变系数广义KdV-Burgers方程的格子Boltzmann模型

Abstract

Abstract:

This paper studies the numerical calculation method of a kind of general Kdv-Burgers equation with variable coefficients. Firstly, a lattice Boltzmann model of the generalized KdV-Burgers equation with variable coefficients is obtained by selecting the equilibrium distribution function and adding the correction function. The model could accurately recover the KdV-Burgers equation without any assumptions. Secondly, this paper studies the temporal and spatial change trend of the nonlinear high-order derivative term in the equation, and compares it with the analytical solution, and then gives an error analysis. Finally, This paper analyzes the precision of the space and time of the equation. According to the simulation experiment results, the model could reach 2nd order accuracy. Numerical results show that the current lattice Boltzmann model is a satisfactory and efficient algorithm.

Key words: Lattice Boltzmann equation, Chapman-Enskog analysis, KdV-Burgers equation, Variable coefficient

CLC Number:

O175

Zongning Zhang,Chunguang Li,jianqiang Dong. General Propagation Lattice Boltzmann Model for a Variable-Coefficient Compound KdV-Burgers Equation[J].Acta mathematica scientia,Series A, 2021, 41(5): 1283-1295.

Trendmd

Figures/Tables 12

References 15

1	Chen S Y , Doolen G D . Lattice Boltzmann method for fluid flows. Annual Review of Fluid Mechanics, 1998, 30, 329- 364 doi: 10.1146/annurev.fluid.30.1.329
2	Ginzburg I . Equilibrium-type and link-type lattice Boltzmann models for generic advection and anisotropic-dispersion equation. Advances in Water Resources, 2005, 28 (11): 1171- 1195 doi: 10.1016/j.advwatres.2005.03.004
3	Li Q H , Chai Z H , Shi B C . Lattice Boltzmann model for a class of convection-diffusionequations with variable coefficients. Computers and Mathematics with Applications, 2015, 5 (4): 548- 561
4	Zhang J Y , Yan G W . Lattice Boltzmann method for one and two-dimensional burgers equation. Physica A: Statistical Mechanics and Its Applications, 2008, 387 (19): 4771- 4786
5	Yan G W . A lattice Boltzmann equation for waves. Journal of Computational Physics, 2000, 161 (1): 61- 69 doi: 10.1006/jcph.2000.6486
6	Liu F , Shi W P , Wu F F . A lattice Boltzmann model for the generalized Boussinesq equation. Applied Mathematics and Computation, 2016, 274 (1): 331- 342
7	Chai Z H , Shi B H , Zheng L . A unified lattice Boltzzmann model for some nonlinear partial differential equations. Chaos, Soliton and Fract, 2008, 36 (4): 874- 882 doi: 10.1016/j.chaos.2006.07.023
8	何雅玲, 王勇, 李庆. 格子Boltzmann方法的理论及应用. 北京: 科学出版社, 2009
	He Y l , Wang Y , Li Q . The Theory and Application of Lattice Boltzmann Method. Beijing: Science Press, 2009
9	默罕默德·阿卜杜勒马吉德. 格子玻尔兹曼方法. 北京: 电子工业出版社, 2015
	Mohamad A A . Lattice Boltzmann Method. Beijing: Electronic Industry Press, 2015
10	Chai Z H , Shi B C . Multiple-relaxation-time lattice Boltzmann method for the Navier-Stokes and nonlinear convection-diffusion equations: modeling, analysis, and elements. Physical Review E, 2020, 102, 023306
11	Hu W Q , Gao Y T , Lan Z Z . Lattice Boltzmann model for a generalized Gardner equation with time-dependent variable coefficient. Applied Mathematical Modelling, 2017, 46, 126- 140 doi: 10.1016/j.apm.2017.01.061
12	Lan Z Z , Hu W Q , Gao Y T . General propagation lattice Boltzmann model for a variable-coefficient compound KdV-Burgers equation. Applied Mathematical Modelling, 2019, 73, 695- 714 doi: 10.1016/j.apm.2019.04.013
13	赖惠林, 马昌凤. 非线性偏微分方程的高阶格子BGK模型. 中国科学G版, 2009, 39 (7): 913- 922
	Lan H L , Ma C F . A higer order lattice BGK model for simulating some nonlinear partial differential equations. Sci China Ser G, 2009, 39 (7): 913- 922
14	Sterling J D , Chen S Y . Stability analysis of lattice Boltzmann methods. Journal of Computational Physics, 1996, 123 (1): 196- 206 doi: 10.1006/jcph.1996.0016
15	Abd-el-Malek M B , Helal M M . Group method solution of the generalized forms of Burgers, Burgers-KDV and Kdv equations with time-dependent variable coefficients. Acta Mesh, 2001, 221, 281- 296

Parameters	t=0.5	t=1.0	t=1.5	t=2.0
E₂	5.1836e-04	4.9568e-04	6.1178e-04	7.6003e-04
E_∞	0.0364	0.0349	0.0478	0.0585
GRE	0.0334	0.0303	0.0369	0.0429

Parameters	t=1.0	t=2.0	t=3.0
E₂	1.0660e-04	7.9380e-05	7.0678e-05
E_∞	0.0230	0.0151	0.0125
GRE	0.0343	0.0267	0.0252

Parameters	t=1.0	t=2.0	t=3.0
E₂	2.2805e-04	2.8858e-04	2.2420e-04
E_∞	0.0043	0.0055	0.0043
GRE	0.0593	0.0750	0.0583

[1]	Lv Xueqin, He Songyan, Wang Shiyu. Combining RKM with FDM for Time Fractional Convection-Diffusion Equations with Variable Coefficients [J]. Acta mathematica scientia,Series A, 2025, 45(1): 153-164.
[2]	Zhang Jinyu,Wang Dan,Geng Yong,Yang Miaomiao,Wang Xiaoli. The Integrability to a (1+1) Dimensional Variable Coefficient Complex Equation [J]. Acta mathematica scientia,Series A, 2023, 43(4): 994-1002.
[3]	YANG Ai-Li, WU Yu-Jiang, SONG Lun-Ji. A Class of Generalized Three Dimensional Convection-Diffusion Equations with Multi-Level Incremental Unknowns Method [J]. Acta mathematica scientia,Series A, 2009, 29(3): 564-572.
[4]	Zhang Jinliang ; Wang Mingliang ; Wang Yueming. The Extended F-expansion Method and the Exact Solutions to the KdV and mKdV Equation with Variable Coefficients [J]. Acta mathematica scientia,Series A, 2006, 26(3): 353-360.

General Propagation Lattice Boltzmann Model for a Variable-Coefficient Compound KdV-Burgers Equation

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

Figures/Tables 12

References 15

Related Articles 4

Metrics

Comments

Recommended 10