Lattice Boltzmann Method Analysis Tool (LBMAT)

A general computational tool for the derivation of equivalent partial differential equations (EPDEs) is presented for the lattice Boltzmann method (LBM) with general collision operators that include single relaxation time (SRT-LBM), multiple relaxation time (MRT-LBM), central LBM (CLBM), or cumulant LBM (CuLBM). The method can be used to recover the advection-diffusion equations (ADEs), Navier-Stokes equations (NSEs), and other problems that could be solved by LBM in all dimensions. The derivation of EPDEs starts with the discrete (lattice) Boltzmann equation for raw moments and uses spatio-temporal Taylor expansion of these moments to obtain a system of partial differential equations. Then, to recover the desired ADEs or NSEs with additional partial differential terms up to a given order, a computationally feasible algorithm is proposed to eliminate higher order moments. The algorithm for the derivation of EPDEs, under the name of LBMAT (Lattice Boltzmann Method Analysis Tool), is implemented in C++ using the GiNaC library for symbolic algebraic computations. In order to optimize memory demands for higher dimension LBM models such as D3Q27, a custom-tailored data structure for storing the terms of partial differential expressions is proposed. The implementation of LBMAT is available to the community as open-source software under the terms and conditions of the GNU general public license (GPL).

Automated summary

A general computational tool for deriving equivalent partial differential equations (EPDEs) in lattice Boltzmann method (LBM) with different collision operators is proposed. The tool can recover various dimensions of advection-diffusion equations (ADEs) and Navier-Stokes equations (NSEs). It starts with the lattice Boltzmann equation and uses spatio-temporal Taylor expansion to obtain a system of partial differential equations. An algorithm is proposed to eliminate higher order moments and recover the desired equations. The tool, named LBMAT, is implemented as open-source software for symbolic algebraic computations.