The Lattice BGK Model for the Poisson Equation

This study deals with the development of the lattice BGK model for the Poisson equation. The lattice BGK method, which was derived from lattice gas automata, is a mesoscopic approach for simulating the fluid flow. We can apply this method to several partial differential equations (PDEs) as a numerical solver without losing its advantages such as noise-free calculation, simple algorithm, and high computational efficiency on parallel computers. We develop the new lattice BGK Poisson solver as an example of the elliptic PDE solver and discuss its fundamental properties. By comparing with the finite element method, we confirm the effectiveness of adopted boundary condition rules. We indicate that the extremely favorable speed-up is achieved in parallel computing and that adjustments of the relaxation parameter accelerate the calculation. Our numerical simulations show the potentiality of the lattice BGK method as the solver for the PDEs besides the hydrodynamic equations.