  Simulation of plasmas in electromagnetic fields requires numerical solution
of a kinetic equation that describes the time evolution of the particle
distribution function. In this paper we propose a finite volume scheme based on
integral relation for Poisson brackets to solve the Liouville equation, the
most fundamental kinetic equation. The proposed scheme conserves the number of
particles, maintains the total-variation-diminishing (TVD) property, and
provides high-quality numerical results. Other types of kinetic equations may
be also formulated in terms of Poisson brackets and solved with the proposed
method including the transport equations describing the acceleration and
propagation of Solar Energetic Particles (SEPs), which is of practical
importance, since the high energy SEPs produce radiation hazards. The proposed
scheme is demonstrated to be accurate and efficient, which makes it applicable
to global simulation systems analyzing space weather.
