1549301742-The_Theory_of_Difference_Schemes__Samarskii

606 Economical Difference Sche1nes for Multidimensional Problems

where h:± is the distance between x and x(±la) (h:± <ha)- The preceding

expression for Aexy is in common usage. If x is one of the regular nodes,

then h:± = h:_ =ha, yielding fornmla (16).

Observe that Aa provides a second-order approxi1nation at the regular

nodes Aau - L 0 u = O(h!), while Aau - Lexu = 0(1) at the irregular ones.

The intuition suggests some things in an attempt to write down the

locally one-dimensional (LOS) scheme in conformity with available con-

structions of Section 3. In working on the segment 0 < t < t 0 we introduce

a grid W 7 = {tj = jr, j = 0, 1,. .. ,j 0 } with step T = t 0 /j 0 and involve

arbitrary functions /,, subject to the normalization condition

ex= 1

In line with established practice we replace the governing multidin1ensional

equation by the chain of the one-dimensional heat conduction equations

( 1 g)

Cl'=l,2, ... ,p, x E G,

with the supplementary conditions

Cl'=l,2, ... ,p, v(ex)=μ(x,t) for xEf 0 ,

where tj+c>/P = (j + Cl/p) T.
Here the boundary conditions for v(ex) 111ay be imposed only on some
part r ex of the entire boundary r consisting of the points of the intersection
of r with possible lines Ca parallel to the axis Oxex and passing through
any inner point x E G. The nodal points x E ih ex belong to that part f ex.
'
If, for exan1ple, G = {O < x < lex} is a parallelepiped, then r ex com-
prises the planes xex = 0 and Xex =lex.
Through the approximation of every heat conduction equation with
number Cl' on the half-interval tj+(a-l)/p < t ::=: tj+a/p by the standard two-
layer weighted scheme we ani ve at the chain of p one-di1nensional sche1nes