Summary
The purpose of this paper is to raise the profile of the Lattice Boltzmann method (LBM) as a computational method for solving fluid flow problems. We put forward the point of view that the method need not be seen as a discretisation of the Boltzmann equation, and also propose an alternative route from microscopic to macroscopic dynamics, traditionally taken via the Chapman-Enskog procedure. In that process the microscopic description is decomposed into processes at different time scales, parametrised with the Knudsen number. In our exposition we use the time step as a parameter for expanding the solution. This makes the treatment here more amenable to numerical analysts. We explain a method by which one may ameliorate the inevitable instabilities arising when trying to solve a convection- dominated problem, entropic filtering.
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References
G. Borg: A condition for the existence of orbitally stable solutions of dynamical systems. Kungl. Tekn. H¨ogsk. Handl. 153, 1960.
C. Chicone: Ordinary Differential Equations with Applications. Springer, 1999.
G.E. Fasshauer: Solving partial differential equations by collocation with radial basis functions. In Surface Fitting and Multiresolution Methods, A.L.M´ehaut´e, C. Rabut, and L. L. Schumaker (eds.), Vanderbilt University Press, 1997, 131–138.
A. Filippov: Differential Equations with Discontinuous Righthand Sides. Kluwer, 1988.
C. Franke and R. Schaback: Convergence order estimates of meshless collocation methods using radial basis functions. Adv. Comput. Math. 8, 1998, 381–399.
P. Giesl: The basin of attraction of periodic orbits in nonsmooth differential equations. ZAMM Z. Angew. Math. Mech. 85, 2005, 89–104.
P. Giesl: Necessary condition for the basin of attraction of a periodic orbit in non-smooth periodic systems. Discrete Contin. Dyn. Syst. 18, 2007, 355–373.
P. Giesl: Construction of Global Lyapunov Functions using Radial Basis Functions. Lecture Notes in Mathematics 1904, Springer-Verlag, Heidelberg, 2007.
P. Giesl: On the determination of the basin of attraction of discrete dynamical systems. J. Difference Equ. Appl. 13, 2007, 523–546.
P. Giesl and H.Wendland: Meshless collocation: error estimates with application to dynamical systems. SIAM J. Numer. Anal. 45, 2007, 1723–1741.
P. Giesl and H. Wendland: Approximating the basin of attraction of timeperiodic ODEs by meshless collocation. Discrete Contin. Dyn. Syst. 25, 2009, 1249–1274.
P. Giesl and H. Wendland: Approximating the basin of attraction of timeperiodic ODEs by meshless collocation of a Cauchy problem. Discrete Contin. Dyn. Syst. Supplement, 2009, 259–268.
Ph. Hartman: Ordinary Differential Equations. Wiley, New York, 1964.
B. Michaeli: Lyapunov-Exponenten bei nichtglatten dynamischen Systemen. PhD Thesis, University of K¨oln, 1999 (in German).
F.J. Narcowich and J.D. Ward: Generalized Hermite interpolation via matrixvalued conditionally positive definite functions. Math. Comput. 63, 1994, 661–687.
H. Wendland: Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv. Comput. Math. 4, 1995, 389–396.
H. Wendland: Error estimates for interpolation by compactly supported radial basis functions of minimal degree. J. Approx. Theory 93, 1998, 258–272.
H. Wendland: Scattered Data Approximation. Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, UK, 2005.
Z. Wu: Hermite-Birkhoff interpolation of scattered data by radial basis functions,Approximation Theory Appl. 8, 1992, 1–10
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Levesley, J., Gorban, A.N., Packwood, D. (2011). A Numerical Analyst’s View of the Lattice Boltzmann Method. In: Georgoulis, E., Iske, A., Levesley, J. (eds) Approximation Algorithms for Complex Systems. Springer Proceedings in Mathematics, vol 3. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16876-5_6
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DOI: https://doi.org/10.1007/978-3-642-16876-5_6
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