1549301742-The_Theory_of_Difference_Schemes__Samarskii

80 Basic Concepts of the Theory of Difference Sche1nes

At the next stage we look for the residual 1/; = u,c -<p on a solution u = u( x)

to the equation u' = f ( x). The traditional way of covering this is to develop

Taylor's series

h2 h3

u(x + h) = u(x) + hu' +Tu"+

6

u"' + O(h^4 )

with the forthcoming substitutions u' = f, u" = J', ... and further refer-

ence to expansions

h h^2

1/; = u' +

2

1/' + (f u"' -<p+ O(h^3 )

= J(x) - <p(x) + ~ u^11 + ~

2

u"' + O(h^3 )

h h^2

= f(x) - <p(x) + 2 J'(x) + (f u"' + O(h^3 ).

Upon substituting <p = J(x) + thf' or '-Pi= (f + thfx)i we obtain the

scheme of order 2 on the solution u = u(x), thus clernonstrating that the

residual 1jJ behaves like O(h^2 ).

By virtue of the relations

u"' = f" = fxx + O(h^2 )

and

I 1 2

J = fx - 2 h fxx + O(h )

we find that

h h^2

1/J = J + 2 fx - 12 fxx + O(h^3 ) - '-P'

thereby clarifying that the scherne Y:i: = <p with the right-hand side

is of order 3 on the solution ll = u(x) (1/J = O(h^3 )).
For the boundary-value problem

u" - qu = -f(x), 0<x<1,

u(O)=O, u(l)=O, q=const,