Vlasov Evoluton of a Gravitational System via a Spectral Method

There are open questions concerning the distribution of clusters in the expanding universe. The coupled Vlasov-Poisson equations govern the evolution of density in mu(position-velocity) space. In the comoving frame, the evolution of the $\mu$-space density $f$ for a one-dimensional gravitational system is governed by the Vlasov-Poisson continuity equation where $a$ is the local acceleration: \[ \frac{\partial f}{\partial t}+v\frac{\partial f}{\partial x}+\frac{\partial af}{\partial v}=0\] Here we introduce a spectral method to obtain a coupled set of ordinary differential equations governing the time dependence of the coefficients. For the bounded position space we utilize a Fourier expansion, whereas for the infinite velocity space we utilize a Hermite expansion. The resulting equations are bilinear and govern the coefficients $\psi_{m,n}(t)$, where $m$ represents the Fourier index and $n$ the Hermite index. By truncating the doubly infinite series they can be integrated numerically to model and simulate the system evolution of $f$, in our case using a traditional fourth-order Runge-Kutta method. We will present the important derivations and preliminary results of the numerical integration.