The computational complexity of traditional Lattice-Boltzmann methods for incompressible fluids

It is well-known that in fluid dynamics an alternative to customary direct solution methods (based on the discretization of the fluid fields) is provided by so-called \emph{particle simulation methods}. Particle simulation methods rely typically on appropriate \emph{kinetic models} for the fluid equations which permit the evaluation of the fluid fields in terms of suitable expectation values (or \emph{momenta}) of the kinetic distribution function $f(\mathbf{r,v},t),$ being respectively $\mathbf{r}$ and\textbf{\}$\mathbf{v}$ the position an velocity of a test particle with probability density $f(\mathbf{r,v},t)$. These kinetic models can be continuous or discrete in phase space, yielding respectively \emph{continuous} or \emph{discrete kinetic models} for the fluids. However, also particle simulation methods may be biased by an undesirable computational complexity. In particular, a fundamental issue is to estimate the algorithmic complexity of numerical simulations based on traditional LBM's (Lattice-Boltzmann methods; for review see Succi, 2001 \cite{Succi}). These methods, based on a discrete kinetic approach, represent currently an interesting alternative to direct solution methods. Here we intend to prove that for incompressible fluids fluids LBM's may present a high complexity. The goal of the investigation is to present a detailed account of the origin of the various complexity sources appearing in customary LBM's. The result is relevant to establish possible strategies for improving the numerical efficiency of existing numerical methods.