1549301742-The_Theory_of_Difference_Schemes__Samarskii

84 Basic Concepts of the Theory of Difference Schemes

where 'P = 'P/ = f ( X;, tj). In giving the difference approxirnation of the

same order for the boundary condition at the point x = 0 we have occasion

to use

_ ou(O,t) ~ o^2 u(O,t) 0(/ 2 )

1lx ' 0 - ux '" + 2 ux '" 2 + 2.

By having recourse to the heat conduction equation for x = 0 we establish

a precise relationship

82 u(O, t) = ou(O, t) _ J( )

(^32) x f) t 0,t,
which implies that
ux(O, t) - h ( ou(O, t) )
2 at - J(O, t)
The expression 011 the left-hand side of this equality approximates the de-
rivative au/ox for x = 0 to O(h^2 ). Replacing ~~ l,,= 0 by the appropriate
difference derivative
u(O, t + r) - u(O, t)
ut ' o = T
we impose the difference boundary condition at the point x = 0:
( 49) Yx o =^1 -2 h Yt o + u Yo - fl1 - ,
' '
whose approximation on a solution of problem ( 4 7) is of accuracy O(h+ r^2 ).
In the case of the implicit scherne Yt = Yx.v + <p the following condition is
acceptable to be an alternative:
(50) Yx, O = ~ h Yt, O + U Yo - Jl 1 '
Exa1nple 3. The second-order hyperbolic equat-ion:
(51)
021l 021l.
f)t2 = f)x2 + .f(x' t)'^0 < x < 1 ,^0 < t < t^0 ,
u(O, t) = u 1 (t), u(l, t) = u 2 (t),
1l () x' 0 = 1lo () x ' ou(O,t) ox -__ 1lo (·) x.