1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference approximation of elementary differential operators 83

since μ 1 + u'(O) - u u(O) = 0. This provides support for the view that

v 1 = O(h). It is necessary to rnake some changes in condition (42) to

achieve the order of approxirnation O(h^2 ). This is due to the fact that

u(x) is just the solution of the original problem (39). From the governing

differential equation the value u^11 (0) can be expressed by

( 44) u^11 ( 0) = q u ( 0) - J ( 0).

Substitution of ( 44) into ( 43) yields

( 45) ux,o- ~h(qu(O)-J(O)) =u'(O)+O(h^2 ),

thereby justifying that the left-hand side of ( 45) approximates to second

order the derivative u'(x) at the point x = 0 on the solution to the equation

u^11 - q u = -f.

From here and ( 42) it follows that the approximation of the boundary

condition

( 46)

is of order 2 on a solution of problem (39).
It is worth emphasizing here that we have succeeded in raising the
order of approximation without enlarging the total number of grid nodes
which will be needed in this connection for approximating the boundary
con di ti on.

Example 2. The boundary-valne proble1n for the heat conduction

equation:

(47)

au 82 u.

Ft= 8x2 + .f(x, t)'^0 < x < 1 ,^0 < t < t^0 ,

u(x, 0) = 1t 0 (x),

8u(O, t)

a =uu(O,t)-μ 1 (t),

x

u ( 1, t) = μ2 ( t).

On the grid w hT arising from Section 2.1 it is simple to follow the explicit
scherne of accuracy O(h^2 + r):

( 48) Yt = Yxx + 'P' y( X, 0) = U 0 ( X) , y( 1, i) = fl2 ( i) ,