1549301742-The_Theory_of_Difference_Schemes__Samarskii

186 Homogeneous Difference Schemes

If q 2 c 1 > 0, then for a solution of the difference problem (21) the maxi-

mum principle provides the estimate II y lie < c^11 11 'P lie with Yo = YN = 0,

which implies on the basis of (23) the uniform convergence of scheme (21)

with the rate O(h^2 ).

The monotone function (21) is quite applicable when r(x) is a fastly

varying function of the variable x and the condition R < 1 fails to hold at

some points with no influence on the accuracy to a considerable extent.

The same procedure with a monotone scheme works on a non-equidis-

tant grid.

One more method available for designing a monotone scheme is related

to equation (18). In preparation for this, let us multiply equation (18) by

the function μ( x) and require that

μ(ku^1 )^1 +rμu' = (k1iu^1 )'.

This is true only if rμ = kμ', so that

μ ( x) = μ 0 exp { J r( t) dt},

0

The equation

(μ k u^1 )^1 - μ q u = -μ f

can be approximated by the conservative monotone scheme

(24) (fa a Yx) x - μ d y = -μ <p, μ =μ(x- ~h),

where, for instance, the members are taken to be a; = k;_ 112 , d; = q;,

'f!;=fi·

The number μ(x) may be very large when the ratio r = r/k grows.

Dividing both sides of the difference equation

by μi = μ 0 exp { J;i r(t) dt }, we obtain the nonconservative, but monotone

scheme

1 - )

h (b; Yx, i - a; Yx, i - qi Yi = -f; , i = 1, 2, ... , N - 1,

(25)