1549301742-The_Theory_of_Difference_Schemes__Samarskii

The alternative-triangular method 709

But it may happen that the coefficients of the governing differential equa-

tion are varying very fastly, but locally in a s111all region so that the spectral

bounds of the operator A change insignificantly.

In the case where G is a rectangle, it is possible to adopt B = R with

further reference to one of the direct methods available for determination

of ktl , thus causing the ratio ~ = c 1 / c 2 and the independence of the total

number of the iterations upon h:

n(E) ~ ~2

1

2

In -.

E

Recall that in the modification of ATM (MATM, see for more detail Section

8) there is no need for involving the operator R and so a proper choice of

parameters { Tn} will be independent of constants c 1 and c 2 both. Here the

operator A arranges itself as a sum A = Ai + A 2.

By having recourse to problem (81) in a unit square we will illus-

trate the perfonnance of MATM in the sequel. In preparation for this, we

introduce a square grid

and write clown on it the equations

vi ~ "Yh = o.

The coefficients a 1 (:e) and a 2 (.e) are given by the fonnulas

a 1 (x) = l + I<o[(x 1 - 0 5)^2 + (x 2 - 0.5)^2 ],

aAx) = 1 + I<o [0.5 - (x 1 - 0.5)^2 - (x 2 - 0.5)^2 ] ,

I<o = const > 0.

A proper choice of the right-hand side <p(x) is stipulated by the fact that
y(x) = x 1 (1-x 1 ) x 2 (1-x 2 ) is the exact solution of the problem concerned.
Also, it will be sensible to introduce

and then take the operators R 1 and R 2 with the values

----Y.u^1 Y:u^2

h h.