1549301742-The_Theory_of_Difference_Schemes__Samarskii

304 Difference Schernes with Constant Coefficients

is the error of approxin1ation (residual) for scheme (II) on the solution

11 = 11( x, t) to equation (I).

Recall the definition of the order of approximation from Chapter 2,

Section 1.3 saying that scheme (II) approximates equation (I) with order

(m, n) or equation (I) is approximated by scheme (II) to O(hm + rn) on

a solution 11 = 11(x, t) to equation (I) if 111/J(x, t) 11( 2 ) = O(hm + rn) or

111/J 11 ( 2 ) < M ( hm + Tn) for all t E wr, where M is a positive constant, not

depending on h and r, and II · 11( 2 ) is a suitably chosen norm on the grid

We proceed to the estimation of the order of approximation for scherne

(II) under the agreement that 11 = 11(x, t) possesses a number of derivatives

in x and t necessary in this connection for performing current and subse-

quent manipulations. Within the notations

11

811

8t ,

I 011

11 = --;::;-- ,

ux

the development of Taylor's series for the function u = 'U(x, t) about the

node (xi, ij+i; 2 ) leads to the expressions:

U = 0. 5 ( U + 11) + 0. 5 ( U - 11) = 0. 5 ( U + 11) + 0. 5 T 11t ,

11= 0.5(tt+11)-0.5r11t,

O"U + (1-o-) 11 = 0.5(u+11) + (o-- 0.5) T11t.

All this enables us to reduce 1/J to 1/J = 0.5 A ( u+11) + ( o--0.5) TA 11t -11t +\O·
Substitution of the set of expansions

L 11

, - 1 ' T2 - 3

11=11+ 2 r11+ S ii+O(r ),

1 ( ~ ) T

2

2 11+11 =fl+S11+ o, 0( T 3) ,