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Methods for the numerical discretization of the Vlasov equation should efficiently use the phase-space discretization and should introduce only enough numerical dissipation to promote stability and control oscillations. A new high-order nonlinear finite-volume algorithm for the Vlasov equation that discretely conserves particle number and controls oscillations is presented. The method is fourth order in space and time in well-resolved regions but smoothly reduces to a third-order upwind scheme as features become poorly resolved. The new scheme is applied to several standard problems for the Vlasov-Poisson system, and the results are compared with those from other finite-volume approaches, including an artificial viscosity scheme and the piecewise parabolic method. It is shown that the new scheme is able to control oscillations while preserving a higher degree of fidelity of the solution than the other approaches.