1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference approximation of elementary differential operators 81

we focus our attention on the three-point difference scheme

Yxx - dy = -<p(x)' x=ih, i=l,2, ... N-1, Yo= YN = 0 ·

\!Ve will show that its order of approximation on a solution u = u( x) can be

made higher without enlarging a pattern under a proper choice of cl and <p.

For a solution u = u(x) of the original proble1n, the residual 1jJ =

uxx - du + <p will be involved in the expansion

h2

u-xx = u^11 + - 12 u(^4 ) + O(h^4 )

with regard to the equation u^11 = q u - J( x ), permitting us to deduce that

h2

·tf; = (q - cl) u + (<;:> - !) + - ttC^4 l + O(h^4 ),

12

thus causing the behaviour like O( h^2 ) for cl = q and <p = f. Substitution
of uC^4 ) = q^11 u - f^11 = q (q ll - f) - fx:r + O(h^2 ) into the formula for 1/; yields

A higher-order approxirnation 1jJ = 0( h^4 ) can be achieved on the solution
u( x) of the initial equation by merely setting

h2

<p = f + 12 (fxx + q J) ·

We now turn to the question of approximations of boundary and initial
conditions on a solution of the original problem. This question is intimately
connected with the statement of a difference problem.

7. Approximations of boundary and initial conditions. From Section 2.5
   it seems clear that the accuracy of a scheme depends on the order of ap-
   proximation on a solution of the original problem. By an approximation we
   mean not only the equation itself, but also the supplementary (boundary
   and initial) conditions.
   In this section we give several examples of raising the order of approx-
   imation for boundary and initial conditions without upgrading a pattern.