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All issues Volume 59 / No 3 (May-June 2025) ESAIM: M2AN, 59 3 (2025) 1271-1299 References
Open Access
Issue
ESAIM: M2AN
Volume 59, Number 3, May-June 2025
Page(s) 1271 - 1299
DOI https://doi.org/10.1051/m2an/2025026
Published online 14 May 2025
  1. H. Babovsky, Die Boltzmann-Gleichung: Modellbildung-Numerik-Anwendungen. Springer Vieweg + Teubner (1998). https://doi.org/10.1007/978-3-663-12034-6. [CrossRef] [Google Scholar]
  2. P. Bachmann, Die analytische Zahlentheorie. Teubner (1923). [Google Scholar]
  3. M. Banda, A. Klar, L. Pareschi, and M. Seaid, Lattice-Boltzmann type relaxation systems and high order relaxation schemes for the incompressible Navier-Stokes equations. Math. Comput. 77 (2008) 943–965. https://doi.org/10.1090/S0025-5718-07-02034-0. [Google Scholar]
  4. C. Bardos, F. Golse, and C.D. Levermore, Fluid dynamic limits of kinetic equations II: Convergence proofs for the Boltzmann equation. Commun. Pure Appl. Math. 46 (1993) 667–753. https://doi.org/10.1002/cpa.3160460503. [CrossRef] [Google Scholar]
  5. S. Bartels, Numerical methods for nonlinear partial differential equations, vol. 47. Springer Cham (2015). https://doi.org/10.1007/978-3-319-13797-1. [CrossRef] [Google Scholar]
  6. T. Bellotti, Truncation errors and modified equations for the lattice Boltzmann method via the corresponding finite difference schemes. ESAIM Math. Model. Num. Anal. 57 (2023) 1225–1255. https://doi.org/10.1051/m2an/2023008. [CrossRef] [EDP Sciences] [Google Scholar]
  7. T. Bellotti, B. Graille, and M. Massot, Finite difference formulation of any lattice Boltzmann scheme. Numer. Math. 152 (2022) 1–40. https://doi.org/10.1007/s00211-022-01302-2. [CrossRef] [MathSciNet] [Google Scholar]
  8. P. L. Bhatnagar, E.P. Gross, and M. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. E 94 (1954) 511–525. https://doi.org/10.1103/PhysRev.94.511. [CrossRef] [Google Scholar]
  9. B.M. Boghosian, J. Yepez, P.V. Coveney, and A. Wagner, Entropic lattice Boltzmann methods. Proc. R. Soc. A 457 (2007) 717–766. https://doi.org/10.1098/rspa.2000.0689. [Google Scholar]
  10. F. Bouchut, F.R. Guarguaglini, and R. Natalini, Diffusive BGK approximations for nonlinear multidimensional parabolic equations. Indiana Univ. Math. J. 49 (2000) 723–749. https://doi.org/10.1512/iumj.2000.49.1811. [CrossRef] [Google Scholar]
  11. F. Caetano, F. Dubois, and B. Graille, A result of convergence for a mono-dimensional two-velocities lattice Boltzmann scheme. Discrete Contin. Dyn. Syst. – S 17 (2024) 3129–3154. https://doi.org/10.3934/dcdss.2023072. [CrossRef] [Google Scholar]
  12. A. Caiazzo, M. Junk, and M. Rheinl¨ander, Comparison of analysis techniques for the lattice Boltzmann method. Comput. Math. Appl., 58 (2009) 883–897. https://doi.org/10.1016/j.camwa.2009.02.011. [CrossRef] [MathSciNet] [Google Scholar]
  13. T.H. Cormen, C.E. Leiserson, R.L. Rivest, and C. Stein, Introduction to Algorithms. The MIT Press (2022). [Google Scholar]
  14. D. Dapelo, S. Simonis, M.J. Krause, and J. Bridgeman, Lattice-Boltzmann coupled models for advection-diffusion flow on a wide range of Péclet numbers. J. Comput. Sci. 51 (2021) 101363. https://doi.org/10.1016/j.jocs.2021.101363. [Google Scholar]
  15. P.J. Dellar, An interpretation and derivation of the lattice Boltzmann method using Strang splitting. Comput. Math. Appl. 65 (2013) 129–141. https://doi.org/10.1016/j.camwa.2011.08.047. [CrossRef] [MathSciNet] [Google Scholar]
  16. F. Dubois, Equivalent partial differential equations of a lattice Boltzmann scheme. Comput. Math. Appl. 55 (2008) 1441–1449. https://doi.org/10.1016/j.camwa.2007.08.003. [CrossRef] [MathSciNet] [Google Scholar]
  17. B.H. Elton, C.D. Levermore, and G.H. Rodrigue, Convergence of Convective-Diffusive Lattice Boltzmann Methods. SIAM J. Numer. Anal. 32 (1995) 1327–1354. https://doi.org/10.1137/0732062. [CrossRef] [MathSciNet] [Google Scholar]
  18. G.B. Folland, Advanced Calculus. Pearson Education (US) (2001). [Google Scholar]
  19. A.N. Gorban, Hilbert’s sixth problem: the endless road to rigour. Philos. Trans. A Math. Phys. Eng. Sci. 376 (2018) 20170238. https://doi.org/10.1098/rsta.2017.0238. [Google Scholar]
  20. Z. Guo, C. Zheng, and B. Shi, Discrete lattice effects on the forcing term in the lattice Boltzmann method. Phys. Rev. E 65 (2002) 046308. https://doi.org/10.1103/PhysRevE.65.046308. [CrossRef] [Google Scholar]
  21. Z. Guo, J. Li, and K. Xu, Unified preserving properties of kinetic schemes. Phys. Rev. E 107 (2023) 025301. https://doi.org/10.1103/PhysRevE.107.025301. [Google Scholar]
  22. M. Haussmann, P. Reinshaus, S. Simonis, H. Nirschl, and M.J. Krause, Fluid-structure interaction simulation of a coriolis mass flowmeter using a lattice Boltzmann method. Fluids 6 (2021) 167. https://doi.org/10.3390/ fluids6040167. [Google Scholar]
  23. X. He and L.-S. Luo, Theory of the lattice Boltzmann method: from the Boltzmann equation to the lattice Boltzmann equation. Phys. Rev. E 56 (1997) 6811–6817. https://doi.org/10.1103/PhysRevE.56.6811. [Google Scholar]
  24. S. Jin, Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review. Riv. Mat. Univ. Parma. 3 (2012) 177–216. https://ins.sjtu.edu.cn/people/shijin/PS/AP.pdf. [Google Scholar]
  25. M. Junk, A finite difference interpretation of the lattice Boltzmann method. Numer. Methods Partial Differ. Equ. 17 (2001) 383–402. https://doi.org/10.1002/num.1018. [CrossRef] [Google Scholar]
  26. M. Junk and A. Klar, Discretizations for the incompressible Navier-Stokes equations based on the Lattice Boltzmann method. SIAM J. Sci. Comput. 22 (2000) 1–19. https://doi.org/10.1137/S1064827599357188. [CrossRef] [MathSciNet] [Google Scholar]
  27. M. Junk and W.-A. Yong, Weighted L2-stability of the Lattice Boltzmann method. SIAM J. Numer. Anal. 47 (2009) 1651–1665. https://doi.org/10.1137/060675216. [CrossRef] [MathSciNet] [Google Scholar]
  28. M. Junk, A. Klar, and L.-S. Luo, Asymptotic analysis of the lattice Boltzmann equation. J. Comput. Phys. 210 (2005) 676–704. https://doi.org/10.1016/j.jcp.2005.05.003. [CrossRef] [MathSciNet] [Google Scholar]
  29. I.V. Karlin, A. Ferrante, and H.C. Öttinger, Perfect entropy functions of the Lattice Boltzmann method. Europhys. Lett. 47 (1999) 182–188. https://doi.org/10.1209/epl/i1999-00370-1. [CrossRef] [Google Scholar]
  30. A. Klar, Relaxation scheme for a Lattice–Boltzmann-type discrete velocity model and numerical Navier–Stokes limit. J. Comput. Phys. 148 (1999) 416–432. https://doi.org/10.1006/jcph.1998.6123. [CrossRef] [MathSciNet] [Google Scholar]
  31. M.J. Krause, Fluid flow simulation and optimisation with Lattice Boltzmann methods on high performance computers – application to the human respiratory system. Doctoral thesis, Karlsruhe Institute of Technology (KIT) (2010). https://doi.org/10.5445/IR/1000019768; https://publikationen.bibliothek.kit.edu/1000019768. [Google Scholar]
  32. M.J. Krause, A. Kummerl¨ander, S.J. Avis, H. Kusumaatmaja, D. Dapelo, F. Klemens, M. Gaedtke, N. Hafen, A. Mink, R. Trunk, J.E. Marquardt, M.-L. Maier, M. Haussmann, and S. Simonis, OpenLB – Open source lattice Boltzmann code. Comput. Math. Appl. 81 (2021) 258–288, 2021. https://doi.org/10.1016/j.camwa.2020.04.033. [CrossRef] [MathSciNet] [Google Scholar]
  33. E. Kummer and S. Simonis, Nonuniqueness of lattice Boltzmann schemes derived from finite difference methods. Ex. Counterex. 7 (2025) 100171. https://doi.org/10.1016/j.exco.2024.100171. [CrossRef] [Google Scholar]
  34. A. Kummerl¨ander, S. Avis, H. Kusumaatmaja, F. Bukreev, D. Dapelo, S. Großmann, N. Hafen, C. Holeksa, A. Husfeldt, J. Jeßberger, L. Kronberg, J. Marquardt, J. Mödl, J. Nguyen, T. Pertzel, S. Simonis, L. Springmann, N. Suntoyo, D. Teutscher, M. Zhong, and M. Krause, OpenLB Release 1.5: Open source lattice Boltzmann code, November (2022). https://doi.org/10.5281/zenodo.6469606. [Google Scholar]
  35. P. Lallemand and L.-S. Luo, Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability. Phys. Rev. E 61 (2000) 6546–6562. https://doi.org/10.1103/PhysRevE.61.6546. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  36. P. Lallemand, L.-S. Luo, M. Krafczyk, and W.-A. Yong, The lattice Boltzmann method for nearly incompressible flows. J. Comput. Phys. 431 (2021) 109713. https://doi.org/10.1016/j.jcp.2020.109713. [Google Scholar]
  37. E. Landau, Handbuch der Lehre von der Verteilung der Primazahlen. BG Teubner (1909). [Google Scholar]
  38. P.D. Lax and R.D. Richtmyer, Survey of the stability of linear finite difference equations. Commun. Pure Appl. Math. 9 (1956) 267–293. https://doi.org/10.1002/cpa.3160090206. [CrossRef] [Google Scholar]
  39. J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63 (1934) 193–248. https://doi.org/10.1007/BF02547354. [Google Scholar]
  40. R.J. LeVeque, Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems. SIAM (2007). https://doi.org/10.1137/1.9780898717839. [CrossRef] [Google Scholar]
  41. P.-A. Masset and G. Wissocq, Linear hydrodynamics and stability of the discrete velocity Boltzmann equations. J. Fluid Mech. 897 (2020) A29. https://doi.org/10.1017/jfm.2020.374. [CrossRef] [Google Scholar]
  42. A. Mink, K. Schediwy, C. Posten, H. Nirschl, S. Simonis, and M.J. Krause, Comprehensive computational model for coupled fluid flow, mass transfer, and light supply in tubular photobioreactors equipped with glass sponges. Energies 15 (2022) 7671. https://doi.org/10.3390/en15207671. [Google Scholar]
  43. S. Mischler, Uniqueness for the BGK-equation in RN and rate of convergence for a semi-discrete scheme. Diff. Integral Equ. 9 (1996) 1119–1138. https://doi.org/10.57262/die/1367871533. [Google Scholar]
  44. J.A. Murdock, Perturbations: Theory and Methods. SIAM (1999). [CrossRef] [Google Scholar]
  45. B. Perthame, Global existence to the BGK model of Boltzmann equation. J. Diff. Equ. 82 (1989) 191–205. https://doi.org/10.1016/0022-0396(89)90173-3. [CrossRef] [Google Scholar]
  46. B. Perthame and M. Pulvirenti, Weighted L∞ bounds and uniqueness for the Boltzmann BGK model. Arch. Ration. Mechan. Anal. 125 (1993) 289–295. https://doi.org/10.1007/BF00383223. [CrossRef] [Google Scholar]
  47. L. Saint-Raymond, Discrete time Navier-Stokes limit for the BGK Boltzmann equation. Commun. Partial Differ. Equ. 27 (2002) 149–184. https://doi.org/10.1081/PDE-120002785. [CrossRef] [Google Scholar]
  48. L. Saint-Raymond, From the BGK model to the Navier-Stokes equations. Ann. Sci. Ec. Norm. Super. 36 (2023) 271–317. https://doi.org/10.1016/S0012-9593(03)00010-7. [Google Scholar]
  49. S. Simonis, Lattice Boltzmann methods for partial differential equations. Doctoral thesis, Karlsruhe Institute of Technology (KIT) (2023). https://doi.org/10.5445/IR/1000161726; https://publikationen.bibliothek.kit.edu/1000161726. [Google Scholar]
  50. S. Simonis, N. Hafen, J. Jeßberger, D. Dapelo, G. Th¨ater, M. J. Krause, Homogenized lattice Boltzmann methods for fluid flow through porous media – Part I: Kinetic model derivation. ESAIM: M2AN 59 (2025) 789–813, 2025. https://doi.org/10.1051/m2an/2025005. [CrossRef] [EDP Sciences] [Google Scholar]
  51. S. Simonis and M. J. Krause, Forschungsnahe Lehre unter Pandemiebedingungen. Mitt. Deutsche Math. Ver. 30 (2022) 43–45. https://doi.org/10.1515/dmvm-2022-0015. [Google Scholar]
  52. S. Simonis and S. Mishra, Computing statistical Navier–Stokes solutions, in Hyperbolic balance laws: interplay between scales and randomness (Oberwolfach Report 21), Edited by R. Abgrall, M. Garavello, M. Lukáčová-Medvid’ová, and K. Trivisa (2024) 634–637. https://doi.org/10.4171/OWR/2024/10. [Google Scholar]
  53. S. Simonis, M. Frank, and M.J. Krause, On relaxation systems and their relation to discrete velocity Boltzmann models for scalar advection–diffusion equations. Philos. Trans. R. Soc. London. Ser. A 378 (2020) 20190400. https://doi.org/10.1098/rsta.2019.0400. [Google Scholar]
  54. S. Simonis, M. Haussmann, L. Kronberg, W. Dörfler, and M.J. Krause, Linear and brute force stability of orthogonal moment multiple-relaxation-time lattice Boltzmann methods applied to homogeneous isotropic turbulence. Philos. Trans. R. Soc. London. Ser. A 379 (2021) 20200405. https://doi.org/10.1098/rsta.2020.0405. [Google Scholar]
  55. S. Simonis, D. Oberle, M. Gaedtke, P. Jenny, and M. J. Krause, Temporal large eddy simulation with lattice Boltzmann methods. J. Comput. Phys. 454 (2022) 110991. https://doi.org/10.1016/j.jcp.2022.110991. [Google Scholar]
  56. S. Simonis, M. Frank, and M. J. Krause, Constructing relaxation systems for lattice Boltzmann methods. Appl. Math. Lett. 137 (2023) 108484. https://doi.org/10.1016/j.aml.2022.108484. [CrossRef] [Google Scholar]
  57. M. Siodlaczek, M. Gaedtke, S. Simonis, M. Schweiker, N. Homma, and M.J. Krause, Numerical evaluation of thermal comfort using a large eddy lattice Boltzmann method. Build. Environ. 192 (2021) 107618. https://doi.org/10. 1016/j.buildenv.2021.107618. [CrossRef] [Google Scholar]
  58. E. Tadmor, A review of numerical methods for nonlinear partial differential equations. Bull. Amer. Math. Soc. 49 (2012) 507–554. https://doi.org/10.1090/S0273-0979-2012-01379-4. [CrossRef] [MathSciNet] [Google Scholar]
  59. S. Ubertini, P. Asinari, and S. Succi, Three ways to lattice Boltzmann: A unified time-marching picture. Phys. Rev. E, 81 (2010) 016311. https://doi.org/10.1103/PhysRevE.81.016311. [CrossRef] [PubMed] [Google Scholar]
  60. G. Wissocq and P. Sagaut, Hydrodynamic limits and numerical errors of isothermal lattice Boltzmann schemes. J. Comput. Phys. 450 (2022) 110858. https://doi.org/10.1016/j.jcp.2021.110858. [CrossRef] [Google Scholar]
  61. D. A. Wolf-Gladrow, Lattice-Gas Cellular Automata and Lattice Boltzmann Models: An Introduction. Lecture Notes in Mathematics. Springer Berlin, Heidelberg (2000). https://doi.org/10.1007/b72010. [CrossRef] [Google Scholar]
  62. W. Zhao and W.-A. Yong, Maxwell iteration for the lattice Boltzmann method with diffusive scaling. Phys. Rev. E 95 (2017) 033311. https://doi.org/10.1103/PhysRevE.95.033311. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]

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