1549301742-The_Theory_of_Difference_Schemes__Samarskii

Homogeneous difference schemes for the heat conduction 475

5. Homogeneous schemes on nonequidistant grids. There is no doubt that
   the users come across nonequidistant grids in both x and t variables in
   practical imple1nentations of s01ne or other proble1ns. All the preceding
   results and esti1nates remain valid for the tvvo-layer sche1ne being used on
   a nonequiclistant in t grid. That is to say, the nonuniforrnity in t has no
   considerable impact in such inatters and it should be taken into account
   only in the selection rule for the step T = Tj depending now on the subscript
   j. The order of approximation in t remains unchanged, but the symbol
   O(rm) will stand either for O(rjm) or for 0((
   1

T~x rj)m), where there is

_J _Jo

no clanger of confusion. The situation in which the grid is nonequiclistant

in x needs investigation by exactly the san1e reasoning as before.

Let wh ={xi, i = 0,1,, ... ,N, Xo = 0, XN = 1} be an arbitrary

grid on the seg111ent 0 < x < 1 with steps hi= xi -xi_ 1 , i = 1,2,. .. ,N.

In line with established priorities from Chapter 3, Section 4 the operator

Lu= -a ( k-au)

ax ax

is approximated by the difference operator

with the san1e coefficients ai as was clone on equidistant grids. This is

acceptable if we take, for example, under the constraint -1 < s < 0 (see

Section 2)

( 45) ai =A [k (x.; + shi)] or ~-A[ l l

ai - k (xi + sh;) ·

The right-hand side is calculated by the simple formula

( 46)

assuming f ( x, t) to be a continuous function of the argument x. .Yhen
some such function n1ay have discontinuities of the first kind at the nodal
points, the obstacles involved can be avoided by setting either

(47) 'Pi hi f;-o + h1+1 f;+o
2 Ii.;