the development of fine-grained structures that are very difficult to follow numerically. These become very important in structure formation scenarios as clusters typically present many matter streams and shell crossings.

Those two main shortcomings make the search for more advanced numerical scheme important. The problem of high-dimensionality could be removed if there were a way to use the density field $\rho(x)$ instead of the probability distribution function $f(x,v)$. This can be done by integrating the first few moments of the Vlasov equation and then use techniques known for hydrodynamical simulations (see e.g. @Hockney1988). This technique is limited by the formal need to integrate all moments and not just the first few ones to obtain an exact solution. Instead of a 6D space, there is now a (formally) infinite number of variables obeying an infinite series of equations. [@Peebles1987], for instance, truncates the series and uses the first two moments (mass conservation, Euler equation) of the collisionless Boltzmann equation to evolve in time the initial perturbations. The framework reaches its limits whenever the velocity dispersion of the fluid becomes important or when shell crossing occurs.

[$N$-body ]{}simulations {#ssec:simulations}
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The other option to solve the system of equations (\[eq:VP\]) is to 