1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference approximation of elementary differential operators 65

formulae for L~~ v, L),^1 ) v and L~~) v the following expressions:

. _ ov(x,t) T 02 v(x,t) O( 2)
Ut - Oi + 2 ot 2 + T

= ov(x,t+r/2) 0( 2)

ot + T '

82 v(x, t) h^2 84 v(x t)

V:;;x = 8x 2 + 12 _8_x_ 4 _ ' + O(h^4 )

82 v(:r, t + r/2)

8x^2

T 83 v(x,t+r/2) O(l 2 2 )

2 OX^2 Oi +^2 + T '

_ o^2 v(x,t+r/2) T 03 v(x,t+r/2) O(l2 2)

Vxx = OX2 + 2 OX2 ot + 2 + T.

The outcomes of such manipulations are:

1) (0) - ov(x, t) - 8

2

v(x, t) 0(12 )

LhT v - ot 8x2 + 2 + T

= Lv(x, t) + O(h^2 + r),

meamng, 1/J(D) = L),^0 ) v - Lv(x, t) = O(h^2 + r);

2. L(l) - ov(x, t + r) - o

(^2) v(x, t + r) 0(12 )
In V - ot OX2 + 2 + T
= Lv(x, t + r) = O(h^2 + r),
meaning, 1/J(l) = L),^1 ) v - Lv(x, t + r) = O(h.^2 + r);
3) (0.5) - ov(x,t+r/2) o
(^2) v(x,t+r/2) 0(12 2)
L hT v - ot - 8x2 +! + T
meaning, 1/J(o^5 ) = L~^0 T^5 ) v - Lv(x, t + r/2) = O(h.^2 + r^2 ).
All this enables us to conclude that the operator L~~) provides the approx-
imation of order 2 with respect to h for any (}, the approximation of order
1 with respect to T for (} = 0 and (} = 1, and the approximation of order 2
with respect to T for (} = 0 .5.