1549301742-The_Theory_of_Difference_Schemes__Samarskii

470 Homogeneous Difference Schemes for Time-Dependent Equations

where vright = u(e + 0, t).

Furthermore, using the expansions behind

un+l = u(O + (1 - ti) h u;·ight + 0.5 (1 - t1)^2 h^2 u;:ight + O(h^3 ),

we find that

wn+I =an+ I ux,n+I = an+I (ti u;eft + (1 - ti) u;·ight) + O(h).

As a corollary to the condition of conjugation [ku'] = 0, we might have

w( ~, t)

kleft '

I w(~, i)

(k u')teft = (k u')right = w(~, t)' uright = k "right '

so that

(29) iun+I = Cln+l ( -k ti - + 1-ti) w(~, t) + O(h).

left krigbt

Upon substituting (26), (28) and (29) into (25) and (27) we finally get

h1/J 11 = Q~a) + O(h),

Qn= [an+l(ktl + ~-tl)-l]w(~,t),

lelt nght

showing the new inembers to be sensible ones. The limiting values emerging
from equation ( 1) are taken to be

( k u')'.·ight = ( U - f)x=~ ,

since [it] = [f] = 0 for .r = ~. TllP next step in this direction is to insert
the assigned values in the preceding expansions with further reference to

• 2 2
  'Pn = fn + O(h + T ) ,