1549301742-The_Theory_of_Difference_Schemes__Samarskii

Other problems 191

We now estimate the accuracy of scheme (38). Substituting into (38)

y = z + u, where u is a solution of problem (26)-(27) and y is a solution of

problem (38), we may set up the problem for the enor z = y - u:

1

Az = - ( ( 7' r -! h) a z,~) r - dz = -1/J , 0 < 7' = ih < 1 '

( 41)

where 1/J and v are the errors of approximation to the equation

( 42)

and the boundary condition

(43) v=

respectively.
The balance equation (30) with regard to ijJ gives

1 o/,*

1/J; = -;:.^1 7r, i + 'Yi '

'

Next, we set r = r; +sh and develop the expansion of integrals involved in
the formula for 1/J; in powers of h:

r;+1;2 1/2

• 1

1

j f(r) r dr = ]:__ j f(r; +sh) (r; +sh) ds

I 7'; 7';

r,:-1;2 -1/2

l /2

j f(r; +sh)

-1/2

1/ 2

ds + ~ j (f; + shf; + O(h^2 )) s ds

-1/2

h2 I 2)

= f; + - f. + O(h.

12 7' '

'