1549301742-The_Theory_of_Difference_Schemes__Samarskii

The summarized approximation 1nethod 635

and

{

O(h^2 + r^2 )

1/J* = °'

°' O(h°' + r^2 )

for

for

Whence it follows that

0 0

,1.i 'f/1+\U2 ,1.J+l/2 -- (^0). 5(£ JU - (^0) , (^5) u+<l .. f )j

. 0
= 0.5 ((Li+ L2) ll - u + f1 + !2)^1 +0.5T1/J2f,'

With equation (63) in view, the first summand becomes zero:

yielding

0. 0.

(81) 1/;{+1/J~=O.

By definition, this means that scheme (75 )-(77) generates a summarized

approximation. The accepted view is to involve the sum

making it possible to de cl uce that

0 0 - 0 0

(82) 1/J2+21/;1 + 1/Jo - = T^2 (1/J 2 )-t,t, - = O(r^2 ).

No wishing to load the book down with full details on this point, we
cite here only final results: LOS of the form (75)-(77) converges in the grid
norm of the space l¥i with the rate O(r + lhl^2 ) if the solution v = u(x, t)
has in Qy continuous derivatives of the first four orders, the derivatives
84 u/ ox! satisfy the Lipshitz condition in t and the right-hand side f is
twice differentiable in t.