1549301742-The_Theory_of_Difference_Schemes__Samarskii

322 Difference Schernes with Constant Coefficients

It is straightforward to verify that at the point x = 1 the boundary condi-

tion

(54)

8u(1, t)

• ax = f32 u(l, t) - μ2(t), (3 2 = const >^0 ,

is approxi1nated to the same order by the difference condition

• h2 -
  zp = f + 12 !"

and replacing !hYt,o and ~hYt,N in (51) and (55) by !hPiYt,o and

~ h p 2 Yt, N, respectively, with Pk = 1 + i h f3k, k = 1, 2, we establish the

difference boundary conditions providing an approximation of O(h^4 + r^2 )

for (} -- (} * -- l - 2 i_ 12 h^2 T-l.

To avoid cumbersome calculations, it will be sensible to introduce

more compact notations

A-y = Yx - f31 Y

~h

and rewrite the difference boundary conditions (51) and (55) in the same

forn1s as approved ·before for scheme (II):

(56)

Yt = A - ( (} y + ( 1 - (}) y) + zp-,

Yt = A+ ( (} y + ( 1 - (}) y) + zp +,

x = 0,

x = 1,

with zp 2μ 1 /h and zp+ = 2P, 2 /h, showing the new notations to be
sensible ones. For (3 1 = {3 2 = 0 these can be viewed as the difference
approxin1ations of the second kind boundary conditions. Also, the order of