1549301742-The_Theory_of_Difference_Schemes__Samarskii

676 Methods for Solving Grid Equations

10.3 THE ALTERNATIVE-TRIANGULAR METHOD

1. Seidel rnethod. As we have mentioned above, implicit schemes are rather
   stable in comparison with explicit ones. Seidel method, being the simplest
   implicit iterative one, is considered firnt. The object of investigation here
   is the system of linear algebraic equations

(1)

or

Au=f

N

L Cl;jUj = fi,

j=l

i=l,2,. .. ,N,

with nonzero diagonal elen1ents aii f:- 0. When this is the case, the iterative

method ascribed to Seidel amounts to

N

(2) +2=

j=i+l

where bj is the kth iteration. Furthermore, starting from l

(k + l)th iteration is to be formed in line with

1, the

g1vmg k+l y. k+l k+l

1. At the next stage, knowmg y 1 and setting ·i = 2, we find y 2

from

N

k+I k+I ~ k

ct21 Y1 + Cl22 Y2 + L.J Cl2j Yj

j=3

The matrix A rearranges itself as a smn

( :J)

where A- is a lower triangle matrix with zero elements on the main diag-
onal: A-= (a£;·), a:;= aij for j < i, a;;= 0 for j > i; A+ is an upper

triangle matrix with zero elements on the rnain diagonal: A+ = ( aij),
aij = aij for j > i, aij = 0 for j < i, and, finally, D is a diagonal matrix:
D = (aiibij ). Here bij stands, as usual, for Kronecker's delta: r5ij = 1 for
j = i and r5ij = 0 for j f:- i.