1549301742-The_Theory_of_Difference_Schemes__Samarskii

The summarized approximation inethod 641

We will pursue the discussion of additive schemes further with re-

gard to problen1 ( 61)-(62) capable of describing the syste1n of hyperbolic

equations

(89)

What has been done is to reduce this systen1 to successive solution of simpler

equations 1noving fron1 cY to CY+ l:

(90)

One possible additive sche1ne

( 91)

CY= 1,2,. .. ,p, (x,t)Ewh Xw 7 ,

Y(a) = 1-t(x, t~), x°' = 0, (" CY= 1, 2,. .. ,p,

y(x, OJ= u 0 (x),

can be obtained through the usual approximations of p equations, where
'Pa = f°'(,t, t~) and t;, = i]+l"/p-u 517 , the coefficients k" 13 are taken at
mo1nent t'.» Ytata is detennined by formula (o7) or formula (68), crP = 0.5
for p = 2 and crP = 1.5 for p = 3.
The second initial condition is approximated by setting

CY= 1,2,. .. ,p-1.

Because of these facts, the describing scheme generates a summarized ap-

proximation

p

W = 2= VJo = 0( T + lhl^2 ) ·

Cf= J