1549301742-The_Theory_of_Difference_Schemes__Samarskii

226 Homogeneous Difference Sche1nes

We introduce the grid wh = {.c; = ih, ·i = 0, 1, ... , N, hN = 1} and

approximate on it the functional J[u] by appeal to one of the auxiliary

modifications

N Xi

J[u] = 8xf

z-1

N a,'i

k(u.')

2

dx + 8 xi[ (qu

2

• 2fa) dx.

With this in mind, we approximate the integrals

xi

.I k(u')^2 dx ~ ai (uf,i)^2 h,

Xi-l

xi

j

' (qu (^2) - 2 Ju) dx ~ h( 2' (qu~ ') - 2 Ju).i + (qu (^2) - 2 fu)i-l ) ,
where ai is a functional depending on k( x) on the seg1nent xi-I < x < xi.
There are many ways of taking care of these restrictions. For instance, we
111ight agree to consider
(41)
Xi
j k(x) dx,
x· z-1
etc.
Thus, instead of J[u] we deal with the functional
N N-l
Ih[u] = L ai (n:e,i)^2 h + L (q; yz - 2 J, Y;) h)
i=l i=l
where y is an arbitrary grid function vanishing for i = 0 and i = N:
Yo = YN = 0. The functional Jh[u] is a function of N - 1 variables y 1 , y 2 ,
... , YN _ 1. Equating the first derivatives
f}Jh
~ uy; = 2 cii+i Y:u ,i ( -1) + 2 ai Yx ,i + 2 qi Yi h - 2 Ji h
to zero leads to the difference equations
( 42) x = ih'