Smooth initialization of lattice Boltzmann schemes

Lattice Boltzmann methods are paradigmatic discrete evolutions with incomplete initial conditions. This is due to the fact that the variables of the (mesoscopic) method outnumber the variables of the (macroscopic) problem to be solved. In such situations, most initializations which are compatible with the given macroscopic data lead to solutions with oscillatory or steep initial layers. In order to reduce such initial effects, we present a general approach to construct initial values which are compatible with the partial information available, and which guarantee a smooth start of the evolution. Our smoothness condition prescribes the unknown initial values as the polynomial backward extrapolation of the values obtained from a few time steps. Specifically for constant and linear extrapolation, we study the consistency, stability and accuracy of the approach in the case of a lattice Boltzmann method for one-dimensional advection. Moreover, the applicability of a simple iteration scheme as the solution method is investigated.