1549301742-The_Theory_of_Difference_Schemes__Samarskii

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where

we find that

Difference Schemes with Constant Coefficients

au

at ' ii.= ui t= t j+1/2 ,

Hence, 1/; = 0( h^2 +r^2 ) for O" = 2-v'2, satisfying the equation i-0"^2 -O"+ ~ =

0. Other ideas are connected with the method of separation of variables,
   whose use permits us to look for a solution to equation (10) with zero
   boundary conditions Yo = YN = 0 as a sum

N-l

yi = L Tj Xk(x),

k=l

By inserting this expression in (10) we obtain the equation related to Tj:

or, what amounts to the same, rj+^1 = qk Ti, where

Furthermore, adopting arguments similar to those being used for the scheme

with weights we arrive at the chain of the inequalities

with p = maxk I qk I. In the further analysis we shall need yet, among other
things, the formula
1 - (1 - ()") r>.;^1

( 1 + ~ O" T >.-; h)

2

,

p

where >.;^1 = 4h-^2 sin^2 (7rh/2). In giving it we will show that maxk lqk i 1s
attained for k = 1. With this ain1, we have occasion to use the function

1-(1-0")p

f(p) = ( ) 2 ,

l+~O"p