Convergence of lattice Boltzmann methods with overrelaxation for a nonlinear conservation law

We approximate a nonlinear multidimensional conservation law by Lattice Boltzmann Methods (LBM), based on underlying BGK type systems with finite number of velocities discretized by a transport-collision scheme. The collision part involves a relaxation parameter ω which value greatly influences the stability and accuracy of the method, as noted by many authors. In this article we clarify the relationship between ω and the parameters of the kinetic model and we highlight some new monotonicity properties which allow us to extend the previously obtained stability and convergence results. Numerical experiments are performed.