  We present a new numerical scheme for solving the advection equation and its
application to the Vlasov simulation. The scheme treats not only point values
of a profile but also its zeroth to second order piecewise moments as dependent
variables, and advances them on the basis of their governing equations. We have
developed one- and two-dimensional schemes and show that they provide quite
accurate solutions compared to other existing schemes with the same memory
usage. The two-dimensional scheme can solve the solid body rotation problem of
a gaussian profile with little numerical diffusion. This is a very important
property for Vlasov simulations of magnetized plasma. The application of the
scheme to the electromagnetic Vlasov simulation of collisionless shock waves is
presented as a benchmark test.
