Abstract

CHAPTER 1. OVERVIEW 21
1.6 Numerical Solution: Finite Difference Method
There are several discretizations one can use to numerically approximate the PDE as
discussed in the introduction. The method we use is a finite difference method. This
method is used in [8], [40]. The simulator we use is based on a code given to our
research group by the authors of [40]. Here, the k domain is truncated from (−∞, ∞)
to (−Kmax,Kmax), and a grid is placed on the (x,k) space[0,L] × (−Kmax,Kmax).
Finite differences are used for derivative terms and quadrature formulas are used to
compute integrals.
To solve this problem numerically, we begin by discretizing the space and momen-
tum domain. The space domain is divided into Nx equally spaced grid points between
x =0andx = L where the grid points are given by:
xm =(m − 1)∆x, for m =1, 2,...,Nx
L
where ∆x = . To handle the wave number domain, we first truncate the do-
Nx − 1
main to (−Kmax,Kmax), where Kmax is chosen so that for any k such that |k| >K,
we should have that f(x, k) ≈ 0. Here, Kmax =0.25 inverse angstroms, which was
chosen by consulting physicists and looking at different distributions. The truncated
k-domain has Nk equally spaced grid points, with
kj = (2j − Nk − 1)∆k
, for j =1, 2,...,Nk
2
where ∆k = 2Kmax
.Notethatkj∈ (−Kmax,Kmax) for j =1, 2,...,Nk.
Nk
We get an approximation to f at the grid points, and we will denote f(xm,kj) ≈
fmj. For the K(f) term, we use a second-order upwind differencing scheme. So we