Modified lattice Boltzmann scheme for nonlinear convection diffusion equations

Abstract To further improve the simulation performance of nonlinear convection diffusion equations (NCDE), in this paper, a modified lattice Boltzmann (LB) scheme is put forward by introducing a new parameter that offers more opportunities in studying NCDE while increases the computation cost hardly, the derivation of LB scheme is investigated in detail, and the numerical experiments are carried out in different NCDE, including Nonlinear Heat Conduction Equation, Burgers–Fisher Equation and Nonlinear Schrodinger Equation. The simulation results obtained not only agree well with the analytical solutions, but also demonstrate that the proposed scheme under the best choice of the parameters can preserve the better stability and higher accuracy in comparison with previous schemes.

[1]  Baochang Shi,et al.  Lattice Boltzmann model for the one-dimensional nonlinear Dirac equation. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  Abdul-Majid Wazwaz,et al.  Soliton solutions for a generalized KdV and BBM equations with time-dependent coefficients , 2011 .

[3]  R. Benzi,et al.  The lattice Boltzmann equation: theory and applications , 1992 .

[4]  Iliya V. Karlin,et al.  Entropic lattice Boltzmann method for simulation of thermal flows , 2006, Math. Comput. Simul..

[5]  Frank T.-C. Tsai,et al.  Non-negativity and stability analyses of lattice Boltzmann method for advection-diffusion equation , 2009, J. Comput. Phys..

[6]  Y. Pomeau,et al.  Lattice-gas automata for the Navier-Stokes equation. , 1986, Physical review letters.

[7]  Baochang Shi,et al.  Lattice Boltzmann model for nonlinear convection-diffusion equations. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  Frank T.-C. Tsai,et al.  Lattice Boltzmann method with two relaxation times for advection–diffusion equation: Third order analysis and stability analysis , 2008 .

[9]  Spectral transition in a sparse model and a class of nonlinear dynamical systems , 2007 .

[10]  K. Noor,et al.  Variational Iteration Method for Re-formulated Partial Differential Equations , 2010 .

[11]  Laila M. B. Assas,et al.  New exact solutions for the Kawahara equation using Exp-function method , 2009, J. Comput. Appl. Math..

[12]  Nikolay K. Vitanov,et al.  Modified method of simplest equation: Powerful tool for obtaining exact and approximate traveling-wave solutions of nonlinear PDEs , 2011 .

[13]  Volker John,et al.  On Finite Element Methods for 3D Time-Dependent Convection-Diffusion-Reaction Equations with Small Diffusion , 2008 .

[14]  B. Amaziane,et al.  Convergence of finite volume schemes for a degenerate convection-diffusion equation arising in flow in porous media , 2002 .

[15]  X. He,et al.  Discretization of the Velocity Space in the Solution of the Boltzmann Equation , 1997, comp-gas/9712001.

[16]  A. Campo,et al.  Approximate Solution of the Nonlinear Heat Conduction Equation in a Semi-Infinite Domain , 2010 .

[17]  Sheng Chen,et al.  A simple lattice Boltzmann scheme for combustion simulation , 2008, Comput. Math. Appl..

[18]  Michael Selzer,et al.  Combined Lattice Boltzmann and phase-field simulations for incompressible fluid flow in porous media , 2010, Math. Comput. Simul..

[19]  Shi Bao-Chang,et al.  A lattice Bhatnagar-Gross-Krook model for a class of the generalized Burgers equations * , 2006 .

[20]  Sauro Succi,et al.  A multi-relaxation lattice kinetic method for passive scalar diffusion , 2005 .

[21]  Washington Taylor,et al.  A Quantum Lattice-Gas Model for the Many-Particle Schroedinger Equation , 1996 .

[22]  Sauro Succi,et al.  The Quantum Lattice Boltzmann Equation: Recent Developments † , 2008 .

[23]  M. Helal,et al.  A comparison between two different methods for solving KdV–Burgers equation , 2006 .

[24]  K. M. Bryden,et al.  Parallel performance and accuracy of lattice Boltzmann and traditional finite difference methods for solving the unsteady two-dimensional Burger's equation , 2006 .

[25]  Chuguang Zheng,et al.  A Lattice BGK Scheme with General Propagation , 2002, J. Sci. Comput..

[26]  Peter M. A. Sloot,et al.  Lattice dependence of reaction-diffusion in lattice Boltzmann modeling , 2000 .

[27]  Fang Liu,et al.  Numerical solutions of two-dimensional Burgers’ equations by lattice Boltzmann method , 2011 .

[28]  Christian Obrecht,et al.  LBM based flow simulation using GPU computing processor , 2010, Comput. Math. Appl..

[29]  R. Salmon,et al.  Shallow water equations with a complete Coriolis force and topography , 2005 .