An improved lattice Boltzmann method for simulating advective-diffusive processes in fluids

Lattice Boltzmann methods are widely used to simulate advective-diffusive processes in fluids. Lattice Bhatnagar-Gross-Krook methods presented in the literature mostly just exhibit first order spatial accuracy and introduce errors proportional to the velocity squared. Formulations proposed to alleviate this have only been partly successful and are valid only in certain specific situations. We present and demonstrate here a formulation that produces no such second order errors. This formulation suggests that a subtle, but important, adjustment is all it takes to improve the accuracy of the method. The key to the improved accuracy of this new model is the non-standard definition of the concentration that relates to the distribution function describing the advection-diffusion in lattice Boltzmann. The main advantage of the algorithm comes to view when simulating situations where fluid density variations appear. The present formulation of the advection-diffusion algorithm will, by taking into account these fluid density variations, drastically reduce the errors produced compared to the standard formulations. We also show how a source term is included in this new formulation without it losing its second order spatial accuracy.