Six-Dimensional Adaptive Simulation of the Vlasov Equations Using a Hierarchical Basis

We present an original adaptive scheme using a dynamically refined grid for the simulation of the six-dimensional Vlasov--Poisson equations. The distribution function is represented in a hierarchical basis that retains only the most significant coefficients. This allows considerable savings in terms of computational time and memory usage. The proposed scheme involves the mathematical formalism of multiresolution analysis and computer implementation of adaptive mesh refinement. We apply a finite difference method to approximate the Vlasov--Poisson equations, although other numerical methods could be considered. Numerical experiments are presented for the $d$-dimensional Vlasov--Poisson equations in the full $2d$-dimensional phase space for $d=1,2$, or 3. The six-dimensional case is compared to a Gadget N-body simulation.