A comparison of three velocity discretizations for the Vlasov equation

Summary form only given. Three different methods of velocity discretization for the Vlasov equation are compared for use in the numerical solution of the one-dimensional Vlasov equation. The first method is a simple central difference on a uniform grid in velocity space; with this method the evaluation of the right hand side of the Vlasov equation requires 8 floating point operations per degree of freedom. The other two methods are weighted residuals methods based on Hermite polynominals: the first of these is based on expansion functions and identical weight functions; the second Hermite based method uses expansion functions and different weight function. These two methods require 12 and 8 floating point operations per degree of freedom respectively. Thus all three methods require the same order of computational work, however, for finite numbers of degrees of freedom the finite difference method conserves only particles, the symmetric Hermite method conserves either particles or momentum, while the asymmetric Hermite method conserves particles, momentum, and energy. The two Hermite methods also exactly recover the Hermite moments of the free streaming solution of the Vlasov equation, and the asymmetric Hermite method exactly solves the spatially uniform Vlasov-Ampere equations (exact plasma oscillations). When used to compute the growth rates of unstable plasma equilibria the finite difference method shows only algebraic convergence, while both Hermite based methods show spectral convergence, with the error decay rate for the asymmetric Hermite method larger than for the symmetric Hermite.