Fine-scale structures and negative-density regions: Comparison of numerical methods for solving the advection equation

A common feature of the Vlasov equation is that it develops fine-scale filamentation as time evolves, as observed, for example, in global nonlinear simulations of the ion-temperature-gradient instability. From a numerical point of view, it is not trivial to simulate nonlinear regimes characterized by increasingly smaller scales, which eventually become smaller than the (finite) grid size. When very small structures occur, higher order interpolation schemes have a tendency to produce overshoots and negative-density regions unless some additional dissipative procedure is applied. Different interpolation schemes for the distribution function are compared and discussed.