Numerical Experiments with the Fokker–Planck Equation in 1D Slab Geometry

ABSTRACT

Several numerical experiments with the steady monoenergetic Fokker–Planck equation in the one-dimensional (1D) slab are performed and discussed. Planar-geometry symmetry is assumed. Results are obtained with a new numerical scheme of order 2 based on finite differences.

KEYWORDS:

• Fokker–Planck equation
• finite differences
• Crank–Nicolson scheme
• two-way diffusion equations
• continuous scattering operator

Notes

1 In Stein and Bernstein (1976), f is our ψ, and . Also, f_μ denotes .

2 Another way to say the same thing: on Q₋ an implicit scheme must be written in “forward form,” while on Q₊ the same scheme must be written in “backward form.”

3 There is no reason to prefer removing the first one versus the last one of these. Therefore, we eliminate all of them and we follow an alternative line of action.

4 These drawbacks of the even scheme for this problem could be corrected, in all of our trials, by taking N ∼ 6I.