Lattice Boltzmann method for general convection-diffusion equations: MRT model and boundary schemes

Abstract

This paper is concerned with the lattice Boltzmann method for general convectiondiffusion equations. For such equations, we develop a multiple-relaxation-time lattice Boltzmann model and show its consistency under the diffusive scaling. The secondorder accuracy of the half-way anti-bounce-back scheme accompanying the present M-RT model is justified based on an elegant relation of the collision matrix. Using the half-way anti-bounce-back scheme as a central step, we further construct some parameterized single-node second-order schemes for curved boundaries. The accuracy of the proposed model and boundary schemes are numerically validated with several nonlinear convection-diffusion equations.

On the collision matrix of the lattice Boltzmann method for anisotropic convection–diffusion equations

“…Nevertheless, we show that the collision matrices of all these MRT models admit an elegant property formulated in [16]. With this property, the half-way anti-bounce-back scheme can be justified to be second-order accurate [17]. Numerical experiments validate our analysis for both two-and three-dimensional anisotropic CDEs.…”

“…Remark 1. It is shown in [17] that under the property (3.8) the second-order accuracy of the half-way antibounce-back scheme…”

“…can be justified for the Dirichlet boundary condition φ(x b , t) = ψ(x b , t), x b ∈ ∂Ω of the isotropic CDE, where f t) is the post-collision distribution. Since the LB model of the isotropic CDE has the same form as (2.2) for the anisotropic ones, the second-order accuracy of the scheme (3.13) can be justified in the same way as in [17] for the anisotropic CDE (2.1).…”

The solution of unconfined groundwater flow with an innovative lattice Boltzmann method

“…Early studies were focused on the solution of advection-diffusion equation with constant coefficients [12,[14][15][16]. In recent years, advection-diffusion equation with variable constant whether in Cartesian or cylindrical coordinates have also been solved using LBM [17][18][19][20]. Of note, the applications of two or multiple relaxation times (MRT) in place of single relaxation time are also taken into account in the numeral simulations using LBM [21][22][23][24].…”

Mass-balance and locality versus accuracy with the new boundary and interface-conjugate approaches in advection-diffusion lattice Boltzmann method

“…However, the advection regime validation is commonly limited to a relatively small P eclet number (about twenty), whereas the ABB completely degrades its second-order accuracy for a mid-grid surface location in an interface-perpendicular plug flow at Pe % 10 2 , 38 because its directional closure relation interferences with the advective projections, 26 overlooked by the later asymptotic analysis. 106 With these ideas in mind, the ABB, the equivalent LI schemes 62,106 and the MR Dirichlet schemes 26 have been recently extended 40 to "linear" [ABB/MPLI/PLI] and "parabolic" accurate [KMR/PP] Dirichlet families, improving their accuracy and parametrization by the grid P eclet number in the presence of the velocity field [PAB/PLI/KMR/PP] and space-variable mass-source, such that every family contains an infinite number of members (coefficients) of equivalent spatial accuracy; only the parametrized LMKC scheme 62 enters the MPLI family. The LI can be operated in-node, whereas the MR requires the next directional fluid neighbor; both LI and MR cope with any discrete-velocity set but require, as a minimum, the TRT collision for their parametrization.…”