1549301742-The_Theory_of_Difference_Schemes__Samarskii

114 Basic Concepts of the Theory of Difference Schemes

This inequality will be aimed at estimating the rate of convergence of

scheme ( 44). We write beforehand the equation for the accuracy of scheme

(44): z = y-u, where u is a solution of problem (43) and y is a solution

of the difference problem (44). Upon substituting y = z + u into (44) it is

plain to set up the problem for z;

( 49) zxx + 1/J(x) = 0, Z 0 =ZN = 0.

Here 1/J(x) = uxx + J(x) is the approximation error of scheme (44). For all

sufficiently smooth functions u(x) it is well-known that 1/;(x) is a quantity

of order O(h^2 ), thus causing the same type of the problem for the function

z(x) as occured for the function y(x). Because of this fact, estimate (48) is

still valid for z( x ):

(50) II z lie< 111/J II I (4J2).

However, 1/J = O(h^2 ) and, consequently, II z lie= II y-u lie< M h^2 , where

Mis a positive constant independent of step h. In agreement with the above

definitions (see Section 1) estimate (50) provides the uniform convergence

of a solution of the difference problem (44) to a solution of the differential

problem (43) with the rate O(h^2 ).

So far we have established an estimate for the rate of convergence in

a very simple problem. It is possible to obtain a similar result for this

problem by means of several other methods that might be even much more

simpler. However, the indisputable merit of the well-developed method

of energy inequalities is its universal applicability: it can be translated

without essential changes to the multidimensional case, the case of variable

coefficients, difference schemes for parabolic and hyperbolic equations and

other situations.

Let us show, for example, that this method leads without any difficul-

ties to the desired result for the case of a non-equidistant grid.

Example 2 Let a non-equidistant grid w h be given on the segment [ 0, 1].

On this grid problem ( 43) can be approximated as follows:

(51) Y.ri: + J(x) = 0, Yo= YN = 0,

(for the notations see Section 1.3, Example 1).
For problem (51) one can derive an a priori estimate of the same type
as estimate ( 48) for problem ( 44). However, in this case such an estimate
fails to provide with a quite reliable idea on the speed of convergence of