1549301742-The_Theory_of_Difference_Schemes__Samarskii

350

where

Difference Schemes with Constant Coefficients

1

ck= (ll 0 , Xk) = J u 0 (x) Xk(x) dx,

0

Granted the grid whT = wh x Wy with wh = {xs =sh, s = 0,1, ... ,N,

hN = 1} and W 7 = { tj = j T, j = 0, 1, ... }, the difference scheme with

weights

y( X, Q) = Uo ( X) , Yo= YN = 0,

where A y = Yxx and O" = 0" 0 + i 0" 1 is a complex number, comes first.

Of our initial concern is the residual

=A il+u +r ( O"--1) Au -u

2 2 t t

= Ail + ( ()" - ~ ) TA u - i u + 0( r^2 )

= (Lu-i:U)+ h

2

2

L^2 u+(()"- ~)rL:U+O(h^4 +r^2 )

1 -

= ( ~~ i + ( O" - ~) T) Lu+ 0( h^4 + r^2 ),

where u = u( x, tj + ~ r), u = au/ at and Lu = a^2 u/ ax^2. It follows from

the foregoing that

O(h^2 + r^2 ) if

1

()" = -2 ,

1 ·h?

1/J = O(h^4 + r^2 ) if

z -

O"=----=O"

2 12 T *'

O(h^2 + r) if

1

0"#2, ()" # ()"* •

The scheme so constructed is of increased accuracy for the parameters

1 i h^2

()" =

2 12 T '

1

O"o = 2 ,

h2

12 T.