinchuan 750021, P.R.China

Funds: The National Natural Science Foundation of China(11471137;11501232;11601012)

Received Date: 2017-06-08
Rev Recd Date: 2017-11-07
Publish Date: 2018-08-15

Abstract

Abstract

For a class of initial value problems of the high-order Boussinesq systems with source terms, a D1Q5 nonstandard lattice Boltzmann model (LBM) with correction functions and source terms was proposed. Different local equilibrium distribution functions and correction functions were selected, and the nonlinear wave equation was recovered by means of the Chapman-Enskog multi-scale analysis and the Taylor expansion technique. Some initial boundary value problems of the Boussinesq systems with analytical solutions were simulated to verify the effectiveness of the LBM. The results show that the numerical solutions agree well with the analytical solutions and the norm errors obtained with the LBM are smaller than those with the modified finite difference method (MFDM). Furthermore, some problems without analytical solutions were numerically studied with the present method and the MFDM. The comparison shows that the numerical solutions from the LBM are in good agreement with those from the MFDM, which validates the effectiveness and stability of the proposed model.
- lattice Boltzmann model,
- Boussinesq equation,
- Chapman-Enskog expansion,
- modified finite difference method

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References(27)

References

[1]	熊志强. Boussinesq方程模型的数值造波方法研究[D]. 硕士学位论文. 天津: 天津大学, 2007.(XIONG Zhiqiang. Numerical wave generation in Boussinesq equation models[D]. Master Thesis. Tianjin: Tianjin University, 2007.(in Chinese))
[2]	王颖. 几类Boussinesq方程的Cauchy问题[D]. 博士学位论文. 成都: 四川大学, 2007.(WANG Ying. The Cauchy problems for some classes of Boussinesq equations[D]. PhD Thesis. Chengdu: Sichuan University, 2007.(in Chinese))
[3]	MILES J W. Solitary waves[J]. Annual Review of Fluid Mechanics,1980,12: 11-43.
[4]	MILEWSKI P A, KELLER J B. Three-dimensional water waves[J]. Studies in Applied Mathematics, 1996,97(2): 149-166.
[5]	TSUTSUMI M, MATAHASHI T. On the Cauchy problem for the Boussinesq type equation[J]. Mathematics Japonica,1991,36: 371-379.
[6]	BONA J, SACHS R. Global existence of smooth solution and stabilityof solitary waves for a generalized Boussinesq equation[J]. Communications in Mathematical Physics,1988,118(1): 12-29.
[7]	KALANTAROV V K, LADYZHENSKAYA O A. The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types[J]. Journal of Mathematical Sciences,1978,10(1): 53-70.
[8]	LEVIN H A, SLEEMAN B D. A note on the nonexistence of global solutions of initial-boundary value problems for the Boussinesq equation u_tt=3u_xxx+u_xx－12(u²)_xx[J].Journal of Mathematical Analysis and Applications,1985,107(1): 206-210.
[9]	HIROTA R. Exact N-solition solutions of the wave equation of long waves in shallow-water and in nonlinear lattices[J]. Journal of Mathematical Physics,1973,14(7): 810-814.
[10]	CLARKSON P. New exact solution of the Boussinesq equation[J]. European Journal of Applied Mathematics,1990,1(3): 279-300.
[11]	PEREGRINE D H. Long waves on a beach[J]. Journal of Fluid Mechanics,1967,27(4): 815-827.
[12]	WEI G, KIRBY J T, GRILLI S T, et al. A fully nonlinear Boussinesq model for surface wave: part I, highly nonlinear unsteady waves[J]. Journal of Fluid Mechanics,1995,294: 71-92.
[13]	李春颖. 基于Boussinesq方程的海岸地区波浪数学模型研究[D]. 硕士学位论文. 天津: 天津大学, 2003.(LI Chunying. Research on numerical model for waves in coastal region based on Boussinesq equations[D]. Master Thesis. Tianjin: Tianjin University, 2003.(in Chinese))
[14]	王涛. 高阶Boussinesq方程的数值模型[D]. 硕士学位论文. 大连: 大连理工大学, 2002.(WANG Tao. The numerical model of the high order Boussinesq equation[D]. Master Thesis. Dalian: Dalian University of Technology, 2002.(in Chinese))
[15]	OLLILA S, DENNISTON C, KARTTUNEN M, et al. Fluctuating Lattice-Boltzmann model for complex fluids[J]. Journal of Chemical Physics,2011,134(6): 549-197.
[16]	FALLAH K, KHAYAT M, BORGHEI M H, et al. Multiple-relaxation-time lattice Boltzmann simulation of non-Newtonian flows past a rotating circular cylinder[J].Journal of Non-Newtonian Fluid Mechanics,2012,177/178: 1-14.
[17]	MAO W, GUO Z L, WANG L. Lattice Boltzmann simulation of the sedimentation of particles with thermal convection[J]. Acta Physica Sinica,2013,62(8): 786-790.
[18]	YANG T Z, JI S, YANG X D, et al. Microfluid-induced nonlinear free vibration of microtubes[J]. International Journal of Engineering Science,2014,76(4): 47-55.
[19]	HE Y B, TANG X H, LIN X Y. Numerical simulation of a class of Fitz Hugh-Nagumo systems based on the lattice Boltzmann method[J].Acta Physica Sinica,2016,65(15): 133-142.
[20]	VARLAMOV V V. Long-time asymptotics of solutions of the second initial-boundary value problem for the damped Boussinesq equation[J]. Abstract and Applied Analysis,1998,2(3/4): 281-289.
[21]	赖惠林. 几类非线性波方程的格子玻尔兹曼模型的数值分析及模拟[D]. 博士学位论文. 福州: 福建师范大学, 2012.(LAI Huilin. Numerical analysis and simulation of lattice Boltzmann models for some nonlinear wave equations[D]. PhD Thesis. Fuzhou: Fujian Normal University, 2012.(in Chinese))
[22]	MAKHANKOV V G. Dynamics of classical solitons(in non-integrable systems)[J]. Physics Reports,1978,35(1): 1-128.
[23]	CLARKSON P A, KRUSKAL M D. New similarity reductions of the Boussinesq equation[J]. Journal of Mathematical Physics,1989,30(10): 2201-2213.
[24]	DEIFT P, TOMEI C, TRUBOWITZ E. Inverse settering and the Boussinesq equation[J]. Pure and Applied Mathematics,1982,35(5): 567-628.
[25]	SHU C W, OSHER S. Efficient implementation of essentially non-oscillatory shock-capturing schemes[J]. Journal of Computational Physics,1998,77(2): 439-471.
[26]	GUO Z L, ZHENG C G, SHI B C. Non-equilibrium extrapolation method for velocity and pressure boundary conditions in the lattice Boltzmann method[J]. Chinese Physics,2002,11(4): 366-374.
[27]	FU Z T, LIU S K, LIU S D, et al. The JEFE method and periodic solutions of two kinds of nonlinear wave equations[J]. Communications in Nonlinear Science and Numerical Simulation,2003,8(2): 67-75.