zbMATH — the first resource for mathematics

A critical study of the compressible lattice Boltzmann methods for Riemann problem. (English) Zbl 1426.76610
Summary: The discrete velocity model proposed by Kataoka and Tsutahara (Phys. Rev. E 69(5):056702, 2004) for simulating inviscid flows is employed. Three approaches for improving the stability and the accuracy of this model, especially for high Mach numbers, are suggested and implemented in this research. First, the TVD scheme (Harten in J. Comput. Phys. 49:357-393, 1983) is used for space discretization of the convective term in the Lattice Boltzmann equation. Next, the modified Lax-Wendroff with artificial viscosity is employed to increase the robustness of the method in supersonic flows. Finally, a combination of TVD and the 2nd order derivative of the distribution function is employed using a differentiable switch. It is found that the recent technique is a more suitable approach for a wide range of Mach numbers. Moreover, the WENO scheme for space discretization has been applied and compared with these newly applied methods.
Reviewer: Reviewer (Berlin)

76M28 Particle methods and lattice-gas methods
76M20 Finite difference methods applied to problems in fluid mechanics
76L05 Shock waves and blast waves in fluid mechanics
Full Text: DOI
[1] Wolf-Gladrow, D.A.: Lattice-Gas Cellular Automata and Lattice Boltzmann Models–An Introduction. Springer, Berlin (2000) · Zbl 0999.82054
[2] Frisch, U., Hasslacher, B., Pomeau, Y.: Lattice gas automata for the Navier–Stokes equations. Phys. Rev. Lett. 56, 1505–1508 (1986) · doi:10.1103/PhysRevLett.56.1505
[3] Succi, S.: The Lattice Boltzmann Equation for Fluid Dynamics and Beyond. Oxford University Press, New York (2001) · Zbl 0990.76001
[4] Benzi, R., Succi, S., Vergassola, M.: The lattice Boltzmann equation: theory and applications. Phys. Rep. 222, 145 (1992) · doi:10.1016/0370-1573(92)90090-M
[5] McNamara, G., Zanetti, G.: Use of the Boltzmann equation to simulate lattice-gas automata. Phys. Rev. Lett. 61, 2332–2335 (1988) · doi:10.1103/PhysRevLett.61.2332
[6] Bhatnagar, P.L., Gross, E.P., Krook, M.: A model for collision processes in gases. I: small amplitude processes in charged and neutral one-component system. Phys. Rev. 94, 511–525 (1954) · Zbl 0055.23609 · doi:10.1103/PhysRev.94.511
[7] Chen, S., Doolen, G.D.: Lattice Boltzmann method for fluid flows. Ann. Rev. Fluid Mech. (1998) · Zbl 0919.76068
[8] Reider, M.B., Sterling, J.D.: Accuracy of discrete-velocity BGK models for the simulation of the incompressible Navier–Stokes equations. Comput. Fluids 24(4), 459–467 (1995) · Zbl 0845.76086 · doi:10.1016/0045-7930(94)00037-Y
[9] He, X.Y., Luo, L.S.: Theory of the lattice Boltzmann method: from the Boltzmann equation to the lattice Boltzmann equation. Phys. Rev. E 56(6), 6811–6817 (1997) · doi:10.1103/PhysRevE.56.6811
[10] Xiaoyi, H., Li-Shi, L.: Theory of the lattice Boltzmann method: from the Boltzmann equation to the lattice Boltzmann equation. Phys. Rev. E 56(6), 6811–6817 (1997) · doi:10.1103/PhysRevE.56.6811
[11] Tosi, F., Ubertini, S., Succi, S., Chen, H., Karlin, I.V.: Numerical stability of entropic versus positivity-enforcing lattice Boltzmann schemes. Math. Comput. Simul. 72, 227–231 (2006) · Zbl 1116.76068 · doi:10.1016/j.matcom.2006.05.007
[12] Li, Y., Shock, R., Zhang, R., Chen, H.: Numerical study of flow past an impulsively started cylinder by the lattice-Boltzmann method. J. Fluid Mech. 519, 273–300 (2004) · Zbl 1065.76166 · doi:10.1017/S0022112004001272
[13] Sofonea, V., Lamura, A., Gonnella, G., Cristea, A.: Finite-difference lattice Boltzmann model with flux limiters for liquid-vapor systems. Phys. Rev. E 70, 046702 (2004) · Zbl 1102.76053 · doi:10.1103/PhysRevE.70.046702
[14] Chen, F., Xu, A.G., Zhang, G.C., Li, Y.J.: Flux limiter lattice Boltzmann for compressible flows. Commun. Theor. Phys. 56, 333–338 (2011) · Zbl 1247.76068 · doi:10.1088/0253-6102/56/2/25
[15] Gan, Y.B., Xu, A.G., Zhang, G.C., Li, Y.J.: Flux limiter lattice Boltzmann scheme approach to compressible flows with flexible specific-heat ratio and Prandtl number. Commun. Theor. Phys. 56, 490–498 (2011) · Zbl 1247.76069 · doi:10.1088/0253-6102/56/3/18
[16] Brownlee, R.A., Gorban, A.N., Levesley, J.: Stability and stabilization of the lattice Boltzmann method. Phys. Rev. E 75, 036711 (2007) · Zbl 1121.76049 · doi:10.1103/PhysRevE.75.036711
[17] Watari, M.: Finite difference lattice Boltzmann method with arbitrary specific heat ratio applicable to supersonic flow simulations. Physica A 382, 502–522 (2007) · doi:10.1016/j.physa.2007.03.037
[18] Alexander, F.J., Chen, H., Chen, S., Doolen, G.D.: Lattice Boltzmann model for compressible fluids. Phys. Rev. A 1992, 46 (1967–1970)
[19] Kim, C., Xu, K., Martinelli, L., Jameson, A.: Analysis and implementation of the gas kinetic BGK scheme for computing inhomogeneous fluid behavior. Int. J. Numer. Methods Fluids 25, 21–49 (1997) · Zbl 0883.76056 · doi:10.1002/(SICI)1097-0363(19970715)25:1<21::AID-FLD515>3.0.CO;2-Y
[20] Kotelnikov, A.D., Montgomery, D.C.: A kinetic method for computing inhomogeneous fluid behavior. J. Comput. Phys. 134, 364–388 (1997) · Zbl 0887.76053 · doi:10.1006/jcph.1997.5720
[21] Shouxin, H., Guangwu, Y., Weiping, S.: A lattice Boltzmann model for compressible perfect gas. Acta Mech. Sin., Engl. Ser. 13(3) (1997) · Zbl 0988.76511
[22] Renda, A., Bella, G., Succi, S., Karlin, I.V.: Thermo hydrodynamics lattice BGK schemes with non-perturbative equilibrium. Europhys. Lett. 41, 279–283 (1998) · doi:10.1209/epl/i1998-00143-x
[23] Vahala, G., Pavlo, P., Vahala, L., Martys, N.S.: Thermal lattice Boltzmann models (TLBM) for compressible flows. Int. J. Mod. Phys. C 9, 1247–1261 (1998) · doi:10.1142/S0129183198001126
[24] Yan, G.W., Chen, Y.S., Hu, S.X.: Simple lattice Boltzmann method for simulating flows with shock wave. Phys. Rev. E 59, 454 (1999) · doi:10.1103/PhysRevE.59.454
[25] De Cicco, M., Succi, S., Bella, G.: Nonlinear stability of compressible thermal lattice BGK model. SIAM J. Sci. Comput. 21, 366–377 (1999) · Zbl 0959.76073 · doi:10.1137/S1064827597319805
[26] Palmer, B.J., Recto, D.R.: Lattice Boltzmann algorithm for simulating thermal flow in compressible fluids. J. Comput. Phys. 161, 1–20 (2000) · Zbl 0969.76075 · doi:10.1006/jcph.2000.6425
[27] Yan, G.W., Song, M.: Recovery of the solutions using a lattice Boltzmann model. Chin. Phys. Lett. 16, 109–110 (1999) · doi:10.1088/0256-307X/16/2/012
[28] Sun, C.: Adaptive lattice Boltzmann model for compressible flows: viscous and conductive properties. Phys. Rev. E 61, 2645–2653 (2000) · doi:10.1103/PhysRevE.61.2645
[29] Sun, C., Hsu, A.T.: Three-dimensional lattice Boltzmann model for compressible flows. Phys. Rev. E 68, 016303 (2003) · doi:10.1103/PhysRevE.68.016303
[30] Sun, C., Hsu, A.: Multi-level lattice Boltzmann model on square lattice for compressible flows. Comput. Fluids 33, 1363–1385 (2004) · Zbl 1113.76435 · doi:10.1016/j.compfluid.2003.12.001
[31] Mason, R.J.: A multi-speed compressible lattice-Boltzmann model. J. Stat. Phys. 107(1/2) (2005) · Zbl 1126.76356
[32] Yan, G., Dong, Y., Liu, Y.: An implicit Lagrangian lattice Boltzmann method for the compressible flows. Int. J. Numer. Methods Fluids 51, 1407–1418 (2006) · Zbl 1158.76418 · doi:10.1002/fld.1170
[33] Yan, G., Zhang, J., Liu, Y., Dong, Y.: A multi-energy-level lattice Boltzmann model for the compressible Navier–Stokes equations. Int. J. Numer. Methods Fluids 55, 41–56 (2007) · Zbl 1119.76049 · doi:10.1002/fld.1440
[34] Yan, G., Zhang, J.: A multi-entropy-level lattice Boltzmann model for the one-dimensional compressible Euler equations. Int. J. Comput. Fluid Dyn. 22(6), 383–392 (2008) · Zbl 1184.76801 · doi:10.1080/10618560802119673
[35] Zhang, J., Yan, G., Shi, X., Dong, Y.: A lattice Boltzmann model for the compressible Euler equations with second-order accuracy. Int. J. Numer. Methods Fluids 60, 95–117 (2009) · Zbl 1159.76040 · doi:10.1002/fld.1883
[36] Ji, C.Z., Shu, C., Zhao, N.: A lattice Boltzmann method-based flux solver and its application to solve shock tube problem. Mod. Phys. Lett. B 23, 313–316 (2009) · Zbl 1419.76520 · doi:10.1142/S021798490901828X
[37] Xu, A.: Two-dimensional finite-difference lattice Boltzmann method for the complete Navier-Stokes equations of binary fluids. Europhys. Lett. 69, 214 (2005) · doi:10.1209/epl/i2004-10334-y
[38] Xu, A.: Finite-difference lattice-Boltzmann methods for binary fluids. Phys. Rev. E 71, 066706 (2005)
[39] Xu, A.: Two-dimensional lattice Boltzmann methods based on Sirovich’s kinetic theory. Prog. Theor. Phys. 162, 197 (2006) · Zbl 1104.82043
[40] Watari, M., Tsutahara, M.: Two-dimensional thermal model of the finite-difference lattice Boltzmann method with high spatial isotropy. Phys. Rev. E 67, 036306 (2003) · doi:10.1103/PhysRevE.67.036306
[41] Kataoka, T., Tsutahara, M.: Lattice Boltzmann method for the compressible Navier–Stokes equations with flexible specific-heat ratio. Phys. Rev. E 69(3), 035701 (2004)
[42] Kataoka, T., Tsutahara, M.: Lattice Boltzmann method for the compressible Euler equations. Phys. Rev. E 69(5), 056702 (2004)
[43] Watari, M., Tsutahara, M.: Supersonic flow simulations by a three-dimensional multispeed thermal model of the finite difference lattice Boltzmann method. Physica A 364, 129–144 (2006) · doi:10.1016/j.physa.2005.06.103
[44] Gan, Y., Xu, A., Zhang, G., Yu, X., Li, Y.: Two-dimensional lattice Boltzmann model for compressible flows with high Mach number. Physica A 387, 1721–1732 (2008) · doi:10.1016/j.physa.2007.11.013
[45] Laney, C.B.: Computational Gasdynamics. Cambridge University Press, Cambridge (1998) · Zbl 0947.76001
[46] Pan, X.F., Xu, A., Zhang, G., Jiang, S.: Lattice Boltzmann approach to high-speed compressible flows. Int. J. Mod. Phys. C 3, 14 (2008) · Zbl 1170.76345
[47] Chen, F., Xu, A.G., Zhang, G.C., Gan, Y.B., Cheng, T., Li, Y.J.: Highly efficient lattice Boltzmann model for compressible fluids: two-dimensional case. Commun. Theor. Phys. 52, 681–693 (2009) (Beijing, China) · Zbl 1253.76114 · doi:10.1088/0253-6102/52/4/25
[48] Wang, Y., He, Y.L., Zhao, T.S., Tang, G.H., Tao, W.Q.: Implicit-explicit finite-difference lattice Boltzmann method for compressible flows. Int. J. Mod. Phys. C 18(12), 1961–1983 (2007) · Zbl 1151.82405 · doi:10.1142/S0129183107011868
[49] Zhang, H.X.: Non-oscillatory and non-free-parameter dissipation difference scheme. Acta Aerodyn. Sin. 6, 143–165 (1988)
[50] Pieraccini, S., Puppo, G.: Implicit explicit schemes for BGK kinetic equations. J. Sci. Comput. 32, 1 (2007) · Zbl 1115.76057 · doi:10.1007/s10915-006-9116-6
[51] Li, Q., He, Y.L., Wang, Y., Tao, W.Q.: Coupled double-distribution-function lattice Boltzmann method for the compressible Navier-Stokes equations. Phys. Rev. E 76, 056705 (2007)
[52] Qu, K., Shu, C., Chew, Y.T.: Alternative method to construct equilibrium distribution functions in lattice-Boltzmann method simulation of inviscid compressible flows at high Mach number. Phys. Rev. E 75, 036706 (2007)
[53] He, Y.L., Wang, Y., Li, Q., Tao, W.Q.: Simulating compressible flows with shock waves using the finite-difference lattice Boltzmann method. Prog. Comput. Fluid Dyn. 9, 3–5 (2009)
[54] Chen, F., Xu, A., Zhang, G., Li, Y.: Multiple-relaxation-time lattice Boltzmann model for compressible fluids. Phys. Lett. A 375, 2129–2139 (2011) · doi:10.1016/j.physleta.2011.04.013
[55] Harten, A.: High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49, 357–393 (1983) · Zbl 0565.65050 · doi:10.1016/0021-9991(83)90136-5
[56] Liu, R., Shu, Q.: Some New Methods in Computational Fluid Dynamics. Science Press, Beijing (2003) (in Chinese)
[57] Nejat, A., Ollivier-Gooch, C.: A high-order accurate unstructured finite volume Newton-Krylov algorithm for inviscid compressible flows. J. Comput. Phys. 227(4), 2582–2609 (2008) · Zbl 1388.76183 · doi:10.1016/j.jcp.2007.11.011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.