A pseudo-kinetic approach for Helmholtz equation

Abstract

A lattice Boltzmann type pseudo-kineticmodel for a non-homogeneous Helmholtz equation is derived in this paper. Numerical results for some model problems show the robustness and efficiency of this lattice Boltzmann type pseudo-kinetic scheme. The computation at each site is determined only by local parameters, and can be easily adapted to solve multiple scattering problems with many scatterers or wave propagation in nonhomogeneous medium without increasing the computational cost.

Keywords

Lattice Boltzmann scheme / Non-homogeneous Helmholtz equation / Discrete velocity

References

[1] Bayliss A, Goldstein C I, Turkel E. An iterative method for the Helmholtz equation. J. Comput. Phys., 1983, 49: 443-457

[2] Benzi R, Succi S, Vergassola M. The lattice Boltzmann equations: theory and applications. Phys. Rep., 1992, 222: 147-197

[3] Chopard B, Luthi P O, Wagen J F. Lattice Boltzmann method for wave propagation in urban microcells. IEEE Proc. Microw. Antennas Propag., 1997, 144(4): 251-255

[4] Clayton R, Engquist B. Absorbing boundary conditions for acoustic and elastic wave equations. Bull. Seis. Soc. America, 1977, 67(6): 1529-1540

[5] Clayton R, Engquist B. Absorbing boundary conditions for wave-equation migration. Geophysics, 1980, 45(5): 895-904

[6] Collins M D, Kuperman W A. Inverse problems in ocean acoustics. Inverse Problems, 1994, 10: 1023-1040

[7] Egorov Y V, Shubin M A. Foundations of the Classical Theory of Partial Differential Equations (Translated by R. Cooke), 2001, New York: Springer-Verlag

[8] Engquist B, Majda A. Absorbing boundary conditions for the numerical simulation of waves. Math. Comput., 1977, 31: 629-651

[9] Fries T P, Belytschko T. The extended/generalized finite element method: An overview of the method and its applications. Int. J. Numer. Meth. Engng, 2010, 84: 253-304

[10] Giladi E, Keller J B. A hybrid numerical asymptotic method for scattering problems. J. Comput. Phys., 2001, 174: 226-247

[11] Givoli D, Patlashenko I. Optimal local non-reflecting boundary conditions. Appl. Numer. Math., 1998, 27: 367-384

[12] He X, Luo L S. Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation. Phys. Rev. E, 1997, 56: 6811-6817

[13] Higdon R L. Numerical absorbing boundary conditions for the wave equation. Math. Comp., 1987, 49(179): 65-90

[14] Ishimaru A. Theory and application of wave propagation and scattering in random media. Proceedings of the IEEE, 1977, 65: 1030-1061

[15] Ishimaru A. Wave Propagation and Scattering in Random Media, Series on Electromagnetic Wave Theory, 1997, New York: IEEE Press

[16] Lindman E L. Free space boundary conditions for time dependent wave equation. J. Comput. Phys., 1975, 18: 66-78

[17] Luthi P O. Lattice Wave Automata: From Radio Waves to Fractures Propagation, 1997

[18] Majda A, Osher S. Reflection of singularities at the boundary. Comm. Pure Appl. Math., 1975, 28: 277-298

[19] Martin P A. Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles, 2006, Cambridge: Cambridge University Press

[20] Obrecht C, Kuznik F, Tourancheau B, Roux J. Multi-GPU implementation of the lattice Boltzmann method. Comp. Math. Appl., 2013, 65(2): 252-261

[21] Perthame B, Vega L. Energy concentration and Sommerfeld condition for Helmholtz and Liouville equations. C. R. Acad. Sci. Paris, Ser. I, 2003, 337: 587-592

[22] Perthame B, Vega L. Morrey-Campanato estimates for Helmholtz equations. J. Funct. Anal., 1999, 164: 340-355

[23] Riegel E, Indinger T, Adams N A. Implementation of a lattice Boltzmann method for numerical fluid mechanics using the nVIDIA CUDA technology. Computer Science-Research and Development, 2009, 23: 241-247

[24] Sommerfeld A. Partial Differential Equations in Physics, 1964, New York: Academic Press

[25] Strouboulis T, Babuskab I, Hidajat R. The generalized finite element method for Helmholtz equation: Theory, computation, and open problems. Computer Methods in Applied Mechanics and Engineering, 2006, 195: 4711-4731

[26] Succi S. The Lattice Boltzmann Equation for Fluid Dynamics and Beyond, 2001, Oxford: Oxford University Press

[27] Taylor M E. Partial differential equations II: Qualitative studies of linear equations, 1996, New York: Springer-Verlag

[28] Tikhonov A N, Samarskii A A. Equations of Mathematical Physics, 1990, New York: Dover Publ.

[29] Tolke J. Implementation of a lattice Boltzmann kernel using the compute unified device architecture developed by nVIDIA. Comput. Visual. Sci., 2010, 13: 29-39

[30] Tubbs K, Tsai F. GPU accelerated lattice Boltzmann model for shallow water flow and mass transport. Int. J. Numer. Meth. Engng., 2011, 86: 316-334

[31] Yan G. A lattice Boltzmann equation for waves. J. Comput. Phys., 2000, 161: 61-69

[32] Zhang J, Yan G, Dong Y. A new lattice Boltzmann model for the Laplace equation. Appl. Math. Comput., 2009, 215: 539-547