The diffusion equation in a 2D homogeneous plane geometry is given by Eq. (2.52).

+ (2.52)

The plane is divided into N and M meshes in x and y directions, respectively (Fig. 2.17). Similarly to the 1D problem, the second-order derivatives can be

Fig. 2.18 Matrix equation forms of finite difference method

approximated by using neutron fluxes at a point (i, j) and its four adjacent ones as the following.

(2.56)

Next, the corresponding simultaneous equations are expressed in matrix form, similarly to the 1D problem except four boundary conditions are used: two conditions in each of the x and y directions.

A 3D problem even in a different coordinate system basically follows the same procedure and a matrix form of ([Л]ф = S) can be taken as shown in Fig. 2.18.

initial guess

і I

^ ф =[D]-1S-[D]-1[B] ф

inner

ф iteration
<^onverged^ ^update ф

convergence

end

In the case of a large-size matrix [A], a direct method such as the Gaussian elimination leads to increased necessary memory size and it is liable to cause accumulation of numerical errors as well. Hence, generally first the matrix [A] is composed into its diagonal element [D] and off-diagonal element [B]. Then ф is guessed and new guesses are calculated iteratively to converge to the true solution (Fig. 2.19). This iterative algorithm, such as found in the Gauss-Seidel method or the successive overrelaxation (SOR) method, is frequently referred to as the inner iteration.