1549301742-The_Theory_of_Difference_Schemes__Samarskii

52 Basic Concepts of the Theory of Difference Schernes

T / N 2 , respectively. After that, we draw the straight lines parallel to the

appropriate axes through the splitting points. The points of the intersection

of those lines are adopted, as usual, as the nodes (xi, tj), which constitute

a widespread grid

with steps hand T along the directions Ox and Ot, respectively (see Fig. 2).

t

T

t J -- ]. T

T

0

.

(x 1 ,tj)

h Xz -- /h x

Figure 2.

The nodes lying on one and the same straight line (horizontal or ver-

tical), the distance between which is equal to the grid step (h or r), are

called adjacent grid nodes.

Exa1nple 3. A non-equidistant grid on a segment. The next example
is devoted to the segment 0 < x < 1 with N subintervals without concern
for how the points 0 < x 1 < x 2 < ... < x N _ 1 < 1 will be chosen. The
nodes {xi, i = 0, ... , N, x 0 = 0, xN = l} constitute what is called a non-
equidistant grid w h [O, l]. The distance between the adjacent nodes, being
a grid step, equals hi = xi - xi-i and depends on the subscript i. Any
distance of this type falls within the category of grid functions. The steps
should satisfy the normalization condition 2=~ 1 h; = 1.