Exact artificial boundary conditions for a lattice Boltzmann method

When using a lattice Boltzmann method on an unbounded (or very large) domain one has to confine this spatial domain to a computational domain. This is realized by introducing so-called artificial boundary conditions. Until recently, characteristic boundary conditions for the Euler equations were considered and adapted to the lattice Boltzmann method. In this work we propose novel discrete artificial boundary conditions which are derived directly for the chosen lattice Boltzmann model, i.e., on the discrete level. They represent the first exact artificial boundary conditions for lattice Boltzmann methods. Doing so, we avoid any detour of considering continuous equations and obtain boundary conditions that are perfectly adapted to the chosen numerical scheme. We illustrate the idea for a one dimensional, two velocity (D1Q2) lattice Boltzmann method and show how the computational efficiency can be increased by a finite memory approach. Analytical investigations and numerical results finally demonstrate the advantages of our new boundary condition compared to previously used artificial boundary conditions.