An accuracy analysis of the cascaded lattice Boltzmann method for the 1D advection-diffusion equation
Robert Straka1,2
, Keerti Vardhan Sharma3
1AGH University of Science and Technology, Department of Heat Engineering and Environment Protection, Faculty of Metals Engineering and Industrial Computer Science, al. Mickiewicza 30, 30-059, Krakow, Poland.
2Czech Technical University in Prague, Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Trojanova 13, 120 00, Praha 2, Czech Republic.
3University of Wyoming, Paris Center of Innovation for Flow Through Porous Media, High Bay Research Facility, 651 N 19th street, Laramie, WY 82072, United States of America.
DOI:
https://doi.org/10.7494/cmms.2020.4.0732
Abstract:
We analyze higher order error terms in a modified partial differential equation of a cascaded lattice Boltzmann method (CLBM) for one conservation law – the advection-diffusion equation. To inspect the behavior of the error terms we derived an equivalent finite difference equation (EFDE), this approach is different from other techniques like the Chapman-Engskog expansion, equivalent partial differential equations or the Maxwell iteration used in the literature. The resulting EFDE is obtained from the recurrence formulas of the lattice Boltzmann equations for the CLBM and is subsequently analyzed by standard analytical techniques. We have found relations of the LBM parameters which could cancel some of the higher order terms, making the method more accurate. The detailed derivation of the EFDE and higher order terms’ pre-factors is the main result of this paper. The resulting explicit form of the error terms are derived and presented.
Cite as:
Straka, R., & Sharma, K. V. (2020). An accuracy analysis of the cascaded lattice Boltzmann method for the 1D advection-diffusion equation. Computer Methods in Materials Science, 20(4), 173–184. https://doi.org/10.7494/cmms.2020.4.0732
Article (PDF):
Keywords:
Cascaded thermal lattice Boltzmann method, High order analysis, Advection-diffusion equation, Equivalent finite difference equation
References:
Asinari, P. (2008). Generalized local equilibrium in the cascaded lattice Boltzmann method. Physical Review E, 78(1), 016701. https://doi.org/10.1103/physreve.78.016701.
Chen, S., & Doolen, G. (1998). Lattice Boltzmann Method for fluid flows. Annual Review of Fluid Mechanics, 30, 329–364. https://doi.org/10.1146/annurev.fluid.30.1.329.
Chopard, B., Falcone, J.L., & Latt, J. (2009). The lattice Boltzmann advection-diffusion model revisited. The European Physical Journal Special Topics, 171(1), 245–249. https://doi.org/10.1140/epjst/e2009-01035-5.
d’Humières, D. (2002). Multiple-relaxation-time lattice Boltzmann models in three dimensions. Philosophical Transactions of The Royal Socciety A: Mathematical, Physical and Engineering Sciences, 360(1792), 437–51. https://doi.org/10.1098/rsta.2001.0955.
De Rosis, A. (2017). Nonorthogonal central-moments-based lattice Boltzmann scheme in three dimensions. Physical Review E, 95, 013310. https://doi.org/10.1103/PhysRevE.95.013310.
Dong, Y., Zhang, J., & Yan, G. (2010). A higher-order moment method of the lattice Boltzmann model for the conservation law equation. Applied Mathematical Modelling, 34(2), 481–494. https://doi.org/10.1016/j.apm.2009.06.024.
Dubois, F. (2008). Equivalent partial differential equations of a lattice Boltzmann scheme. Computers & Mathematics with Applications, 55(7), 1441–1449. https://doi.org/10.1016/j.camwa.2007.08.003.
Fei, L., & Luo, K.H. (2018). Cascaded lattice Boltzmann method for incompressible thermal flows with heat sources and general thermal boundary conditions. Computers and Fluids, 165, 89–95. https://doi.org/10.1016/j.compfluid.2018.01.020.
Fučík, R., & Straka, R. (2021). Equivalent finite difference and partial differential equations for the lattice Boltzmann method. Computers & Mathematics with Applications, 90, 96–103. https://doi.org/10.1016/j.camwa.2021.03.014.
Fučík, R., Eichler, P., Straka, R., Pauš, P., Klinkovský, J., & Oberhuber, T. (2018). On optimal node spacing for immersed boundary-lattice Boltzmann method in 2D and 3D. Computers & Mathematics with Applications, 77(4), 1144–1162. https://doi.org/10.1016/j.camwa.2018.10.045.
Geier, M.C. (2006). Ab initio derivation of the cascaded lattice Boltzmann automaton [PhD. Thesis]. University of Freiburg, Germany. https://freidok.uni-freiburg.de/data/2860/.
Geier, M., & Pasquali, A. (2018). Fourth order Galilean invariance for the lattice Boltzmann method. Computers & Fluids, 166, 139–151. https://doi.org/10.1016/j.compfluid.2018.01.015.
Geier, M., Schönherr, M., Pasquali, A., & Krafczyk, M. (2015). The Cumulant lattice Boltzmann equation in three dimensions: Theory and validation. Computers & Mathematics with Applications, 70(4), 507–547. https://doi.org/10.1016/j.camwa.2015.05.001.
Geier, M., Pasquali, A., & Schönherr, M. (2017). Parametrization of the cumulant lattice Boltzmann method for fourth order accurate diffusion Part I: Derivation and validation. Journal of Computational Physics, 348, 862–888. https://doi.org/10.1016/j.jcp.2017.05.040.
Ginzburg, I. (2012). Truncation Errors, Exact and Heuristic Stability Analysis of Two-Relaxation-Times Lattice Boltzmann Schemes for Anisotropic Advection-Diffusion Equation. Communications in Computational Physics, 11(5), 1439–1502. https://doi.org/10.4208/cicp.211210.280611a.
Hajabdollahi, F., & Premnath, K.N. (2018). Symmetrized Operator Split Schemes for Force and Source Modeling in Cascaded Lattice Boltzmann Methods for Flow and Scalar Transport. Physical Review E, 97(6), 063303. https://doi.org/10.1103/PhysRevE.97.063303.
Holdych, D.J., Noble, D.R., Georgiadis, J.G., & Buckius, R.O. (2004). Truncation error analysis of lattice Boltzmann methods. Journal of Computational Physics, 193(2), 595–619. https://doi.org/10.1016/j.jcp.2003.08.012.
Sharma, K.V., Straka, R., & Tavares, F.W. (2017). New Cascaded Thermal Lattice Boltzmann Method for Simulations of Advection-Diffusion and Convective Heat Transfer. International Journal of Thermal Sciences, 118, 259–277. https://doi.org/10.1016/j.ijthermalsci.2017.04.020.
Suga, S. (2006). Numerical scheme obtained from lattice Boltzmann equations for advection diffusion equations. International Journal of Modern Physics C, 17(11), 1563–1577. https://doi.org/10.1142/S0129183106010030.
Suga, S. (2009). Stability and accuracy of lattice Boltzmann schemes for anisotropic advection-diffusion equations. International Journal of Modern Physics C, 20(4), 633–650. https://doi.org/10.1142/S0129183109013856.
Suga, S. (2010). An accurate multi-level finite difference scheme for 1D diffusion equations derived from the lattice Boltzmann method. Journal of Statistical Physics, 140(3), 404–503. https://doi.org/10.1007/s10955-010-0004-y.
Wang, J., Wang, D., Lallemand, P., & Luo, L.-S. (2013). Lattice Boltzmann simulations of thermal convective flows in two dimensions. Computers & Mathematics with Applications, 65(2), 262–286. https://doi.org/10.1016/j.camwa.2012.07.001.
Wolf-Gladrow, D.A. (2005). Lattice-gas cellular automata and lattice Boltzmann models. An introduction. Springer. https://doi.org/10.1007/b72010.
Zhao, F. (2013). Optimal relaxation collisions for lattice Boltzmann methods. Computers & Mathematics with Applications, 65(2), 172–185. https://doi.org/10.1016/j.camwa.2011.06.005.
Zhao, W., & Yong, W.-A. (2017). Maxwell iteration for the lattice Boltzmann method with diffusive scaling. Physical Review E, 95(3), 033311. https://doi.org/10.1103/PhysRevE.95.033311.