1549301742-The_Theory_of_Difference_Schemes__Samarskii

Heat conduction equation with constant coefficients

or

cry Yi-1 - (2 err+ 1) Yi+ cry Y;+1 =-Fi, i = 1, 2, ... , N - 1,

(37) Fi= ( 1-O")/ Yi-l + (1-2(1-O")/) Yi+ (1-O")/ Y;+ 1 + rzpi,

T

I = h2 ' i=l,2, ... ,N-1.

315

The theorem we have mentioned above asserts that for a solution to the

difference equation

A; Yi-1 - C; Yi+ B; Yi+1 =-Fi, i = 1, 2, ... , N - 1,

Yo= 0, YN = 0' I Bi If- 0'

the estimate

llvllc < II F D II c

is valid if and only if Di = IC; 1-1Ai1-1 Bi I > 0. For problem (36) these

conditions (I Ai If- 0, I B; If- 0, Di > 0) are satisfied for O" > 0 and D; = 1,

so that a solution to equation (37) can be most readily evaluated as follows:

llYlle<llFlle for O">O,

(Yi = F; for O" = 0). A simple observation that

if 1-2(1-0")1>0,

may be useful in the further establishment of the inequality

(38)

which is valid for scheme (36) under the constraint

(39)

Summation of (38) over j'
problem (16) solution

h2

T <

• 2(1-0")

0,1,2, ... ,j leads to the estimate for the

(40) II Yj+^1 lie <II Y^11 lie+ t^7 II~' lie·
j'=O