First a non-multiplying medium (e. g., water) is considered for simplicity of the equation, though practical systems are composed of various materials with different cross sections. In addition, it is assumed that all neutrons can be characterized by a single energy (one-group problem). Therefore, the cross section does not depend on location and in this case the diffusion equation is given by Eq. (2.41).

ву2ф-ъаф+8=о

V2 (the Laplacian operator) is dependent on the coordinate system and in the Cartesian coordinate system it is represented as Eq. (2.42).

In an infinitely wide medium in y and z directions, the diffusion equation reduces to the 1D form.

(2.43)

Here a uniform neutron source (s1 = s2 = s3 ••• = s) in a finite slab of width 2d is considered as shown in Fig. 2.16 and the finite difference method to solve for neutron flux distribution is discussed. The reflective boundary condition is applied to the center of the slab and one side of width d is divided into equal N regions (spatial meshes). The second-order derivative of Eq. (2.43) can be
approximated[3] by using neutron flux фі at a mesh boundary і and neutron fluxes at both adjacent sides as follows;

(2.44)

(2.46)

where

(2.47)

In the case of N = 5, the following simultaneous equations are gotten.

афо+Ьфі+аф2=Si

< аф1+Ьф2+афз=82 (248)

аф2+Ьфз+аф4=8з

Since there are six unknowns (ф0 to ф5) among four equations in Eq. (2.48), another two conditions are needed to solve the simultaneous equations. Bound­ary conditions at both ends (x = 0 and x = d) of the slab are given by the following.

Vacuum boundary condition (an extrapolated distance is assumed zero):

ф(х = ё) = 0 -> фв = 0 (2.49)

Fig. 2.17 Finite difference in 2D plane geometry

і ~ 1 і і +1 N Divisions in Width A x

Reflective boundary condition:

(2.50)

Equation (2.51) is obtained by inserting Eqs. (2.49) and (2.50) into Eq. (2.48) and rewriting in matrix form.

A 1D problem generally results in a tridiagonal matrix equation of (N — 1) x (N — 1) system which can be directly solved using numerical methods such as the Gaussian elimination method. Since the finite difference method introduces the approximation as Eq. (2.44), it is necessary to choose a sufficiently small mesh spacing. The spatial variation of neutron flux is essentially characterized by the extent of neutron diffusion such as the neutron mean free path. Hence the effect of the mesh size on nuclear characteristic evaluations of a target reactor core must be understood and then the mesh size is optimized relative to the computation cost.