# American Institute of Mathematical Sciences

February  2009, 23(1&2): 221-248. doi: 10.3934/dcds.2009.23.221

## Third order equivalent equation of lattice Boltzmann scheme

 1 Numerical Analysis and Partial Differential Equations, Department of Mathematics, University Paris Sud, Bat. 425, F-91405 Orsay Cedex, France

Received  September 2007 Published  September 2008

We recall the origin of lattice Boltzmann scheme and detail the version due to D'Humières [8]. We present a formal analysisof this lattice Boltzmann scheme in terms of a single numericalinfinitesimal parameter. We derive third order equivalent partial differential equation of this scheme.Both situations of single conservation law and fluid flow with mass and momentum conservations are detailed. We apply our analysis to so-called D1Q3 and D2Q9 lattice Boltzmann schemes in one and two space dimensions.
Citation: François Dubois. Third order equivalent equation of lattice Boltzmann scheme. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 221-248. doi: 10.3934/dcds.2009.23.221
 [1] Barbara Kaltenbacher. Mathematics of nonlinear acoustics. Evolution Equations & Control Theory, 2015, 4 (4) : 447-491. doi: 10.3934/eect.2015.4.447 [2] Pavel Eichler, Radek Fučík, Robert Straka. Computational study of immersed boundary - lattice Boltzmann method for fluid-structure interaction. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 819-833. doi: 10.3934/dcdss.2020349 [3] Zhonghua Qiao, Xuguang Yang. A multiple-relaxation-time lattice Boltzmann method with Beam-Warming scheme for a coupled chemotaxis-fluid model. Electronic Research Archive, 2020, 28 (3) : 1207-1225. doi: 10.3934/era.2020066 [4] Rainer Buckdahn, Ingo Bulla, Jin Ma. Pathwise Taylor expansions for Itô random fields. Mathematical Control & Related Fields, 2011, 1 (4) : 437-468. doi: 10.3934/mcrf.2011.1.437 [5] Adrien Dekkers, Anna Rozanova-Pierrat, Vladimir Khodygo. Models of nonlinear acoustics viewed as approximations of the Kuznetsov equation. Discrete & Continuous Dynamical Systems, 2020, 40 (7) : 4231-4258. doi: 10.3934/dcds.2020179 [6] Yaozhong Hu, Yanghui Liu, David Nualart. Taylor schemes for rough differential equations and fractional diffusions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3115-3162. doi: 10.3934/dcdsb.2016090 [7] Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515 [8] Pooja Girotra, Jyoti Ahuja, Dinesh Verma. Analysis of Rayleigh Taylor instability in nanofluids with rotation. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021018 [9] Pedro M. Jordan, Barbara Kaltenbacher. Introduction to the special volume Mathematics of nonlinear acoustics: New approaches in analysis and modeling''. Evolution Equations & Control Theory, 2016, 5 (3) : i-ii. doi: 10.3934/eect.201603i [10] H. T. Banks, R.C. Smith. Feedback control of noise in a 2-D nonlinear structural acoustics model. Discrete & Continuous Dynamical Systems, 1995, 1 (1) : 119-149. doi: 10.3934/dcds.1995.1.119 [11] Ivan C. Christov. Nonlinear acoustics and shock formation in lossless barotropic Green--Naghdi fluids. Evolution Equations & Control Theory, 2016, 5 (3) : 349-365. doi: 10.3934/eect.2016008 [12] Jiangfeng Huang, Zhiliang Deng, Liwei Xu. A Bayesian level set method for an inverse medium scattering problem in acoustics. Inverse Problems & Imaging, 2021, 15 (5) : 1077-1097. doi: 10.3934/ipi.2021029 [13] Sonja Cox, Arnulf Jentzen, Ryan Kurniawan, Primož Pušnik. On the mild Itô formula in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2217-2243. doi: 10.3934/dcdsb.2018232 [14] Peter Seibt. A period formula for torus automorphisms. Discrete & Continuous Dynamical Systems, 2003, 9 (4) : 1029-1048. doi: 10.3934/dcds.2003.9.1029 [15] Hans F. Weinberger, Xiao-Qiang Zhao. An extension of the formula for spreading speeds. Mathematical Biosciences & Engineering, 2010, 7 (1) : 187-194. doi: 10.3934/mbe.2010.7.187 [16] Gamaliel Blé, Carlos Cabrera. A generalization of Douady's formula. Discrete & Continuous Dynamical Systems, 2017, 37 (12) : 6183-6188. doi: 10.3934/dcds.2017267 [17] Masaaki Fukasawa, Jim Gatheral. A rough SABR formula. Frontiers of Mathematical Finance, 2022, 1 (1) : 81-97. doi: 10.3934/fmf.2021003 [18] Tae Gab Ha. On the viscoelastic equation with Balakrishnan-Taylor damping and acoustic boundary conditions. Evolution Equations & Control Theory, 2018, 7 (2) : 281-291. doi: 10.3934/eect.2018014 [19] Fei Jiang, Song Jiang, Weiwei Wang. Nonlinear Rayleigh-Taylor instability for nonhomogeneous incompressible viscous magnetohydrodynamic flows. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1853-1898. doi: 10.3934/dcdss.2016076 [20] Djédjé Sylvain Zézé, Michel Potier-Ferry, Yannick Tampango. Multi-point Taylor series to solve differential equations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1791-1806. doi: 10.3934/dcdss.2019118

2020 Impact Factor: 1.392