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. © 2007 Elsevier Ltd. All rights reserved.
Stiebler, M., Tölke, J., & Krafczyk, M. (2008). Advection-diffusion lattice Boltzmann scheme for hierarchical grids. Computers and Mathematics with Applications, 55(7), 1576–1584. https://doi.org/10.1016/j.camwa.2007.08.024