uniform mesh size for x, y, and z (that is, Ax = Ay = Az = h), we have the
approximation
SWi,j,k = i,j,k+l Oij,k-1
2h
for the partial derivative O//oz at the point (xi, yj, Zk). The second partial
derivative 0291/0x2 at the point (xi, yj, zk) has an approximation denoted by
OijPi,k:
2,jk = _i +l,j,k 2~ ,j,k + Vi-l,j,k
8 Wi~j,k 2 ---- ^----
Similarly, we use approximations 19ij,k and 930ij,k to the second partial deriva-
tives 92?/Oy2 and 02/0z2 :
a2q, Oij+1,k 21)0ijk + Oij-lk
2' Oi,j,k h- 2
and
i,,k i,j,k+1 20i,j,k + i,j,k-l
)3,,k -h2
respectively. We define the discrete Laplacian operator Ah as follows:
3
A ,j,k (21 ijk-
l=1
After replacing A by Ah and O9i/Oz by the discrete azo4 in Equation (3.11), we
obtain the following finite difference scheme.
-Ahvzn k As j., =zh dk n 9,,p (3.12)
-A ij, + AkA ,,3k = (1 P) h29z,3,k + P ? ] (3,12)
Equation (3.12) yields a matrix-vector form
A n+1 AA n" = h [(1 p)B "n + tB"'++] (3.13)
where 4 denotes the vector with components i,j,k. The matrix A is symmetric
and positive definite and the structure of the matrix A is a block tridiagonal