1549301742-The_Theory_of_Difference_Schemes__Samarskii

230 Homogeneous Difference Schemes

factors: an error in calculations of pattern functionals, an error in specifying

coefficients of a differential equation, rounding errors, etc.

vVe say that a scheme is stable with respect to coefficients (co-

st able) if a solution of the boundary-value problen1 has slight variations

under sn1all perturbations of the scheme coefficients. In order to avoid

misunderstanding, we focus our attention on the scheme with coefficients

a, d, t_p

0 < x = ih < 1, y(O) = 0, y(l) = 0,

and the same scheme with perturbed coefficients ii, d, <p (for the sake of

simplicity the values y(O) and y(l) remain unperturbed)

0 < x = ih < 1, j/(O) = 0, jj(l) = 0,

under the conditions

a(x)>c 1 >0, ii(x) > c 1 > 0,

(3)

d(x)>O, cl(x)>O, c 1 = const > 0 ,

with constant c 1 independent of a grid.

We estimate the difference z = y-y in tenns of perturbed coefficients.

Substituting y = z + y into (2) and taking into account (1), we get

(4) z 0 =ZN= 0,

where

(5) iJ! = <p- cp +(A - A) y = <p - cp +((ii - a) Yx),, - (cl - d) y.

From ( 4) it is easily seen that iJ! is representable by

(6)

where

(7)

and TJ is determined from the conditions T/x = <p - t_p - ( d - d)y and 171 = 0,
so that

i-1

(8) TJi= L h[(<pk-cpd-(dk-dk)Yk], i = 2, 3, ... , N, TJ 1 = 0.

k=l