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On Triangular Lattice Boltzmann Schemes for Scalar Problems

Published online by Cambridge University Press:  03 June 2015

François Dubois  and
Pierre Lallemand
François Dubois*
Affiliation:
Conservatoire National des Arts et Métiers, Department of Mathematics, Paris, and Department of Mathematics, University Paris-Sud, Bât. 425, F-91405 Orsay Cedex, France
Pierre Lallemand
Affiliation:
Centre National de la Recherche Scientifique, Paris, France
*
Corresponding author.Email:[email protected]
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Abstract

We propose to extend the d’Humieres version of the lattice Boltzmann scheme to triangular meshes. We use Bravais lattices or more general lattices with the property that the degree of each internal vertex is supposed to be constant. On such meshes, it is possible to define the lattice Boltzmann scheme as a discrete particle method, without need of finite volume formulation or Delaunay-Voronoi hypothesis for the lattice. We test this idea for the heat equation and perform an asymptotic analysis with the Taylor expansion method for two schemes named D2T4 and D2T7. The results show a convergence up to second order accuracy and set new questions concerning a possible super-convergence.

Keywords

65-0565Q9982C20Laplacian operatorheat equationd’Humieres schemeD2T4D2T7
Type
Research Article
Information
Communications in Computational Physics , Volume 13 , Issue 3 , March 2013 , pp. 649 - 670
Copyright
Copyright © Global Science Press Limited 2013

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References

[1]Arnoldi, W.E.The principle of minimized iteration in the solution of the matrix eigenvalue problem, Quarterly of Applied Mathematics, Vol. 9, pp. 17–25, 1951. See the “Arpack” soft-wave at http://www.caam.rice.edu/software/ARPACK.Google Scholar
[2]Beale, J.T., Majda, A.Vortex methods I: Convergence in three dimensions, Vortex methods II: Higher order accuracy in two and three dimensions, Mathematics of Computation, Vol. 39, pp. 1-27 and 29–52, 1982.Google Scholar
[3]Bouzidi, M., Firdaouss, M., Lallemand, P.Momentum transfer of a Boltzmann-lattice fluid with boundaries, Physics of Fluids, Vol. 13, no 11, pp. 3452–3459, 2001.CrossRefGoogle Scholar
[4]Chang, S.C.A critical analysis of the modified equation technique of Warming and Hyett, Journal of Computational Physics, Vol. 86, pp. 107–126, 1990.CrossRefGoogle Scholar
[5]Chen, H.Volumetric formulation of the lattice Boltzmann method for fluid dynamics: Basic concept, Physical Review E, Vol. 58, pp. 3955–3963, 1998.Google Scholar
[6]Champier, S., Gallouet, T., Herbin, R.Convergence of an Upstream Finite Volume Scheme for a Nonlinear Hyperbolic Equation on a Triangular Mesh, Numerische Mathematik, Vol. 66, pp. 139–157, 1993.Google Scholar
[7]Ciarlet, P.G., Raviart, P.A.General Lagrange and Hermite interpolation in IR” with applications to finite element methods, Archive for Rational Mechanics and Analysis, Vol. 46, pp. 177–199, 1972.Google Scholar
[8]Cottet, G.H., Mas-Gallic, S.A particle method to solve the Navier-Stokes system, Numerische Mathematik, Vol. 57, pp. 805–827, 1990.Google Scholar
[9]d’Humières, D.Generalized Lattice-Boltzmann Equations, in Rarefied Gas Dynamics: Theory and Simulations, vol. 159 of AIAA Progress in Aeronautics and Astronautics, pp. 450–458, 1992.Google Scholar
[10]Dubois, F.Equivalent partial differential equations of a lattice Boltzmann scheme, Computers and Mathematics with Applications, Vol. 55, pp. 1441–1449, 2008.Google Scholar
[11]Dubois, F.Introduction au Schéma de Boltzmann sur Réseau, Master degree Lectures given at Université Paris-Sud, winter 2009-2010, unpublished. See http://www.math.u-psud.fr/~fdubois/cours/lbs-2010.html.Google Scholar
[12]Dubois, F., Lallemand, P.On lattice Boltzmann scheme, finite volumes and boundary conditions, Progress in Computational Fluid Dynamics, Vol. 8, pp. 11–24, DOI: 10.1504/PCFD.2008.018075, 2008.Google Scholar
[13]Dubois, F., Lallemand, P.Towards higher order lattice Boltzmann schemes, Journal of Statis-tical Mechanics: Theory and Experiment, P06006, doi: 10.1088/1742-5468/2009/06/P06006, 2009.Google Scholar
[14]Frisch, U., Hasslacher, B., Pomeau, Y., SLattice gas automata for the Navier Stokes equation, Physical Review Letters, Vol. 56, no 14, pp. 1505–1508, 1986.CrossRefGoogle ScholarPubMed
[15]George, P.L.Automatic Mesh Generation: Applications to Finite Element Methods, John Wiley & Sons, Inc. New York, USA, 1992.Google Scholar
[16]Ginzburg, I., Verhaeghe, F., d’Humieres, D.Two-relaxation-time lattice Boltzmann scheme: about parametrization, velocity, pressure and mixed boundary conditions, Communications in Computational Physics, Vol. 3, pp. 427–478, 2008.Google Scholar
[17]Griffiths, D., Sanz-Serna, J.On the scope of the method of modified equations, SIAM Journal on Scientific and Statistical Computing, Vol. 7, pp. 994–1008, 1986.Google Scholar
[18]Harlow, F.H., Ellington, M.A., Reid, J.H.The particle-in-cell computing method for fluid dynamics, Methods in Computational Physics, Vol. 3, (Alder, B., Fernbach, S., Rotenberg, M. Eds), pp. 319–343, Academic Press, New York, 1964.Google Scholar
[19]Hénon, M.Viscosity of a Lattice Gas, Complex Systems, Vol. 1, pp. 763–789, 1987.Google Scholar
[20]Junk, M., Klar, A., Luo, L.S.Asymptotic analysis of the lattice Boltzmann equation, Journal of Computational Physics, Vol. 120, pp. 676–704, 2005.Google Scholar
[21]Junk, M., Yong, W.A.Weighted L2-Stability of the Lattice Boltzmann Method, SIAM Journal on Numerical Analysis, Vol. 47, pp. 1651, 2009.CrossRefGoogle Scholar
[22]Karlin, I.V., Succi, S., Orszag, S.Lattice Boltzmann Method for Irregular Grids, Physical Review Letters, Vol. 82, pp. 5245–5248, 1999.Google Scholar
[23]Klales, A., Cianci, D., Needell, Z., Meyer, D.A., Love, P.J.Lattice gas simulations of dynamical geometry in two dimensions, Physical Review E, Vol. 82, pp. 046705, 2010.Google Scholar
[24]Lallemand, P., Luo, L.S.Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability, Physical Review E, Vol. 61, pp. 6546–6562, June 2000.Google Scholar
[25]Lerat, A., Peyret, R.Noncentered Schemes and Shock Propagation Problems, Computers and Fluids, Vol. 2, pp. 35–52, 1974.CrossRefGoogle Scholar
[26]Love, P.J., Cianci, D.From the Boltzmann equation to fluid mechanics on a manifold, Philosophical Transactions of the Royal Society A, Vol. 369, pp. 2362–2370, 2011.Google Scholar
[27]McCartin, B.J.Eigenstructure of the Equilateral Triangle, Part I: The Dirichlet Problem, SIAM Review, Vol. 45, pp. 267–287, 2003.CrossRefGoogle Scholar
[28]Mas Gallic, S., Raviart, P.A.A Particle Method for First order Symmetric Systems, Nu-merische Mathematik, Vol. 51, pp. 323–352, 1987.Google Scholar
[29]Peng, G., Xi, H., Duncan, C., Chou, S. H.A Finite Volume Scheme for the Lattice Boltzmann Method on Unstructured Meshes, Physical Review E, Vol. 59, pp. 4675–4682, 1999.Google Scholar
[30]Pontrelli, G., Ubertini, S., Succi, S.The unstructured lattice Boltzmann method for non-Newtonian flows, Journal of Statistical Mechanics: Theory and Experiment, P06005, doi: 10.1088/1742-5468/2009/06/P06005, 2009.Google Scholar
[31]Qian, Y., d’Humières, D., Lallemand, P., Lattice BGK Models for Navier-Stokes Equation, Europhysics Letters, Vol. 17, pp. 479–484, 1992.Google Scholar
[32]Raviart, P.A.An analysis of particle method, in Numerical Methods in Fluid Mechanics (Brezzi, F. Ed.), Lecture Notes in Mathematics, Vol. 1127, pp. 243–324, Springer Verlag, 1985.Google Scholar
[33]Roache, P.J.Computational Fluid Dynamics, Hermosa Publishers, Albuquerque, 1972.Google Scholar
[34]Sone, Y.Asymptotic theory of flow of rarefied gas over a smooth boundary I, in Rarefied Gas Dynamics, Trilling, L. and Wachmann, H.Y. Eds, Academic press, New York, pp. 243–253, 1969.Google Scholar
[35]Ubertini, S., Bella, G., Succi, S.Lattice Boltzmann method on unstructured grids: Further developments, Physical Review E, Vol. 68, 016701, 2003.Google Scholar
[36]Ubertini, S., Succi, S., Bella, G.Lattice Boltzmann schemes without coordinates, Philosophical Transactions of the Royal Society A, Vol. 362, pp. 1763–1771, 2004.Google Scholar
[37]van der Sman, R.G.M., Ernst, M.H.Diffusion Lattice Boltzmann scheme on an Orthorhombic Lattice, Journal of Statistical Physics, Vol. 94, pp. 203–217, 1999.CrossRefGoogle Scholar
[38]van der Sman, R.G.M., Ernst, M.H.Convection-Diffusion Lattice Boltzmann scheme for Ir-regular Lattices, Journal of Computational Physics, Vol. 160, pp. 1–17, 2000.Google Scholar
[39]van der Sman, R.G.M.Lattice Boltzmann scheme for Diffusion on Triangular Grids, Sloot, P.M.A.et al. (Eds), Lecture Notes in Computational Science, Vol. 2657, pp. 1072–1081, Springer-Verlag Berlin, Heidelberg, 2003Google Scholar
[40]van der Sman, R.G.M.Diffusion on unstructured triangular grids using Lattice Boltzmann, Future Generation Computer Systems, Vol. 20, pp. 965–971, 2004.CrossRefGoogle Scholar
[41]Vijayasundaram, G.Transonic flow simulations using an upstream centered scheme of Go-dunov in finite elements, Journal of Computational Physics, Vol. 63, pp. 416–433, 1986.CrossRefGoogle Scholar
[42]Warming, R.F., Hyett, B.J.The modified equation approach to the stability and accuracy analysis of finite difference methods, Journal of Computational Physics, Vol. 14, pp. 159–179, 1974.Google Scholar