We introduce the semi-Lagrangian Vlasov method, which computes the distribu
tion function of the particles on a grid in phase space, to beam propagatio
n in a uniform focusing channel. With this new tool, we study halo formatio
n in a mismatched thermal beam, and the evolution of an initial semi-Gaussi
an beam. For the latter problem comparisons are made with the Particle-In-C
ell code WARP. (C) 2001 Elsevier Science B.V, All rights reserved.