1549301742-The_Theory_of_Difference_Schemes__Samarskii

680 Methods for Solving Grid Equations

with a known right-hand side Fk. In both cases the matrix B falls within

the category of triangle matrices, since B = A-+ D and B = A-+ t D do

arise during the course of Seidel and upper relaxation methods, respectively.

Due to this property every new iteration Yk+l can be obtained by making

a minimal number of iterations. For the model problem concerned, the

number of the necessary iterations is equivalent to the total number N of

the grid nodes.

Proper guidelines for the well-founded choice of the operator B are

provided by the following requirements:

• a minimal number of iterations;


• the operator B is economical, the work and storage require O(N).
  The meaning of the latter property for difference second-order equa-
  tions of the elliptic type is that the equation Byk+i = Fk with a
  known right-hand side Fk must be solved in a mini1nal number of
  operations.
  The main result in Section 2 regarding the optimal set of parameters
  { Tk} can be generalized directly for implicit schemes with B f:. E as follows:

k = 0, 1, ... , n - 1, Yo EH given,

under the conditions

(12) B=B>O, A=A>O, 11 B<A<1 2 B, 11 >0,

whose key role is to specify the primary family (6) of iterative methods

at the very beginning. But this is not the case for the upper relaxation

method and Seidel method both in connection with the property that the

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

In mastering the difficulties involved, the intention is to use the ho-

1nogeneous equation related to the correction wk = B-^1 (Ayk - f):

( 13)

The preceding scheme is equivalent to the explicit scheme

(14)

Xk+I - Xk

-~--+ C xk = 0, k = 0, 1, ... , n - 1, X 0 EH given,