  Existing approaches to solving the Vlasov equation treat the system as a
partial differential equation on a phase space grid, and track in either an
Eulerian, Lagrangian, or semi-Lagrangian picture. We present an alternative
approach, which treats the Vlasov equation as a conservative flow on phase
space, and derives its equations of motion using particle-pushing algorithms
akin to particle-in-cell methods. Deposition to the grid is determined from the
convolution of local basis functions. This approach has the benefit of allowing
flexible definitions in the grid, which are decoupled from how the phase space
flow evolves. We present numerical examples and comment on the various
properties of the algorithm.
