Analysis of the central-moments-based lattice Boltzmann method for the numerical modelling of the one-dimensional advection-diffusion equation: Equivalent finite difference and partial differential equations

Abstract

This work presents a detailed theoretical analysis of the multiple-relaxation-time (MRT) lattice Boltzmann method (LBM), formulated on central moment (CM) space, for the numerical modelling of the one-dimensional advection-diffusion equation (ADE) with a constant velocity and diffusion coefficient, based on the D1Q3 lattice. Other LBM collision operators, such as single-relaxation-time Bhatnagar–Gross–Krook (BGK), regularized (REG) and MRT in raw moment (RM) space are also considered in this study. Without recurring to asymptotic analyses, such as the Chapman–Enskog expansion, we investigate the approximation of the MRT-CM with respect to the ADE by deriving its equivalent finite difference (EFD) scheme, which obeys an explicit four-level finite difference scheme at discrete level. Its steady-state limit follows a standard central differencing scheme for the steady ADE, yet with possible artefacts in the effective diffusion coefficient. Then, through the Taylor expansion of the EFD scheme, a detailed accuracy analysis, based on the equivalent partial differential (EPD) equation, reveals the leading order truncation errors associated with each collision model under study. Although MRT-CM and MRT-RM models have similar error structures, the former has a much reduced and simpler form, particularly in the dispersion error term, which might explain the improved Galilean invariance of the CM model. Through a suitable combination of the MRT free parameters (either in RM or CM bases), it is possible to improve its accuracy from second- to fourth-order. After that, we study the necessary and sufficient stability conditions of the MRT-CM, and its relation with other collision operators, based on the von Neumann stability analysis of the derived EFD schemes. Unexpectedly, the MRT-CM appears to support a narrower stability domain than the MRT-RM model, particularly at higher advection velocities, which can be tracked down to the inclusion of additional terms in the stability condition of the former that scale with higher order polynomials of the advection velocity. Finally, some numerical tests for the ADE on 1D unbounded domains are conducted, which confirm this work theoretical conclusions on the MRT-CM performance.