On the stability structure for lattice Boltzmann schemes

The stability structure for lattice Boltzmann schemes has been introduced in Banda et al. (2006) [16], Junk and Yong (2007) [14] to analyze the stability of numerical algorithms. The first purpose of this paper is to discuss the stability structure from the perspective of matrix analysis. Its second goal is to illustrate and apply the results to different classes of lattice Boltzmann collision operators. In particular we formulate an equivalence condition–just recently also reported in Yong (2008) [18]–that guarantees the existence of a pre-stability structure. It is then illustrated by several examples, how this equivalence condition can be effectively employed for the systematic verification and construction of stable collision operators. Finally, we point out some shortcomings of the stability structure approach arising in certain cases.