Generic boundary conditions for lattice Boltzmann models and their application to advection and anisotropic dispersion equations

We address a “multi-reflection” approach to model Dirichlet and Neumann time-dependent boundary conditions in lattice Boltzmann methods for arbitrarily shaped surfaces. The multi-reflection condition for an incoming population represents a linear combination of the known population solutions. The closure relations are first established for symmetric and anti-symmetric parts of the equilibrium functions, independently of the nature of the problem. The symmetric part is tuned to build second- and third-order accurate Dirichlet boundary conditions for the scalar function specified by the equilibrium distribution. The focus is on two approaches to advection and anisotropic-dispersion equations (AADE): the equilibrium technique when the coefficients of the expanded equilibrium functions match the coefficients of the transformed dispersion tensor, and the eigenvalue technique when the coefficients of the dispersion tensor are built as linear combinations of the eigenvalue functions associated with the link-type collision operator. As a particular local boundary technique, the “anti-bounce-back” condition is analyzed. The anti-symmetric part of the generic closure relation allows to specify normal flux conditions without inversion of the diffusion tensor. Normal and tangential constraints are derived for bounce-back and specular reflections. The bounce-back closure relation is released from the non-physical tangential flux restriction at leading orders. Solutions for the Poisson equation and for convection-diffusion equations are presented for isotropic/anisotropic configurations with specified Dirichlet and Neumann boundary conditions.