A higher-order moment method of the lattice Boltzmann model for the conservation law equation. (English) Zbl 1185.76845
Summary: We propose a higher-order moment method in the lattice Boltzmann model for the conservation law equation. In contrast to the lattice Bhatnagar-Gross-Krook (BGK) model, the higher-order moment method has a wide flexibility to select equilibrium distribution function. This method is based on so-called a series of partial differential equations obtained by using multi-scale technique and Chapman-Enskog expansion. According to Hirt’s heuristic stability theory, the stability of the scheme can be controlled by modulating some special moments to design the third-order dispersion term and the fourth-order dissipation term. As results, the conservation law equation is recovered with higher-order truncation error. The numerical examples show the higher-order moment method can be used to raise the accuracy of the truncation error of the lattice Boltzmann scheme for the conservation law equation.
MSC:
| 76M28 | Particle methods and lattice-gas methods |
| 35L67 | Shocks and singularities for hyperbolic equations |
| 35Q20 | Boltzmann equations |
References:
| [1] | Frisch, U.; Hasslacher, B.; Pomeau, Y., Lattice gas automata for the Navier-Stokes equations, Phys. Rev. Lett., 56, 1505-1508 (1986) |
| [2] | Qian, Y. H.; d’humieres, D.; Lallemand, P., Lattice BGK model for Navier-Stokes equations, Europhys. Lett., 17, 6, 479-484 (1992) · Zbl 1116.76419 |
| [3] | Chen, H. D.; Chen, S. Y.; Matthaeus, M. H., Recovery of the Navier-Stokes equations using a lattice Boltzmann gas method, Phys. Rev. A, 45, 5339-5342 (1992) |
| [4] | Benzi, R.; Succi, S.; Vergassola, M., The lattice Boltzmann equations: theory and applications, Phys. Rep., 222, 147-197 (1992) |
| [5] | Chen, S. Y.; Doolen, G. D., Lattice Boltzmann method for fluid flows, Annu. Fluid Mech., 3, 314-322 (1998) |
| [6] | Luo, L.-S., Theory of the lattice Boltzmann method: lattice Boltzmann method for nonideal gases, Phys. Rev. E, 62, 4982 (2000) |
| [7] | Shan, X. W.; Chen, H. D., Lattice Boltzmann model of simulating flows with multiple phases and components, Phys. Rev. E, 47, 1815 (1993) |
| [8] | Shan, X. W.; Chen, H. D., Simulation of non-ideal gases liquid-gas phase transitions by the lattice Boltzmann equation, Phys. Rev. E, 49, 2941 (1994) |
| [9] | Swift, M.; Osborn, W.; Yeomans, J., Lattice Boltzmann simulation of nonideal fluids, Phys. Rev. Lett., 75, 830 (1995) |
| [10] | Gustensen, K.; Rothman, D. H.; Zaleski, S., Lattice Boltzmann model of immiscible fluids, Phys. Rev. A, 43, 4320-4327 (1991) |
| [11] | Premnath, K. N.; Abraham, J., Three-dimensional multi-relaxation lattice Boltzmann models for multiphase flows, J. Comput. Phys., 224, 539-559 (2007) · Zbl 1116.76066 |
| [12] | Dawson, S. P.; Chen, S. Y.; Doolen, G., lattice Boltzmann computations for reaction-diffusion equations, J. Chem. Phys., 98, 1514-1523 (1993) |
| [13] | Holdych, D. J.; Georgiadis Buckius, R. O., Magration of a van der Waals bubble: lattice Boltzmann formulation, Phys. Fluids, 13, 817 (2001) · Zbl 1184.76228 |
| [14] | Swift, M. R.; Orlandini, E.; Osborn, W. R., Lattice Boltzmann simulations of liquid-gas and binary systems, Phys. Rev. E, 54, 5041 (1996) |
| [15] | Maier, R. S.; Bernard, R. S.; Grunau, D. W., Boundary conditions for the lattice Boltzmann method, Phys. Fluids, 6, 1788 (1996) · Zbl 1027.76632 |
| [16] | Succi, S.; Foti, E.; Higuera, F. J., 3-Dimensional flows in complex geometries with the lattice Boltzmann method, Europhys. Lett., 10, 433 (1989) |
| [17] | Ladd, A., Numerical simulations of particle suspensions via a discretized Boltzmann equation. Part 2. Numerical results, J. Fluids Mech., 271, 311 (1994) |
| [18] | Filippova, O.; Hanel, D., Lattice Boltzmann simulation of gas-particle flow in filtes, Comput. Fluids, 26, 697-712 (1997) · Zbl 0902.76077 |
| [19] | Yan, G. W.; Chen, Y. S.; Hu, S. X., Simple lattice Boltzmann model for simulating flows with shock wave, Phys. Rev. E, 59, 454-459 (1999) |
| [20] | Sun, C. H., Lattice-Boltzmann model for high speed flows, Phys. Rev. E, 58, 7283-7287 (1998) |
| [21] | Alexander, F. J.; Chen, H.; Chen, S., Lattice Boltzmann model for compressible fluids, Phys. Rev. A, 46, 1967-1970 (1992) |
| [22] | De Cicco, M.; Succi, S.; Balla, G., Nonlinear stability of compressible thermal lattice BGK model, SIAM J. Sci. Comput., 21, 366-377 (1999) · Zbl 0959.76073 |
| [23] | Mason, R. J., A compressible lattice Boltzmann model, Bull. Am. Phys. Soc., 45, 168-170 (2000) |
| [24] | Mason, R. J., A multi-speed compressible lattice Boltzmann model, J. Stat. Phys., 107, 385-400 (2002) · Zbl 1126.76356 |
| [25] | Yan, G. W.; Dong, Y. F.; Liu, Y. H., An implicit Lagrangian lattice Boltzmann method for the compressible flows, Int. J. Numer. Methods Fluids, 51, 1407-1418 (2006) · Zbl 1158.76418 |
| [26] | Kataoka, T.; Tsutahara, M., Lattice Boltzmann method for the compressible Euler equations, Phys. Rev. E, 69, 5 (2004), Art No. 056702 |
| [27] | Kataoka, T.; Tsutahara, M., Lattice Boltzmann method for the compressible Navier-Stokes equations with flexible specific-heat ratio, Phys. Rev. E, 69, 3 (2004), Art No. 035701 |
| [28] | Yan, G. W.; Zhang, J. Y.; Liu, Y. H., 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 |
| [29] | Yan, G. W., A lattice Boltzmann equation for waves, J. Comput. Phys., 161, 61-69 (2000) · Zbl 0969.76076 |
| [30] | Yan, G. W., Studies of Burgers equation using a lattice Boltzmann method, Acta Mech. Sinica, 31, 143-151 (1999), (in Chinese) |
| [31] | Yan, G. W.; Song, M., Recovery of the solitons using a lattice Boltzmann model, Chinese Phys. Lett., 16, 109-110 (1999) |
| [32] | Yan, G. W.; Yuan, L., Lattice Bhatnagar-Gross-Krook model for the Lorenz attractor, Physica D, 154, 43-50 (2001) · Zbl 1015.76068 |
| [33] | Zhou, J. G., Lattice Boltzmann Methods for Shallow Water Flows (2000), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, Germany |
| [34] | Ginzburg, I., Variably saturated flow described with the anisotropic lattice Boltzmann methods, J. Comput. Fluids, 25, 831-848 (2006) · Zbl 1177.76314 |
| [35] | Chai, Z. H.; Shi, B. C., A novel lattice Boltzmann model for the Poisson equation, Appl. Math. Modell., 32, 2050-2058 (2008) · Zbl 1145.82344 |
| [36] | Succi, S., Numerical solution of the Schrödinger equation using discrete kinetic theory, Phys. Rev. E, 53, 1969-1975 (1996) |
| [37] | Succi, S., Lattice quantum mechanics: an application to Bose-Einstein condensation, Int. J. Mod. Phys. C, 9, 1577-1585 (1998) |
| [38] | Palpacelli, S.; Succi, S., Numerical validation of the quantum lattice Boltzmann scheme in two and three dimension, Phys. Rev. E, 75, 066704 (2007) |
| [39] | Palpacelli, S.; Succi, S.; Spiggler, R., Ground-state computation of Bose-Einstein condensates by an imaginary-time quantum lattice Boltzmann scheme, Phys. Rev. E, 76, 036712 (2007) |
| [40] | Zhong, L. H.; Feng, S. D.; Dong, P.; Gao, S. T., Lattice Boltzmann schemes for the nonlinear Schrödinger equation, Phys. Rev. E, 74, 036704 (2006) |
| [41] | Frisch, U.; d’Humières, D.; Hasslacher, B.; Lallemand, P.; Pomeau, Y.; Rivet, J. P., Lattice gas hydrodynamics in two and three dimensions, Complex Sys., 1, 649-707 (1987) · Zbl 0662.76101 |
| [42] | d’Humières, D., Generalized lattice-Boltzmann equations, AIAA rarefied gas dynamics: theory and simulations, Prog. Astronaut. Aeronaut., 59, 450-548 (1992) |
| [43] | Yan, G. W.; Chen, Y. S.; Hu, S. X., A lattice Boltzmann method for KdV equation, Acta Mech. Sinica, 14, 18-26 (1998) |
| [44] | Ginzburg, I., Equilibrium-type and link-type lattice Boltzmann model for generic advection and anisotropic-dispersion equation, Adv. Water Resour., 28, 1171-1195 (2005) |
| [45] | Lallemand, P.; Luo, L. S., Theory of the lattice Boltzmann method: dispersion, dissipation, isotropy, Galilean invariance and stability, Phys. Rev. E, 61, 6546-6562 (2000) |
| [46] | d’Humières, D.; Ginzburg, I.; Krafczyk, M.; Lallemand, P.; Luo, L. S., Multiple-relaxation time lattice Boltzmann models in three-dimension, Phil. Trans. Royal Soc. Lond. A, 360, 437-451 (2002) · Zbl 1001.76081 |
| [47] | Chapman, S.; Cowling, T. G., The Mathematical Theory of Non-uniform Gas (1970), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0098.39702 |
| [48] | Holdych, D. J.; Noble, D. R.; Georgiadis, J. G., Truncation error analysis of lattice Boltzmann methods, J. Comput. Phys., 193, 595-619 (2004) · Zbl 1040.76052 |
| [49] | Shu, C. W.; Osher, Stanley, efficient implementation of essentially non-oscillatory shock capturing schemes, J. Comput. Phys., 77, 439-471 (1988) · Zbl 0653.65072 |
| [50] | Shu, C. W.; Osher, Stanley, efficient implementation of essentially non-oscillatory shock capturing schemes. II., J. Comput. Phys., 83, 32-78 (1989) · Zbl 0674.65061 |
| [51] | Harten, A., ENO schemes with subcell resolution, J. Comput. Phys., 83, 148-184 (1989) · Zbl 0696.65078 |
| [52] | Nessyahu, H.; Tadmor, E., Non-oscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys., 87, 408-419 (1990) · Zbl 0697.65068 |
| [53] | Hirt, C. W., Heuristic stability theory for finite-difference equations, J. Comput. Phys., 2, 339-355 (1968) · Zbl 0187.12101 |
| [54] | Wang, J.; Liu, R. X., A new approach to design high-order schemes, J. Comput. Appl. Math., 134, 59-67 (2001) · Zbl 0985.65104 |
| [55] | Yang, H., An artificial compression method for ENO schemes: the slope modification method, J. Comput. Phys., 89, 125-160 (1990) · Zbl 0705.65062 |
| [56] | Frisch, U., Relation between the lattice Boltzmann equation and the Navier-Stokes Equation, Physica D, 47, 231-232 (1991) |
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