1549301742-The_Theory_of_Difference_Schemes__Samarskii

66 Basic Concepts of the Theory of Difference Schemes

82 v 82 v

Example 5 L v = Bt 2 - Bx 2 • In that case the values of a grid function at

three instants of time t - T, t and t + T are aimed at selecting the difference

operator LhT, The five-point pattern plotted in Fig. 5.a,b,c is the best

possible for our purposes.

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

(x-h,t)

( x, t)

(x,t-r)

a

(x,t+r)

( x, t)

(x-h,t-r) (x,t-r) (:r+h,t-r)

b

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

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

(x,t) (x+h,t)

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

c d

Figure 5.

One of the possible approximations on the pattern 5.c with using the
value v 55 x on the middle layer t is of the form

(23)

where

(

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

Vft X, t - T 2