A Numerical Method for Solution of the Generalized Liouville Equation

A numerical method for the time evolution of systems described by Liouville-type equations is derived. The algorithm uses a lattice of numerical markers, which follow exactly Hamiltonian trajectories, to represent the operatord/dt in moving (i.e., Lagrangian) coordinates. However, nonconservative effects such as particle drag, creation, and annihilation are allowed in the evolution of the physical distribution function, which is itself represented according to a ?fdecomposition. Further, the method is suited to the study of a general class of systems involving the resonant interaction of energetic particles with plasma waves. Detailed results are presented for both the classic bump-on-tail problem and the beam-driven TAE instability. In both cases, the algorithm yields exceptionally smooth, low-noise evolution of wave energy, especially in the linear regime. Phenomena associated with the nonlinear regime are also described.