1549301742-The_Theory_of_Difference_Schemes__Samarskii

The summarized approxim.ation method 613

With regard to problem (32) for w, it will be sensible to rewrite state-

ment (32) in the canonical form (28)

1 wj+a/p + ..!:. wj+(a-1)/p + 1n*j+a/p

+ }*! _ /! 1 , 1.0:-l T ra '

a

cx=l,2, ... ,p,

wj+a/p = 0 for x E lh,a, w(x, 0) = 0,

meaning w = 0 on the boundary S of the grid 12:

w(P) - (^0) for PE 8.
The right-hand side <p differs frorn zero only at such nodes ( x, t'),
where x E w;,. The trace of the homogeneous boundary condition w = 0
should be clearly seen in
1
h2 ' where h = 1nax °' hu.
Applying Theorern 4 fr01n Chapter 4, Section2 yields
(35)
ll
n1ax ly(P)I ~ max <p(x ' t')ll < max h^2 ll'Pllr·.
n+s t' Ew' T D 1 ~· -' t' Ew' T _,
In the esti1nation of the function c we write down equation (:31) in the
canonical form (28) by regarging P = .r to nodal points of the p-dinwnsional
grid wh:
·uj+a/p = 0 fo1· x E,..,, I h.a v(J~,0)=0.