1549301742-The_Theory_of_Difference_Schemes__Samarskii

498 Hornogeneous Difference Schemes for Time-Dependent Equations

of the grid at hand. As a final result we get the homogeneous difference

scheme with weights

y 1 =A([)y(a)+<p(x,t), x=ih, i=l,2, ... ,N, t=(j+0.5)r,

l/(x, 0) = uo(a:), YN+1 = Y1, Yo = YN,

where Ay = (a(x, t) y,Jx and the coefficients a and <pare given by the usual

formulas. For example, it is fairly common to deal with

Cl;= '"i-1/2'

The solution can uniquely be found from the above conditions. This sche111e

has the approximation order O((o-- 0.5)r+ r^2 + h^2 ).

In these concerns, there arises the proble111 for determination of y =

AiYi-I - Bi'[;i + A;+1Yi+i = -F;, i = L 2, ... , N,

Yo= " YN, " C; = A; + A;+ I +^1 J

which can be solved by the cyclic elimination method established in Chapter

1, Section 2.

In an attempt to cover all the issues, we should raise the questions

of stability and accuracy for the approximation just established. With

this ai1n, \Ve introduce the space H of all grid functions y(;ri) given for

i = 1, 2,, .. , N, N + 1 and satisfying the condition of periodicity YN+l = y 1 ,

YN = y 0. An inner product and associated nonn in that space are defined

by (v, w) = 'L~ 1 v;wJi and II v II= ~-

The operator A is specified by the relation

Ay = -Ay for y EH.

It seen1s dear that Green's fonnulas are certainly true in the case when
the operator A is defined in such a way. Moreover, A = A* > 0. All
this provides the sufficient background for the possible applications of the
general stability theory outlined in Chapter 6, within the framework of
which the scheme concerned is unconditionally stable for o- > 0.5.
For o- = 1 the maximmn principle is in full force for any T and h,
due to which the resulting scheme is uniformly stable with respect to the
initial data and the right-hand side, What is more, the uniform convergence
occurs with the rate O(r^2 + h^2 ).