FOKKER-PLANCK EQUATIONS OF STOCHASTIC ACCELERATION - A STUDY OF NUMERICAL-METHODS

Stochastic wave-particle acceleration may be responsible for producing suprathermal particles in many astrophysical situations. The process can be described as a diffusion process through the Fokker-Planck equa tion. If the acceleration region is homogeneous and the scattering mea n free path is much smaller than both the energy change mean free path and the size of the acceleration region, then the Fokker-Planck equat ion reduces to a simple form involving only the time and energy variab les. In an earlier paper(Park & Petrosian 1995, hereafter Paper 1), we studied the analytic properties of the Fokker-Planck equation and fou nd analytic solutions for some simple cases. In this paper, we study t he numerical methods which must be used to solve more general forms of the equation. Two classes of numerical methods are finite difference methods and Monte Carlo simulations. We examine six finite difference methods, three fully implicit and three semi-implicit, and a stochasti c simulation method which uses the exact correspondence between the Fo kker-Planck equation and the Ito stochastic differential equation. As discussed in Paper I, Fokker-Planck equations derived under the above approximations are singular, causing problems with boundary conditions and numerical overflow and underflow. We evaluate each method using t hree sample equations to test its stability, accuracy, efficiency, and robustness for both time-dependent and steady state solutions. We con clude that the most robust finite difference method is the fully impli cit Chang-Cooper method, with minor extensions to account for the esca pe and injection terms. Other methods suffer from stability and accura cy problems when dealing with some Fokker-Planck equations. The stocha stic simulation method, although simple to implement, is susceptible t o Poisson noise when insufficient test particles are used and is compu tationally very expensive compared to the finite difference method.