1549301742-The_Theory_of_Difference_Schemes__Samarskii

Asymptotic stability^335

The usual trick amounts to eliminating yJ+^1 /^2 from (8) so as to reduce the

governing difference equations to the following ones:

As a corollary to this fact, we see that

(10)

Having substituted into (10) the identity of artificial character

yj+1 -yj

yj+1 = yj + T ----

T

or y = y + T Yt, we recast the scheme at hand as

(11)

The accurate account of the approximation error of scheme (11) on a
solution u = u(x, t) to equation (1) can be done using

(12)

Upon substituting into (12) the expansions

u u -2-+ u -'rut -2- = '1l _ - 2 r (Ju -(}-t + O( T 2) 1

h2

Au= Lu+ l2 L^2 u + O(h^4 ),