1549301742-The_Theory_of_Difference_Schemes__Samarskii

468 Homogeneous Difference Schemes for Time-Dependent Equations

Taking into account the relations Lu + f - it = 0 and

(19) Au=Lu+O(h^2 ), <p=f+O(r^2 +h^2 ),

we arrive at

(20) 'ljJ = (o- - 0.5) TL :U + O(r^2 + h^2 ).

This supports the view that the order of approximation for a given value of

o- coincides with the order of approximation established before for the unit

constant coefficient k( x) 1:

'l/;=0(r^2 +h^2 ) for o-=0.5,

·if;=O(r+h^2 ) for o--j.0.5.

Along these lines, the error of approximation zj = yJ - uj, where yj is

the solution of problem (7) and u = u(x, t) is the solution of the governing

problem ( 1 )-(3), can be most readily eval uatecl with the aid of the equation

Zt = A ( (} z + ( l - (}) z) + 'ljJ ) x E w h , t = j T > 0 )

z(x, 0) = 0, z(O, t) = z(l, t) = 0.

Some progress can be achieved by having recourse to relation ( 11), leaving

us with the estin1ate

(21) llzj+l11A < l (t T 11'1/Jj'11^2 ) l/

2

J"2E j'=O

Using the results obtained in Chapter 2, Section 3, namely the relations

llzll c -<^2 l (1 ) ( ,. ~:r )2] 1/2 -- 1.^2 V(Ao ~)^7 "-")

and the esti1nate c 1 llzll~ < 11.::11~, we derive the inequality

A

( 22) (

j ) l /2

llz~+

1

llc < 2 J~sc 1 j~ rll i/Jj' 11

2

,

thereby justifying by virtue of representation ( 20) that scheme (7) converges

unifonnly with the rate O(h^2 + rrnu), where

·1n = {

2

(J 1

for o-= 0.5,
for o- -:j:. 0.5,
provided that the conditions hold under which the describing scheme is of
accuracy 0( h^2 + rmu) and o- > 0- 0 for 0 < E < 1. A similar conclusion was
drawn in Chapter 5 for the relevant equation with a constant coefficient
k( x ).