A common problem with direct Vlasov Solvers is ensuring that the distribution function remains positive. A related problem is to guarantee that the numerical scheme does not introduce false oscillations in velocity space. In this paper we use a variety of schemes to assess the importance of these issues and to determine an optimal strategy for Eulerian split approached, to Vlasov Solvers. From these tests we conclude that maintaining positivity is less important than correctly dissipating the Fine-scale structure which arises naturally in the solution to many Vlasov problems, Furthermore we show that there are distinct advantages to using hi.-h-order schemes, i.e., third order rather than second. A natural choice which satisfies all of these requirements is the piecewise parabolic method (PPM), which is applied here to Vlasov's equation for the first time. (C) 2002 Elsevier Science (USA).