应用数学进展  >> Vol. 6 No. 3 (May 2017)

一种用于反应扩散方程的格子Runge-Kutta-Boltzmann模型
A Lattice Runge-Kutta-Boltzmann Model for the Reaction Diffusion Equation

DOI: 10.12677/AAM.2017.63047, PDF, HTML, XML, 下载: 937  浏览: 1,181  国家自然科学基金支持

作者: 闫铂, 王建朝:吉林建筑大学土木工程学院,吉林 长春;闫广武:吉林大学数学学院,吉林 长春

关键词: 格子Boltzmann模型Runge-Kutta公式反应扩散方程Latctice Boltzmann Model Runge-Kutta Scheme Reaction Diffusion Equation

摘要: 本文构建了一个求解反应扩散方程的格子Runge-Kutta-Boltzmann模型。通过使用经典的Runge-Kutta公式,得到了四阶截断误差。通过Chapmann-Enskog展开和多尺度展开技术,获得了不同时间尺度的系列偏微分方程和修正的反应扩散方程。数值结果表明,本文的模型可以用来求解反应扩散方程。
Abstract: A lattice Runge-Kutta-Boltzmann model for the reaction diffusion equations is constructed in this paper. By using the classical Runge-Kutta formula, we obtain four-order accuracy of truncation error. The Chapman-Enskog expansion and multi-scale technique are employed in order to obtain a series of equations in different time scales and modify partial differential equations of the reaction diffusion equations. Numerical tests show that the scheme can be used to simulate the reaction diffusion equations.

文章引用: 闫铂, 王建朝, 闫广武. 一种用于反应扩散方程的格子Runge-Kutta-Boltzmann模型[J]. 应用数学进展, 2017, 6(3): 408-416. https://doi.org/10.12677/AAM.2017.63047

参考文献

[1] Frisch, U., Hasslacher, B. and Pomeau, Y. (1986) Lattice Gas Automata for the Navier-Stokes Equations. Physical Review Letters, 56, 1505-1508.
https://doi.org/10.1103/PhysRevLett.56.1505
[2] Chen, S.Y. and Doolen, G.D. (1998) Lattice Boltzmann Method for Fluid Flows. Annual Review of Fluid Mechanics, 30, 329-364.
[3] Succi, S. (2001) The Lattice Boltzmann Equation for Fluid Dynamics and Beyond. Oxford University Press, New York.
[4] Benzi, R., Succi, S. and Vergassola, M. (1992) The Lattice Boltzmann Equations: Theory and Applications. Physics Reports, 222, 147-197.
[5] Ladd, A. (1994) Numerical Simulations of Particulate Suspensions via a Discretized Boltzmann Equation. Part II. Journal of Fluid Mechanics, 271, 311-339.
https://doi.org/10.1017/S0022112094001783
[6] Chen, S.Y., Chen, H.D., Martíınez, D. and Matthaeus, W. (1991) Lattice Boltzmann Model for Simulation of Magneto-Hydrodynamics. Physical Review Letters, 67, 3776-3779.
https://doi.org/10.1103/PhysRevLett.67.3776
[7] Gunstensen, A.K. (1992) Lattice-Boltzmann Studies of Multiphase Flow through Porous Media. MIT, Boston.
[8] Chen, S.Y., Dawson, S.P., Doolen, G., Jenecky, D. and Lawniczak, A. (1995) Lattice Methods for Chemically Reacting Systems. Computers & Chemical Engineering, 19, 617-646.
[9] Yan, G.W. (2000) A Lattice Boltzmann Equation for Waves. Journal of Computational Physics, 161, 61-69.
https://doi.org/10.1006/jcph.2000.6486
[10] Yan, G.W. and Zhang, J.Y. (2009) A Higher-Order Moment Method of the Lattice Boltzmann Model for the Korteweg-de Vries Equation. Mathematics and Computers in Simulation, 79, 1554-1565.
[11] Velivelli, A.C. and Bryden, K.M. (2006) Parallel Performance and Accuracy of Lattice Boltzmann and Traditional Finite Difference Methods for Solving the Unsteady Two Dimensional Burger’s Equation. Physica A, 362, 139-145.
[12] Zhang, J.Y. and Yan, G.W. (2008) Lattice Boltzmann Methods for One and Two-Dimensional Burger’s Equation. Physica A, 387, 4771-4786.
[13] Succi, S. (1993) Lattice Boltzmann Equation for Quantum Mechanics. Physica D, 69, 327-332.
[14] Palpacelli, S. and Succi, S. (2007) Numerical Validation of the Quantum Lattice Boltzmann Scheme in Two and Three Dimension. Physical Review E, 75, Article ID: 066704.
https://doi.org/10.1103/PhysRevE.75.066704
[15] Zhong, L.H., Feng, S.D., Dong, P. and Gao, S.T. (2006) Lattice Boltzmann Schemes for the Nonlinear Schrodinger Equation. Physical Review E, 74, Article ID: 036704.
https://doi.org/10.1103/PhysRevE.74.036704
[16] Chai, Z.H. and Shi, B.C. (2007) A Novel Lattice Boltzmann Model for the Poisson Equation. Applied Mathematical Modelling, 32, 2050-2058.
[17] Wang, M.R., Wang, J.K. and Chen, S.Y. (2007) Roughness and Cavitations Effect on Electro-Osmotic Flows in Rough Microchannels Using the Lattice Poisson-Boltzmann Methods. Journal of Computational Physics, 226, 836-851.
[18] King, J.T. (1984) Introduction to Numerical Computation. McGraw-Hill Inc., New York.
[19] Chapman, S. and Cowling, T.G. (1970) The Mathematical Theory of Non-Uniform Gas. Cambridge University Press, Cambridge.
[20] Yan, G.W. and Yuan, L. (2000) Lattice Boltzmann Solver of Rössler Equation. Communications in Nonlinear Science & Numerical Simulation, 5, 64-68.