1549301742-The_Theory_of_Difference_Schemes__Samarskii

Convergence and accuracy of homogeneous conservative schemes 161

This can be done with the aid of the balance equation upon integrating

equation (1) over x from X;_ 112 to X;+ 1 ; 2 :

"''i+1/2 x;+1/2

~ [(ku^1 );+1;2-(ku^1 );- 1 ;2]-~ J q(x)u(x)dx+~ J f(x)dx=O.

Xi-1/2 Xi-1/2

vVith the new variables= (x - x;)/h, the balance equation becomes

1/2 1/2

((kv^1 );- 1 ;2)x - j q(x; +sh) u(x; +sh) ds + j f(x; +sh) ds = 0.

-1/2 -1/2

Upon subtracting the last equation from (4) we establish formulae (6)-(8)

and deduce that if k(x), q(x), J(x) E cC^2 l, then

(9)

For this, we proceed as usual. What is available are the useful expansions

1 I h2 II O(h3)

U; = ui-1/2 + 2 h ui-1/2 + 8 ui-1/2 + '

1 I h2 II O(h3)

ui-1 = ui-1/2 - 2 h ui-1/2 + 8 ui-1/2 + '

which emerge from the chain of the relations

a;= A[k(x; +sh)]= A[k(x,_ 112 + (s + ~) h)]

= k; 112 + h k; 112 A[s + ~] + O(h^2 ) = k;_ 112 + O(h^2 ).

A simple observation that for any v( x) E C(^2 )

1/2

j v(x, +sh) ds = v; + O(h^2 )

-1/2