Stability analysis of the lattice Boltzmann schemes for solving the diffusion equation

Authors

  • G.V. Krivovichev

Keywords:

lattice Boltzmann method
linear diffusion equation
stability with respect to initial conditions
von Neumann method

Abstract

The one-parameter families of lattice Boltzmann schemes for solving the linear diffusion equation in the cases of D2Q5, D2Q7 and D2Q9 velocity sets are considered. The comparison of various schemes proposed in previous studies is performed. The stability analysis of schemes is performed in the space of parameters. The stability with respect to initial conditions is studied by the von Neumann method. The optimal parameter values for which the absolute values of the largest-in-magnitude eigenvalues of the transition matrix are minimal are found.

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Published

2013-04-05

Issue

Vol. 14 (2013): Issue 1.

Section

Section 1. Numerical methods and applications

Author Biography

G.V. Krivovichev

St Petersburg University,
Faculty of Applied Mathematics and Control Processes
• Associate Professor

References

  1. Wolf-Gladrow D.A. Lattice-gas cellular automata and lattice Boltzmann models - an introduction. Berlin: Springer-Verlag, 2005.
  2. Succi S. The Lattice Boltzmann equation for fluid dynamics and beyond. Oxford: Oxford Clarendon, 2001.
  3. Chen S., Doolen G.D. Lattice Boltzmann method for fluid flows // Annual Review of Fluid Mechanics. 1998. 30. 329-364.
  4. Nourgaliev R.R., Dinh T.N., Theofanous T.G., Joseph D. The lattice Boltzmann equation method: theoretical interpretation, numerics and implications // Int. J. of Multiphase Flow. 2003. 29. 117-169.
  5. Dellar P.G. Lattice kinetic schemes for magnetohydrodynamics // J. of Computational Physics. 2002. 179. 95-126.
  6. Dellar P.G. Moment equations for magnetohydrodynamics // J. of Statistical Mechanics: Theory and Experiment. 2009. 9. 06003-060025.
  7. Kupershtokh A.L. A lattice Boltzmann equation method for real fluids with the equation of state known in tabular form only in regions of liquid and vapor phases // Computers and Mathematics with Applications. 2011. 61. 3537-3548.
  8. Kupershtokh A.L. Criterion of numerical instability of liquid state in LBE simulations // Computers and Mathematics with Applications. 2010. 59. 2236-2245.
  9. Куперштох А.Л. Трехмерное моделирование двухфазных систем типа жидкость-пар методом решеточных уравнений Больцмана на GPU // Вычислительные методы и программирование. 2012. 13. 130-138.
  10. Куперштох А.Л. Трехмерное моделирование методом LBE на гибридных GPU-кластерах распада бинарной смеси жидкого диэлектрика с растворенным газом на систему парогазовых каналов // Вычислительные методы и программирование. 2012. 13. 384-390.
  11. Zhao Z., Huang P., Li Y., Li J. A lattice Boltzmann method for viscous free surface waves in two dimensions // Int. J. for Numerical Methods in Fluids. 2013. 71. 223-248.
  12. Guo Z., Zhao T.S. Lattice Boltzmann model for incompressible flows through porous media // Physical Review E. 2002. 66. 036304-1-036304-9.
  13. Pan C., Luo L.S., Miller C.T. An evaluation of lattice Boltzmann schemes for porous media flow simulation // Computers and Fluids. 2006. 35. 898-909.
  14. Грачев Н.Е., Дмитриев А.В., Сенин Д.С. Моделирование динамики газа при помощи решеточного метода Больцмана // Вычислительные методы и программирование. 2011. 12. 227-231.
  15. Бикулов Д.А., Сенин Д.С., Демин Д.С., Дмитриев А.В., Грачев Н.Е. Реализация метода решеточных уравнений Больцмана для расчетов на GPU-кластере // Вычислительные методы и программирование. 2012. 13, № 1. 221-227.
  16. Rinaldi P.R., Dari E.A., Venere M.J., Clansse A. A lattice Boltzmann solver for 3D fluid simulation on GPU // Simulation Modelling Practice and Theory. 2012. 25. 163-171.
  17. Xiong Q.G., Li B., Xu J., Fang X.J., Wang X.W., Wang L.M., He X.F., Ge W. Efficient parallel implementation of the lattice Boltzmann method on large clusters of graphics processing units // Computer Science and Technology. 2012. 57, № 7. 707-715.
  18. Zhang J., Yan G., Dong Y. A new lattice Boltzmann model for the Laplace equation // Applied Mathematics and Computation. 2009. 215. 539-547.
  19. Yan G., Zhang J. A higher-order moment method of the lattice Boltzmann model for the Korteweg de Vries equation // Mathematics and Computers in Simulation. 2009. 79. 1554-1565.
  20. Lai H.L., Ma C.F. A higher order lattice BGK model for simulating some nonlinear partial differential equations // Science in China Series G: Physics, Mechanics and Astronomy. 2009. 52, № 7. 1053-1061.
  21. Ma C.F. A new lattice Boltzmann model for KdV-Burgers equation // Chinese Physics Letters. 2005. 22, № 9. 2313-2315.
  22. Yan G., Yuan L. Lattice Bhatnagar-Gross-Krook model for the Lorenz attractor // Physica D. 2001. 154. 43-50.
  23. Wang Z., Shi B., Xiang X., Chai Z., Lu J. Lattice Boltzmann method for n-dimensional nonlinear hyperbolic conservation laws with the source term // Chaos. 2011. 21. 013120-1-013120-8.
  24. Wolf-Gladrow D.A. A lattice Boltzmann equation for diffusion // J. of Statistical Physics. 1995. 79, № 5/6. 1023-1032.
  25. Chen S., Tolke J., Geller S., Krafczyk M. Lattice Boltzmann model for incompressible axisymmetric flows // Physical Review E. 2008. 78. 046703-1-046703-8.
  26. Huber C., Parmigiani A., Chopard B., Manga M., Bachmann O. Lattice Boltzmann model for melting with natural convection // Int. J. of Heat and Fluid Flow. 2008. 29. 1469-1480.
  27. Huber C., Cassata W.S., Renne P.R. A lattice Boltzmann model for noble gas diffusion in solids: the importance of domain shape and diffusive anisotropy and implications for thermochronometry // Geochimica et Cosmochimica Acta. 2011. 75. 2170-2186.
  28. Hirabayashi M., Chan Y., Ohashi H. The lattice BGK model for the Poisson equation // JSME Int. J. Ser. B. 2001. 44, № 1. 45-52.
  29. Ponce Dawson S., Chen S., Doolen G.P. Lattice Boltzmann computations for reaction-diffusion equations // J. of Chemical Physics. 1993. 98, № 2. 1514-1523.
  30. Blaak R., Sloot P.M. A. Lattice dependence for reaction-diffusion in lattice Boltzmann modelling // Computer Physics Communications. 2000. 129. 256-266.
  31. Mishra S.C., Mondal B., Kush T., Krishna R.S. R. Solving transient heat conduction problems on uniform and non-uniform lattices using the lattice Boltzmann method // Int. Communications in Heat and Mass Transfer. 2009. 36. 322-328.
  32. Wang J., Wang M., Li Z. Lattice evolution solution for the nonlinear Poisson-Boltzmann equation in confined domains // Communications in Nonlinear Science and Numerical Simulation. 2008. 13. 575-583.
  33. Wang J., Wang M., Li Z. Lattice Poisson-Boltzmann simulations of electro-osmotic flows in microchannels // J. of Colloid and Interface Science. 2006. 296. 729-736.
  34. Wang J., Wang M., Chen S. Roughness and cavitations effects on electro-osmotic flows in rough microchannels using the lattice Poisson-Boltzmann methods // J. of Computational Physics. 2007. 226. 836-851.
  35. Kameli H., Kowsary F. Solution of inverse heat conduction problem using the lattice Boltzmann method // International Communications in Heat and Mass Transfer. 2012. 39. 1410-1415.
  36. He X., Chen S., Doolen G.D. A novel thermal model for the lattice Boltzmann method in incompressible limit // J. of Computational Physics. 1998. 146. 282-300.
  37. He X., Luo L.-S. A priori derivation of the lattice Boltzmann equation // Physical Review E. 1997. 55, № 6. R6333-R6336.
  38. He X., Luo L.-S. Theory of the lattice Boltzmann method: from the Boltzmann equation to the lattice Boltzmann equation // Physical Review E. 1997. 56, № 6. 6811-6817.
  39. Федоренко Р.П. Введение в вычислительную физику. Долгопрудный: Изд. дом «Интеллект», 2008.
  40. Wang H., Yan G., Yan B. Lattice Boltzmann model based on the rebuilding-divergency method for the Laplace equation and the Poisson equation // J. of Scientific Computing. 2011. 46. 470-484.
  41. Самарский А.А., Попов Ю.П. Разностные методы решения задач газовой динамики. М.: Книжный дом «ЛИБРОКОМ», 2009.
  42. Самарский А.А., Вабищевич П.Н. Вычислительная теплопередача. М.: Книжный дом «ЛИБРОКОМ», 2009.
  43. Рихтмайер Р., Мортон К. Разностные методы решения краевых задач. М.: Мир, 1972.
  44. Smith B., Boyle J., Dongarra J., Garbow B., Ikebe Y., Klema V., Moler C. Matrix eigensystem routines. EISPACK Guide. Lecture Notes in Computer Science. Vol. 6. Berlin: Springer-Verlag, 1976.
  45. He X., Luo L.S. Lattice Boltzmann model for the incompressible Navier-Stokes equation // J. of Statistical Physics. 1997. 88, № 3/4. 927-944.
  46. Cheng M., Yao Q., Luo L.S. Simulation of flow past a rotating circular cylinder near a plane wall // Int. J. of Computational Fluid Dynamics. 2006. 20, № 6. 391-400.
  47. Fares E. Unsteady flow simulation of the Ahmed reference body using a lattice Boltzmann approach // Computers and Fluids. 2006. 35. 940-950.
  48. d’Humieres D., Ginzburg I., Krafczyk M., Lallemand P., Luo L.S. Multiple-relaxation-time lattice Boltzmann models in three dimensions // Philosophical Transactions of Royal Society of London A. 2002. 360. 437-451.
  49. Van der Sman R.G. M., Ernst M.H. Diffusion lattice Boltzmann scheme on a orthorhombic lattice // J. of Statistical Physics. 1999. 94, № 1/2. 203-217.
  50. Van der Sman R.G. M. Finite Boltzmann schemes // Computers and Fluids. 2006. 35. 849-854.
  51. Rasin I., Succi S., Miller W. A multi-relaxation lattice kinetic method for passive scalar diffusion // J. of Computational Physics. 2005. 206. 453-462.
  52. Kuzmin A., Ginzburg I., Mohamad A.A. The role of the kinetic parameter in the stability of two-relaxation-time advection-diffusion lattice Boltzmann schemes // Computers and Mathematics with Applications. 2011. 61. 3417-3442.