Equivalent finite difference and partial differential equations for the lattice Boltzmann method

A general method for the derivation of equivalent finite difference equations (EFDEs) and subsequent equivalent partial differential equations (EPDEs) is presented for a general matrix lattice Boltzmann method (LBM). The method can be used for both the advection diffusion equations and Navier-Stokes equations in all dimensions. In principle, the EFDEs are derived using a recurrence formula. A computational algorithm is proposed for generating sequences of matrices and vectors that are used to obtain EFDEs coefficients. For all DdQq velocity models, the algorithm is proven to be finite and all coefficients are obtained after q iterations. The resulting EFDEs and EPDEs are derived for selected velocity models and include the single relaxation time, multiple relaxation times, and cascaded LBM collision operators. The algorithm for the derivation of EFDEs and EPDEs is implemented in C++ using the GiNaC library for symbolic algebraic computations. Its implementation is available under the terms and conditions of the GNU general public license (GPL).

Automated summary

The presented method is applicable to a general matrix lattice Boltzmann method for deriving equivalent finite difference equations and subsequent equivalent partial differential equations, which can be used for advection diffusion equations and Navier-Stokes equations. The algorithm for obtaining coefficients and deriving equations is implemented for different velocity models, ensuring finiteness and convergence after a certain number of iterations. The implementation is done in C++ with the GiNaC library and is available under the GNU general public license (GPL).