1549301742-The_Theory_of_Difference_Schemes__Samarskii

Methods for designing difference schen:1es 229

by means of the relation

h

J

1 l

( qu - J) dx =

2

h ( qu - f)o + 2 h ( qu - J)i ,

0

etc. Accepting, for instance, V; = D;, io, 0 < i 0 < N, and recalling that

vx,i = 0 for i <'lo and i > i 0 + 1, vx,io = l/h and v;;;,,: 0 + 1 = -l/h, we find

for i = i 0 that

or (ay,r)x - dy = -f,

Ifv; = D;, 0 , then vx,i = (-1/h)b;, 1 and identity (49) yields

(-l/h)a1Yx,l+8'1 Yo - P1 = 0

or al Yr, 1 = O'l Yo - P1' Likewise, for vi = oi,N

With these, we arrive at the difference boundary-value problem

0 < x = ih < 1,

• 
  
  
  
  • a1 Y~,_o = 0'1 Yo - fl1,
  
  
  
  

3.9 STABILITY WITH RESPECT TO COEFFICIENTS

1. Stability of difference schemes with respect to coefficients. In solv-
   ing some or other problems for a differential equation it may happen that
   coefficients of the equation are specified not exactly, but with some error be-
   cause they may be determined by means of some computational algorithms
   or physical measurernents, etc. Coefficients of a homogeneous difference
   scheme are functionals of coefficients of the relevant differential equation.
   An error in determining coefficients of a scherne may be caused by various