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 operator d/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 delta f decomposition. 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. (C) 1996 Academic Press, Inc.