1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference approxirnation of elementary differential operators 63

where

8(1+0.5s)

3

for - 2 < - s -< -1 ,

8 (1 +0.5s)

3

-4(1 +s)^3 for -l<s<O

g(s) =

• 
  
  
  
  • ,
    8 ( 1 - 0 5 s)
    3
  
  
  
  


• 4 ( 1 - s )^3 for O<s<l,
  8(1-0.5s)
  3
  for l<s<2.

The reader is invited to do it on his/her own.

Example 4 L v = ~~ - ~:~, v = v(x, t). Let (x, t) be a fixed point in

the plane ( x, t) and let h > 0 and T > 0 be two arbitrary numbers taken

as steps in the sequel. A difference approximation LhT of the operator L is

connected with a proper choice of the pattern.

vVe begin by placing approximations of the simplest type for which

the pattern consists of the four points (Fig. 4.a), so that LhT is certainly

expressed by

( 19)

(O) _ v(x,t+r)-v(x,t)

L In V - --------

T

v( x + h, t) - 2 v( x, t) + v( x - h, t)

h2

A suitably chosen symbolism may symplify the form of writing various

difference expressions and makes our exposition more transparent. This is

acceptable if we agree to consider

v=v(x,t), v=v(x,t+r), v=v(x,t-r).

Within these notations, the difference derivative with respect tot becomes

v(x, t + r) - v(x, t) v-v

(20)

T T

By virtue of relations (7) and (20) we write down (19) in the form

( 19')

In the preceding constructions of L),^0 ) we have taken the value of V;;;;i; at the
moment t, that is, on the lower layer. On the pattern depicted in Fig. 4.b it