The lattice BGK model for the Poisson equation

This study deals with the development of the lattice BGK model for the Pois son equation. The lattice BGK method, which was derived from lattice gas au tomata, is a mesoscopic approach for simulating the fluid flow. We can appl y this method to several partial differential equations (PDEs) as a numeric al solver without losing its advantages such as noise-free calculation, sim ple algorithm, and high computational efficiency on parallel computers. We develop the new lattice BGK Poisson solver as an example of the elliptic PD E solver and discuss its fundamental properties. By comparing with the fini te element method, we confirm the effectiveness of adopted boundary conditi on rules. We indicate that the extremely favorable speed-up is achieved in parallel computing and that adjustments of the relaxation parameter acceler ate the calculation. Our numerical simulations show the potentiality of the lattice BGK method as the solver for the PDEs besides the hydrodynamic equ ations.