1549301742-The_Theory_of_Difference_Schemes__Samarskii

The alternative-triangular inethod 677

Within these notations, Seidel rnethod can be written as follows:

(4)

• k+l + k.
  (A +D) y +A y =f,

Alternative forms of such a two-layer scheme are

and

(5)

k+I k k

(A-+ D) ( y - y) +A y = f.

Further identification with the preceding canonical form reveals

k+I k

(6)

y - y k 0

B +Ay=f, k=O,l, ... ,n-1, YEH,

Tk+I

with B = A- + D and Tk = l. This sche1ne is certainly implicit. The

matrix B concerned is triangular and, hence, it is not symn1etric, because

the operator is non-self-adjoint: B f:- B*.

On the basis on the model problem

Ay=-<p, xEwh, Yl,,h=O,

arising fro1n Section 2, Seidel method acquires the form

so that

(7)

k+I k+I k k 9

Yi,-!+ Yi2-1 +Yi,+1 +Y; 2 +1 +h-t.p

4

In such a setting it see111s reasonable to begin operations at tlw node

.... k+I k+l.
i 1 1, i 2 = 1 rn tern1s of known values y i,-I and y ; 2 _ 1 and the nght-
hand side of (7) at the boundary nodes (0, 1) and ( 1, 0). With knowledge

of ktl at the node i 1 = I, i 2 = 1 the values kt! are yet to be determined
along the lower row by setting i 2 = 2, 3, ... for i 1 = 1 with further transition
to the rows with i 1 = 2, 3, .... During the course of Seidel method it is
possible to get the values ktl at all grid nodes.