1549301742-The_Theory_of_Difference_Schemes__Samarskii

4 Preliminaries

and offer below some of its approximations with new notations Vi= v(xi),

Vi+l = v(xi+d and x,:+1 = ;ri + h, where h > 0 is the distance from Xi to

Xi+ h, namely

( ~~)

~ u(xi + h) - u(xi) Vi+l - 'Ui ,6. Ui

."C=Xi h h h

( ~~) x=xi

~ Vi - Hi-1 \J 1li

h h

or

( ~~ )

~ 1li+1-V·i-l ,6. U; + \J U.;

x=.ri 2h 2h

All of the preceding difference expressions approach ~~ as h ----+ 0. The

symbol ~ rneans, as usual, approximation or correspondence. In what

follows we say that the expression

,6. iii U.i + 1 - '1li

h h

approximates the first derivative ~~ = u'.

The very definition implies that the equation

boy·

h z = f' i, f; = f(x;),

is a first-order difference equation in one or another form

b. Yi = hf; or Yi+l = Yi + hf;.

There is no difficulty to solve this equation for a given initial value y 0 •
It should be noted here that a second-order difference equation also may
appear in approximating a first-order differential equation, for example, in
connection with these expressions

v.(xi+l) = u(x;) + h u.'(:r;) + ~ h^2 v.^11 (x;) + ~ h^3 u.^111 (a:i) + O(h^4 ),

u.(;ri-1) = u(x;) - h u'(x;) + ~ h^2 v^11 (xi) - t h^3 u.^111 (x1) + O(h^4 ),