1549301742-The_Theory_of_Difference_Schemes__Samarskii

The alternative-triangular method 703

lines x°' = x~"), i°' = 0,±1,±2, ... , Ct= 1,2, the points xi= (x~ii),x~i^2 ))

constitute the basic pattern R~ in the plane (x 1 , x 2 ). Any such point xi

belonging to G is called an inner node of the grid, the total collection of

which is denoted by wh, that is, wh ={xi E GnR~}.

The intersection of the domain G and any straight line passing through

a point x E wh and in parallel to the axis Ox°' consists of the interval

,3.a( xi). The endpoints of this interval are called boundary nodes in the

x°'-direction. A set of all boundary nodes in the x°'-direction is designated

by /a. The boundary ih of the grid w h is just the union ih = / 1 U / 2 ,

making it possible to write clown wh = wh u ,,,.

Furthermore, let wc~(xf3 ), ,13 = :3 - ct, et = 1, :2, be a set of nodal

points from the interval ,3.°'; let w;t" ( x f3) be a set containing w °' and the

right endpoint of the interval ,3.°'; let w°'(xf3) be a set containing w°'(xf3)

and both endpoints of the interval ,3.°'. For an inner node x E w°'(xf3),

we denote by xC+lc.) and x(-lc.) adjacent nodes belonging to w°'(xf3). If

xC+^1 "J E '°''it may happen that this node does not coincide with the node

xCi"'+l) of the grid at hand. The accepted view is connected with steps

h°'! ( x) of the grid as possible spacings between the nodal points x E wh and

the grid nodes xC±l") E wh.

What is more, at all inner nodes of the grid wh the mean steps are

taken to be

vVhen the pattern of interest happens to be nonuniform in either of the
directions x °', that is,

i°' = 0,±1,±2, ... '

the n1ean steps n°' = h°' bec0111e constant and n! at a near-boundary zone
differ fr0111 h °':

With these ingredient.s, a reasonable difference scheme is

(81) Ay = -<p(.r), x E wh, y(:r) = p(x), x E /ii,