Advection–diffusion lattice Boltzmann scheme for hierarchical grids

In this paper we describe an extension of a recently developed lattice Boltzmann method for solving the advection–diffusion equation. Our proposed approach allows to couple grids of different grid resolutions and includes a staggered timestepping scheme, interpolations in space and time and finally a scaling step ensuring the continuity of the desired macroscopic quantities across the grid interface. After validating the basic lattice Boltzmann method on a uniform grid by a convergence study of analytic problems we demonstrate the consistency of our approach by solving benchmark problems and comparing results on uniform grids and multiply locally refined grids.