1549301742-The_Theory_of_Difference_Schemes__Samarskii

The su1nn1arized approxiination inethod 597

where fco:(.T:, t), Cl'= 1, 2, ... ,p, are arbitrary functions with the same smo-

othness properties as the function f( x, t), satisfying, in addition, the nor-

malization condition

!1 + f2 + ... + f p = f.

In working on the seg1nent 0 < t < t 0 with an equidistant grid

W 7 = {tj = jr, j = 0, 1, ... ,j 0 }

with step T it seems reasonable to divide every interval into p parts by

recording the points ij+cx/p = tj + Cl'T /p, Cl'= 1, 2, ... ,p-1. Half-intervals

tj+(a-l)p < t < tj+cx/p denoted by 6cx are made up, as usual, by those

points. By successive solution of the equations starting from a= 1, 2, ...

(6)

under the additional assu1nptions

ct= l, :2, ... ,Ji,

we find the values v(x, tj) = v(p)(x, tj), j = 0, 1, 2, ... , j 0 , with further

treatment as a solution of the problem concerned. For the sake of simplicity,

it is supposed that we have on the boundary the homogeneous boundary

condition of the first kind and this should confine no generality of further

motivations.

Every equation Pavcx = 0 or

( 6') Cl'=l,2, ... ,p,

is replaced by the newly formed difference sche1ne in which au/ at and Lex
are approximated by the appropriate difference expressions of the general
form on a grid w h with steps h 1 , h 2 , ... , hp

(8) ct=l,2, .. .,p.

In the simplest case the resulting two-layer scheme is aimed at con-
necting the values

Y(cx) = ~+cx/p and Y( cx-1) _ -^0 Y j+(cx-l)/p ·