1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference approximation of elementary differential operators 79

holds with constant M > 0 independent of I h I and n > 0 both.

We say the difference scheme (37) is of the nth approximation order

if

Denoting by fh and (Luh the values of f(x) and Lu(x) on the grid wh

and taking into account that (f - Luh = 0, we can rewrite 1/Jh as

1/Jh = ('Ph - Lh uh) - ( fh - (Luh)

=('Ph - fh) + ((Luh - Lh uh)

= o/,(1) 'Ph + (^0) 'Ph /.C^2 l J
thus rearranging the approximation error 1/Jh of a scheme as a sum of the
error of approximation 1/;~^1 ) = 'Ph - .f h of the right-hand side and the error
of approximation 1/;~^2 ) = (Luh - Lh uh of the differential operator.
Since 1/Jh is the error of approximation in the class of solutions to
a differential equation, the condition 111/Jh llc 2 h) = O(I h In) holds true if
neither 1/;~^1 ) nor 1/;~^2 ) is of order n. An example in Section 1.3 confirms this
statement.
We now should raise the question: how does the accuracy order of
a scheme depend on the approximation order on a solution? Because the
error zh = Yh -uh solves problern (38) with the right-hand side ·tf;h (and vh),
the link between the order of accuracy and the order of approximation is
stipulated by the character of dependence of the difference problem solution
upon the right-hand side. Let zh depend on 1/Jh and vh continuously and
uniforrnly in h. In other words, if a scherne is stable, its order of accuracy
coincides with the order of approximation.
A rigorous definition of stability of a difference scheme will be formu-
lated in the next section. The improvement of the approxirnation order for
a difference scheme on a solution of a differential equation will be of great
importance since the scientists wish the order to be as high as possible.

6. The attainable order of approximation of a difference sche1ne. As we
   have stated in Section 2.3, the order of approximation to a differential
   operator on a solution of a differential equation can be made higher without
   enlarging a pattern. For convenience in analysis, we take into consideration
   only two different difference schemes which will be associated with problem
   (32) for u' = f(x) and u(O) = u 0. Our starting point is the difference
   scheme of the form

Yx = 'P, Yo = Uo ·