The summarized approxiination method 595scheme on the odd layers and the implicit one on the even layers. The
outco1ne of this is(4)y2j+l - y2j.
-~-- = 2 (1-(}) Ay^21 ,
T
y2j+2 - y2j+l.
------= 2() Ay^2 1+^2 , j = 0, 1, 2, ... ,
Twhere (} > 0 is an arbitrary para111eter. In order to calculate the residuals
2j+l - 2j
"l/!1 = u u - 2(1-(})Au2j,
Twe insert in the1n the expressionsAu=Lu+O(h^2 ), Lu -- u '
in1plying that·1/J~ = 0( T"^0 + h" 0) ,
Fro1n such reasoning it seems clear that ·1/; 1
(} i- 0.5, but for any(} there is no doubt that
0(1) and 1/; 2 0(1) forCollection of equations ( 4) permits us to eliminate the value yj +^1 , leaving
us with the weighted scheme with step 2r: