1549301742-The_Theory_of_Difference_Schemes__Samarskii

502 Homogeneous Difference Schernes for Tirne-Dependent Equations

n1ain idea behind this approach was explained earlier in Chapter 3, Section

4 and Section 1 of the present chapter).

For convenience in analysis, the residual ·i/J is representable by

( 13) 1/J = 7li: + 1/J*,

where

(14)

OU 2 h; of o^3 u

( k -0 x ) i-l/2. + (J T aiUftx,i + -8 ( -0 x - 0t2 (^0) x ) i-1/2. '
(15) 4i'=O(r^2 +h").
Apparently, the current situation needs certain clarification. Having inte-
grated equation (1) with respect to x at a fixed moment t = tj from xi-l/ 2
to a;i+1/2
(16) (k ou)
OX x=1:,-0.5
'"i+l/2

• j f(x,tj) dx- dx = 0,
  we then divide this identity by !ii' subtract the resulting expressions from
  the right-hand side of representation (12) for the residual 1/J and, finally,
  get
  (17) ( k -0 01l). ) - Uft,i +'Pi
  x i-1/2 i: t
  '
  where the coefficients ai and 'Pi are given for fixed t = tj by the same
  formulas as stated in Chapter 3, Section 4.
  Let xi be a discontinuity point of both functions k and f. To avoid
  cumbersome calculations, the usual practice involves the simplest formulas
  for finding ai and 'Pi:
  (18) Ji± = f(x; ± 0).