Numerical Analysis for Time-Averaged Equilibrium of Lattice Boltzmann Method in Heat Diffusion and Sod Shock Tube Problem

The Bhatnagar-Gross-Krook (BGK) model in the lattice Boltzmann method (LBM) is being widely used for simulating fluid flow and heat transfer due to its simplicity and parallelization capability. However, improving its computational efficiency and accuracy remains an ongoing challenge. This work introduces a time-averaged equilibrium distribution function (TAE) within the BGK model of collision operator in LBM, aiming to explore its performance in heat diffusion and compressible flows, such as the Sod shock tube. The TAE-LBM is tested on the heat diffusion problem using different values of the numerical relaxation coefficient, and the results show that convergence is achieved with significantly fewer time steps compared to standard LBM and the finite difference method (FDM), while maintaining reasonable accuracy. Higher coefficient values improve accuracy but reduce convergence speed. In Sod shock tube simulation, the TAE approach achieves the lowest root mean square error (RMSE) compared to the finite volume method using the Harten-Lax-van Leer scheme with monotonic upstream-centred for conservation laws reconstruction (FVM-HLL-MUSCL) both first-order Runge-Kutta and second-order Runge-Kutta.