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High order spectral difference lattice Boltzmann method for incompressible hydrodynamics. (English) Zbl 1378.76095
Summary: This work presents a lattice Boltzmann equation (LBE) based high order spectral difference method for incompressible flows. In the present method, the spectral difference (SD) method is adopted to discretize the convection and collision term of the LBE to obtain high order \((\geq 3)\) accuracy. Because the SD scheme represents the solution as cell local polynomials and the solution polynomials have good tensor-product property, the present spectral difference lattice Boltzmann method (SD-LBM) can be implemented on arbitrary unstructured quadrilateral meshes for effective and efficient treatment of complex geometries. Thanks to only first oder PDEs involved in the LBE, no special techniques, such as hybridizable discontinuous Galerkin method (HDG), local discontinuous Galerkin method (LDG) and so on, are needed to discrete diffusion term, and thus, it simplifies the algorithm and implementation of the high order spectral difference method for simulating viscous flows. The proposed SD-LBM is validated with four incompressible flow benchmarks in two-dimensions: (a) the Poiseuille flow driven by a constant body force; (b) the lid-driven cavity flow without singularity at the two top corners-Burggraf flow; and (c) the unsteady Taylor-Green vortex flow; (d) the Blasius boundary-layer flow past a flat plate. Computational results are compared with analytical solutions of these cases, and convergence studies of these cases are also given. The designed accuracy of the proposed SD-LBM is clearly verified.

76M28 Particle methods and lattice-gas methods
76M22 Spectral methods applied to problems in fluid mechanics
Full Text: DOI
[1] He, X.; Luo, L.-S., A priori derivation of the lattice Boltzmann equation, Phys. Rev. E, 55, R6333-R6336, (1997)
[2] He, X.; Luo, L.-S., Theory of lattice Boltzmann method: from the Boltzmann equation to the lattice Boltzmann equation, Phys. Rev. E, 56, 6811-6817, (1997)
[3] Cao, N.; Shen, S. Y.; Jin, S.; Matinez, D., Physic symmetry and lattice symmetry in the lattice Boltzmann method, Phys. Rev. E, 55, R21-R24, (1997)
[4] Guo, Z.; Zhao, T. S., Explicit finite-difference lattice Boltzmann method for curvilinear coordinates, Phys. Rev. E, 67, (2003)
[5] Reider, M.; Sterling, J., Accuracy of discrete-velocity BGK models for the simulation of the incompressible Navier-Stokes equations, Comput. Fluids, 24, 459-467, (1995) · Zbl 0845.76086
[6] Ubertini, S.; Bella, G.; Succi, S., Lattice Boltzmann method on unstructured grids: further developments, Phys. Rev. E, 68, (2003)
[7] Stiebler, M.; Tölke, J.; Krafczyk, M., An upwind discretization scheme for the finite volume lattice Boltzmann method, Comput. Fluids, 35, 814-919, (2006) · Zbl 1177.76329
[8] Patil, D.; Lakshmisha, K., Finite volume TVD formulation of lattice Boltzmann simulation on unstructured mesh, J. Comput. Phys., 228, 5262-5279, (2009) · Zbl 1280.76054
[9] W. Li, M. Kaneda, K. Suga, A stable, low diffusion up-wind scheme for unstructured finite volume lattice Boltzmann method, in: The 4th Asian Symposium on Computational Heat Transfer and Fluid Flow, June 3-6, Hong Kong, China, 2014.
[10] Li, W.; Luo, L.-S., Finite volume lattice Boltzmann method for nearly incompressible flows on arbitrary unstructured meshes, Commun. Comput. Phys., 20, 301-324, (2016) · Zbl 1373.76256
[11] Li, W.; Luo, L.-S., An implicit block LU-SGS finite-volume lattice-Boltzmann scheme for steady flows on arbitrary unstructured meshes, J. Comput. Phys., 327, 503-518, (2016) · Zbl 1373.76255
[12] Lee, T.; Lin, C.-L., A characteristic Galerkin method for discrete Boltzmann equation, J. Comput. Phys., 171, 336-356, (2001) · Zbl 1017.76043
[13] Lee, T.; Lin, C., An Eulerian description of the streaming process in the lattice Boltzmann equation, J. Comput. Phys., 185, 445-471, (2003) · Zbl 1047.76106
[14] Li, Y.; LeBoeuf, E. J.; Basu, P. K., Least-squares finite-element scheme for the lattice Boltzmann method on an unstructured mesh, Phys. Rev. E, 72, (2005)
[15] Guo, Z.; Xu, K.; Wang, R., Discrete unified gas kinetic scheme for all Knudsen number flows: low-speed isothermal case, Phys. Rev. E, 88, (2013)
[16] Guo, Z.; Wang, R.; Xu, K., Discrete unified gas kinetic scheme for all Knudsen number flows. II. thermal compressible case, Phys. Rev. E, 91, (2015)
[17] Zhu, L.; Wang, P.; Guo, Z., Performance evaluation of the general characteristics based off-lattice Boltzmann scheme and DUGKS for low speed continuum flows, J. Comput. Phys., (2016)
[18] Shi, X.; Lin, J.; Yu, Z., Discontinuous Galerkin spectral element lattice Boltzmann method on triangular element, Int. J. Numer. Methods Fluids, 42, 1249-1261, (2003) · Zbl 1033.76046
[19] Düster, A.; Demkowicz, L.; Rank, E., High-order finite elements applied to the discrete Boltzmann equation, Int. J. Numer. Methods Eng., 67, 1094-1121, (2006) · Zbl 1113.76049
[20] Min, M.; Lee, T., A spectral-element discontinuous Galerkin lattice Boltzmann method for nearly incompressible flows, J. Comput. Phys., 230, 245-259, (2011) · Zbl 1427.76189
[21] Zadehgol, A.; Ashrafizaadeh, M.; Musavi, S., A nodal discontinuous Galerkin lattice Boltzmann method for fluid flow problems, Comput. Fluids, 105, 58-65, (2014) · Zbl 1391.76646
[22] Hejranfar, K.; Ezzatneshan, E., Implementation of a high-order compact finite-difference lattice Boltzmann method in generalized curvilinear coordinates, J. Comput. Phys., 267, 28-49, (2014) · Zbl 1349.76475
[23] Hejranfar, K.; Hajihassanpour, M., Chebyshev collocation spectral lattice Boltzmann method for simulation of low-speed flows, Phys. Rev. E, 91, (2015), 25 pp · Zbl 1390.76719
[24] Burggraf, O. R., Analytical and numerical studies of the structure of steady separated flows, J. Fluid Mech., 24, 113-151, (1966)
[25] Green, A. E.; Taylor, G. I., Mechanism of the production of small eddies from larger ones, Proc. R. Soc. A, 158, 499-521, (1937) · JFM 63.1358.03
[26] Schlichting, H.; Gersten, K., Boundary-layer theory, (2000), Springer Netherlands
[27] Qian, Y.; d’Humières, D.; Lallemand, P., Lattice BGK models for Navier-Stokes equation, Europhys. Lett., 17, 479-484, (1992) · Zbl 1116.76419
[28] He, X.; Luo, L.-S., Lattice Boltzmann model for the incompressible Navier-Stokes equation, J. Stat. Phys., 88, 927-944, (1997) · Zbl 0939.82042
[29] Chapman, S.; Cowling, T. G., The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction and diffusion in gases, (1970), Cambridge University Press · JFM 65.1541.01
[30] Chorin, A. J., A numerical method for solving incompressible viscous flow problems, J. Comput. Phys., 2, 12-26, (1967) · Zbl 0149.44802
[31] Guo, Z.; Shu, C., Lattice Boltzmann method and its applications in engineering, Advances Computational Fluid Dynamics, vol. 3, (2013), World Scientific · Zbl 1278.76001
[32] Cockburn, B.; Gopalakrishnan, J.; Lazarov, R., Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal., 47, 1319-1365, (2009) · Zbl 1205.65312
[33] Nguyen, N.; Peraire, J.; Cockburn, B., A hybridizable discontinuous Galerkin method for the incompressible Navier-Stokes equations, (Proceedings of the 48th AIAA Aerospace Sciences Meeting and Exhibit, Orlando, Florida, (2010)), AIAA Paper 2010-362
[34] Nguyen, N. C.; Peraire, J.; Cockburn, B., An implicit high-order hybridizable discontinuous Galerkin method for the incompressible Navier-Stokes equations, J. Comput. Phys., 230, 1147-1170, (2011) · Zbl 1391.76353
[35] Cockburn, B., The hybridizable discontinuous Galerkin methods, (Proceedings of the International Congress of Mathematicians, vol. 4, (2014)), 2749-2775 · Zbl 1228.65221
[36] Bassi, F.; Rebay, S., A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J. Comput. Phys., 131, 267-279, (1997) · Zbl 0871.76040
[37] Cockburn, B.; Kanschat, G.; Schötzau, D.; Schwab, C., Local discontinuous Galerkin methods for the Stokes system, SIAM J. Numer. Anal., 40, 319-343, (2002) · Zbl 1032.65127
[38] Xu, Y.; Shu, C.-W., Local discontinuous Galerkin methods for high-order time-dependent partial differential equations, Commun. Comput. Phys., 7, 1-46, (2010) · Zbl 1364.65205
[39] Kopriva, D. A.; Kolias, J. H., A conservative staggered-grid Chebyshev multidomain method for compressible flow, (1995), Technical Report, DTIC Document · Zbl 0847.76069
[40] Kopriva, D. A.; Kolias, J. H., A conservative staggered-grid Chebyshev multidomain method for compressible flows, J. Comput. Phys., 125, 244-261, (1996) · Zbl 0847.76069
[41] Kopriva, D. A., A conservative staggered-grid Chebyshev multidomain method for compressible flows. II. A semi-structured method, J. Comput. Phys., 128, 475-488, (1996) · Zbl 0866.76064
[42] Kopriva, D. A., A staggered-grid multidomain spectral method for the compressible Navier-Stokes equations, J. Comput. Phys., 143, 125-158, (1998) · Zbl 0921.76121
[43] Sun, Y.; Wang, Z. J.; Liu, Y., High-order multidomain spectral difference method for the Navier-Stokes equations on unstructured hexahedral grids, Commun. Comput. Phys., 2, 310-333, (2007) · Zbl 1164.76360
[44] Cockburn, B., Discontinuous Galerkin methods, Z. Angew. Math. Mech., 83, 731-754, (2003) · Zbl 1036.65079
[45] Hesthaven, J. S.; Warburton, T., Nodal discontinuous Galerkin methods: algorithms, analysis, and applications, (2008), Springer-Verlag New York · Zbl 1134.65068
[46] Barth, T. J., Recent developments in high order k-exact reconstruction on unstructured meshes, (1993), 0668
[47] Li, W.; Ren, Y.-X., High-order k-exact WENO finite volume schemes for solving gas dynamic Euler equations on unstructured grids, Int. J. Numer. Methods Fluids, 70, 742-763, (2012)
[48] Wang, Z. J.; Fidkowski, K.; Abgrall, R.; Bassi, F.; Caraeni, D.; Cary, A.; Deconinck, H.; Hartmann, R.; Hillewaert, K.; Huynh, H.; Kroll, N.; May, G.; Persson, P.-O.; van Leer, B.; Visbal, M., High-order CFD methods: current status and perspective, Int. J. Numer. Methods Fluids, 72, 811-845, (2013)
[49] Titarev, V. A.; Toro, E. F., ADER: arbitrary high order Godunov approach, J. Sci. Comput., 17, 609-618, (2002) · Zbl 1024.76028
[50] Jameson, A., A proof of the stability of the spectral difference method for all orders of accuracy, J. Sci. Comput., 45, 348-358, (2010) · Zbl 1203.65198
[51] Liang, C.; Cox, C.; Plesniak, M., A comparison of computational efficiencies of spectral difference method and correction procedure via reconstruction, J. Comput. Phys., 239, 138-146, (2013)
[52] Toro, E. F., Riemann solvers and numerical methods for fluid dynamics: A practical introduction, (2013), Springer-Verlag Berlin, Heidelberg
[53] Roe, P. L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43, 357-372, (1981) · Zbl 0474.65066
[54] Cockburn, B.; Shu, C.-W., Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput., 16, 173-261, (2001) · Zbl 1065.76135
[55] Gottlieb, S.; Shu, C.-W., Total variation diminishing Runge-Kutta schemes, Math. Comput., 67, 73-85, (1998) · Zbl 0897.65058
[56] Shu, C.-W., Total-variation-diminishing time discretizations, SIAM J. Sci. Stat. Comput., 9, 1073-1084, (1988) · Zbl 0662.65081
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