Bt. Park et V. Petrosian, FOKKER-PLANCK EQUATIONS OF STOCHASTIC ACCELERATION - A STUDY OF NUMERICAL-METHODS, The Astrophysical journal. Supplement series, 103(1), 1996, pp. 255-267
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.