Abstract
The connection of relaxation systems and discrete velocity models is essential to the progress of stability as well as convergence results for lattice Boltzmann methods. In the present study we propose a formal perturbation ansatz starting from a scalar one-dimensional target equation, which yields a relaxation system specifically constructed for its equivalence to a discrete velocity Boltzmann model as commonly found in lattice Boltzmann methods. Further, the investigation of stability structures for the discrete velocity Boltzmann equation allows for algebraic characterizations of the equilibrium and collision operator. The methods introduced and summarized here are tailored for scalar, linear advection–diffusion equations, which can be used as a foundation for the constructive design of discrete velocity Boltzmann models and lattice Boltzmann methods to approximate different types of partial differential equations.
This article is part of the theme issue ‘Fluid dynamics, soft matter and complex systems: recent results and new methods’.
Footnotes
References
- 1.
Bouchut F . 1999Construction of BGK models with a family of kinetic entropies for a given system of conservation laws. J. Stat. Phys. 95, 113–170. (doi:10.1023/A:1004525427365) Crossref, ISI, Google Scholar - 2.
Junk M . 1999On the construction of discrete equilibrium distributions for kinetic schemes. Institut für Techno- und Wirtschaftsmathematik Kaiserslautern, ITWM Report 14. Visit: http://kops.uni-konstanz.de/handle/123456789/25480. Google Scholar - 3.
Bhatnagar P, Gross EP, Krook M . 1954A model for collision processes in gases I: small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94, 511–525. (doi:10.1103/PhysRev.94.511) Crossref, ISI, Google Scholar - 4.
Aregba-Driollet D, Natalini R . 2000Discrete kinetic schemes for multidimensional systems of conservation laws. SIAM J. Numer. Anal. 37, 1973–2004. (doi:10.1137/S0036142998343075) Crossref, ISI, Google Scholar - 5.
Liu T-P . 1987Hyperbolic conservation laws with relaxation. Commun. Math. Phys. 108, 153–175. (doi:10.1007/BF01210707) Crossref, ISI, Google Scholar - 6.
Jin S, Liu H . 1998Diffusion limit of a hyperbolic system with relaxation. Methods Appl. Anal. 5, 317–334. (doi:10.4310/MAA.1998.v5.n3.a6) Google Scholar - 7.
Jin S, Xin Z . 1995The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Commun. Pure Appl. Math. 48, 235–276. (doi:10.1002/cpa.3160480303) Crossref, ISI, Google Scholar - 8.
Rheinländer MK . 2007Analysis of lattice-Boltzmann methods: asymptotic and numeric investigation of a singularly perturbed system. PhD thesis, Universität Konstanz. Google Scholar - 9.
Graille B . 2014Approximation of mono-dimensional hyperbolic systems: a lattice Boltzmann scheme as a relaxation method. J. Comput. Phys. 266, 74–88. (doi:10.1016/j.jcp.2014.02.017) Crossref, ISI, Google Scholar - 10.
Banda MK, Yong W -A, Klar A . 2006A stability notion for lattice Boltzmann equations. SIAM J. Sci. Comput. 27, 2098–2111. (doi:10.1137/040606211) Crossref, ISI, Google Scholar - 11.
Yong W-A . 2001Basic aspects of hyperbolic relaxation systems. In Advances in the theory of shock waves (eds H Freistühler, A Szepess), pp. 259–305. Boston, MA: Springer. Google Scholar - 12.
Junk M, Yong W-A . 2009Weighted L2-stability of the lattice Boltzmann method. SIAM J. Numer. Anal. 47, 1651–1665. (doi:10.1137/060675216) Crossref, ISI, Google Scholar - 13.
Yong W-A . 2009An onsager-like relation for the lattice Boltzmann method. Comput. Math. Appl. 58, 862–866. (doi:10.1016/j.camwa.2009.02.010) Crossref, ISI, Google Scholar - 14.
Rheinländer MK . 2010On the stability structure for lattice Boltzmann schemes. Comput. Math. Appl. 59, 2150–2167. (doi:10.1016/j.camwa.2009.08.040) Crossref, ISI, Google Scholar - 15.
Otte P, Frank M . 2016Derivation and analysis of lattice Boltzmann schemes for the linearized Euler equations. Comput. Math. Appl. 72, 311–327. (doi:10.1016/j.camwa.2015.12.004) Crossref, ISI, Google Scholar - 16.
LeVeque RJ . 2007Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems, vol. 98. Siam Publications online. (doi:10.1137/1.9780898717839) Crossref, Google Scholar - 17.
Trefethen LN . 1996Finite difference and spectral methods for ordinary and partial differential equations. Cornell University - Department of Computer Science and Center for Applied Mathematics. Google Scholar - 18.
Otte P, Frank M . 2019A structured approach to the construction of stable linear lattice Boltzmann collision operator. Comput. Math. Appl. 79, 1447–1460. in press (doi:10.1016/j.camwa.2019.09.009) Crossref, ISI, Google Scholar - 19.
Caetano F, Dubois F, Graille B . 2019A result of convergence for a mono-dimensional two-velocities lattice Boltzmann scheme. (http://arxiv.org/abs/1905.12393v1). Google Scholar - 20.
Gaedtke M, Wachter S, Rädle M, Nirschl H, Krause MJ . 2018Application of a lattice Boltzmann method combined with a Smagorinsky turbulence model to spatially resolved heat flux inside a refrigerated vehicle. Comput. Math. Appl. 76, 2315–2329. (doi:10.1016/j.camwa.2018.08.018) Crossref, ISI, Google Scholar - 21.
Mink A, Thäter G, Nirschl H, Krause MJ . 2016A 3D lattice Boltzmann method for light simulation in participating media. J. Comput. Sci. 17, 431–437. (doi:10.1016/j.jocs.2016.03.014) Crossref, ISI, Google Scholar - 22.
Rao R, Subba Rao M . 2003A simple multidimensional relaxation scheme based on characteristics and interpolation. In 16th AIAA Computational Fluid Dynamics Conference, Orlando, FL, June 23–26 2003, p. 3535. AIAA. Google Scholar - 23.
Shen W, Zhang C, Zhang J . 2011Relaxation method for unsteady convection–diffusion equations. Comput. Math. Appl. 61, 908–920. (doi:10.1016/j.camwa.2010.12.039) Crossref, ISI, Google Scholar - 24.
Bouchut F, Guarguaglini FR, Natalini R . 2000Diffusive BGK approximations for nonlinear multidimensional parabolic equations. Indiana Univ. Math. J. 49, 723–750. (doi:10.1512/iumj.2000.49.1811) Crossref, ISI, Google Scholar - 25.
Morton KW . 1996Numerical solution of convection-diffusion problems. London, UK: Chapman & Hall. Google Scholar - 26.
Bianchini R . 2018Uniform asymptotic and convergence estimates for the Jin–Xin model under the diffusion scaling. SIAM J. Math. Anal. 50, 1877–1899. (doi:10.1137/17M1152395) Crossref, ISI, Google Scholar - 27.
Bouchut F . 2005Stability of relaxation models for conservation laws. In European Congress of Mathematics, pp. 95–101. European Mathematical Society. Google Scholar - 28.
Krüger T, Kusumaatmaja H, Kuzmin A, Shardt O, Silva G, Viggen EM . 2017The lattice Boltzmann method. Cham, Switzerland: Springer International Publishing. Google Scholar - 29.
- 30.
Coreixas C, Chopard B, Latt J . 2019Comprehensive comparison of collision models in the lattice Boltzmann framework: theoretical investigations. Phys. Rev. E 100, 033305. (doi:10.1103/PhysRevE.100.033305) Crossref, PubMed, ISI, Google Scholar - 31.
Chopard B, Falcone J-L, Latt J . 2009The lattice Boltzmann advection-diffusion model revisited. Eur. Phys. J. Spec. Top. 171, 245–249. (doi:10.1140/epjst/e2009-01035-5) Crossref, ISI, Google Scholar - 32.
He X, Luo L-S . 1997Theory of the lattice Boltzmann method: from the Boltzmann equation to the lattice Boltzmann equation. Phys. Rev. E 56, 6811–6817. (doi:10.1103/PhysRevE.56.6811) Crossref, ISI, Google Scholar - 33.
Junk M, Klar A, Luo L-S . 2005Asymptotic analysis of the lattice Boltzmann equation. J. Comput. Phys. 210, 676–704. (doi:10.1016/j.jcp.2005.05.003) Crossref, ISI, Google Scholar - 34.
Ubertini S, Asinari P, Succi S . 2010Three ways to lattice Boltzmann: a unified time-marching picture. Phys. Rev. E 81, 016311. (doi:10.1103/PhysRevE.81.016311) Crossref, ISI, Google Scholar - 35.
Krause MJ . 2010Fluid flow simulation and optimisation with lattice Boltzmann methods on high performance computers: application to the human respiratory system. PhD thesis, Karlsruhe Institute of Technology (KIT). Google Scholar - 36.
Dellar PJ . 2013An interpretation and derivation of the lattice Boltzmann method using Strang splitting. Comput. Math. Appl. 65, 129–141. (doi:10.1016/j.camwa.2011.08.047) Crossref, ISI, Google Scholar - 37.
Junk M, Yang Z . 2015L2 convergence of the lattice Boltzmann method for one dimensional convection-diffusion-reaction equations. Commun. Comput. Phys. 17, 1225–1245. (doi:10.4208/cicp.2014.m369) Crossref, ISI, Google Scholar - 38.
Mojtabi A, Deville MO . 2015One-dimensional linear advection–diffusion equation: analytical and finite element solutions. Comput. Fluids 107, 189–195. (doi:10.1016/j.compfluid.2014.11.006) Crossref, ISI, Google Scholar - 39.
Haussmann M, Simonis S, Nirschl H, Krause MJ . 2019Direct numerical simulation of decaying homogeneous isotropic turbulence—numerical experiments on stability, consistency and accuracy of distinct lattice Boltzmann methods. Int. J. Mod. Phys. C 30, 1–29. (doi:10.1142/S0129183119500748) Crossref, ISI, Google Scholar - 40.
Suga S . 2006Numerical schemes obtained from lattice Boltzmann equations for advection diffusion equations. Int. J. Mod. Phys. C 17, 1563–1577. (doi:10.1142/S0129183106010030) Crossref, ISI, Google Scholar - 41.
Weiß J-P . 2006Numerical analysis of lattice Boltzmann methods for the heat equation on a bounded interval. Karlsruhe, Germany: Univ.-Verlag Karlsruhe. Google Scholar - 42.
Junk M, Rheinlander MK . 2008Regular and multiscale expansions of a lattice Boltzmann method. Prog. Comput. Fluid Dyn. Int. J. 8, 25–37. (doi:10.1504/PCFD.2008.018076) Crossref, ISI, Google Scholar - 43.
Dubois F, Graille B, Rao SVR . 2019A stability property for a mono-dimensional three velocities scheme with relative velocity. (https://arxiv.org/abs/1911.12215v1). Google Scholar - 44.
Rheinländer MK . 2008Stability and multiscale analysis of an advective lattice Boltzmann scheme. Prog. Comput. Fluid Dyn. Int. J. 8, 56–68. (doi:10.1504/PCFD.2008.018079) Crossref, ISI, Google Scholar


