1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference approximation of elementary differential operators 75

in correspondence with problem (32). Here the right-hand side 'Pi can be

specified in a nurnber of different ways, for instance, by the formulae

'Pi = j ( X;) '

provided the condition 'Pi - f; = O( h) holds.

Thus, given y 0 = u 0 , the solutions can be found from the recurrence

relation Yi+i =Yi+ h 'Pi, i = 0, 1, 2 ....

Example 2. The Cauchy probleni for a syste1n of first-orde1' dif-

ferential equations:

t > 0' u(O) = u 0 ,

where A = (ai 1 ,,) is a square n x n-rnatrix and u = (u 1 , ..• , un) is an n-

dimensional vector. As we will see later, it seems reasonable to introduce

the grid w 7 = {tj = jr, j = 0, 1, 2, ... } with step r. One of the possibilities

is Euler's difference scheme

yj+1_yj.

----+A y^1 = 0,

T

j = 0, 1, 2, ... ' Y j -- ( Y 1 j , Y 2 j , · · · , Yn j) ·

The difference problem under consideration will be completely posed if the

subsidiary inforrnation is available on the vector yj for j = 0 with the

initial condition y^0 = u 0. The value y j +i is successively calculated by the

explicit formula

yj+i = yj - rAyj.

This is one way of gaining experience with Euler's scherne that can be

accepted and used in theory and practice.

Example 3. The boundary-value problem:

(33) u"(x)=-f(x), 0<x<1, u(0)=μ 1 , u(1)=μ 2 •

We take once again the equidistant grid

wh = {xi= ih, i = 0, 1, ... , N, hN = 1}

and set up on it the difference problem

Yxx = -<p or

i= 1,2,. .. ,N-1,

Yj+1 - 2 Yi - Yj-1

h2 = -<p; J

Thus emerged the system of algebraic equations with a tridiagonal matrix.
Because of this form, the elimination method may be useful (see Chapter 1,
Section 1).