The summarized approxiination methodallowing Theorems 1-2 to hold.
2) A quasilinear equation of parabolic type.
The complete posing of problem (15) includes
8 01l
La tl = OXa ( ka(x, t, u) OXo:), 0 <Ci ~ k 0 ,Two possible ways of approxin1ating the operator Lo: are:a) AL, Y(o:) = (ao:(x,l, 0.5 (Y(c>) + y(~)la))) Yxa) :Va I t = tj+l/2 I
617b) AcrY(u)= (acr(;r,l,0.5(y(o-l)+Yi:~':i)Jyx,,).1" t=ij+1/2·
In the first case a nonlinear equation is ai1ned at detennining Y( o: l that
can be solved by one or another iterative n1ethod. Every iteration can be
found during the course of the elimination.
In the second case we obtain a linear equation related to Y( o:) and then
use the elimination method for solving it. The uniform convergence with
the rate 0( T + h^2 ) takes place under the extra restrictions concerning the
boundedness of the derivatives c!2ko:/8tt^2 , 82 ko:/8xo: au, 82 ko:/ax;.
Locally one-dimensional sche111es find a wide range of applications in
solving the third boundary-value proble1n. If, for example, G is a rectangle
of sides li and l 2 or a "step-shaped" domain, then equations (21) should be
written not only at the inner nodes of the grid, but also on the appropriate
boundaries. When the boundary condition 8tt/8xi = a-;-tt+v}-l is in1posed
on the side Xi = 0 of the rectangle {O ~ xo: ~lo:, a= 1, 2}, the main idea
behind this approach is to write for n = 1 equations (21) at the node J:i = 0
as well. This can be dom' by setting
A ·1 - Y C^1 J:r:^1 - O"(-)y i C^1 l
1 Y(1) - 0.5 hiassuring the unifonn convergence of the describing locally one-di1nensional
scheme with the rate O(r + lhl^2 ).
- Additive sche111es. The general fornmlations and statements. Consid-
erable effort is devoted to a discussion of additive schemes after introducing
the notion of summarized approximation. vVith this aim, we recall the no-
tion of the n-layer difference schen1e as a difference equation with respect
tot of order n - 1 with operator coefficients:
n-1
2= Cμ(tj) y(tj+l - f3 r) = f(tj),
f3=CI