1549301742-The_Theory_of_Difference_Schemes__Samarskii

Heat conduction equation with constant coefficients^321

9. The third kind boundary conditions. The first kind boundary conditions
   we have considered so far are satisfied on a grid exactly. In Chapter 2 we
   have suggested one effective method, by means of which it is possible to
   approximate the third kind boundary condition for the forward difference
   scheme ( O" = 1) and the explicit scheme ( O" = 0) and generate an approxi-
   mation of 0( T + h^2 ). Here we will handle scheme (II) with weights, where
   O" is kept fixed. In preparation for this, the third kind boundary condition

(50)

ou(O, t)

ox = (3 1 u(O, t) = ~t 1 (t), (3 1 = const >^0 ,

will be imposed at the point x = 0, providing later the difference boundary

condition on the four-point pattern with the nodes (0, tj+I), (h, tj+I), (0, tj)

and (h, tj)· Once supplied by the difference condition

where 10 = f (0, t j +I/2) and PI = μ 1 ( tj +I/2), we will show that it approx-

imates condition (50) on a solution u = u(x, t) to equation (3) subject to

condition (50) and the order of approximation is the same as we obtained

in approximating equation (3) by scheme (II) for a given value of O".

Upon substituting y = z + u into (51) we find that

(52) O" (zx - (JI z)o + (1 - O") (zx - (JI z)o =! h Zt,o - VI,

where VI= O" (ux - f31 u)o + (1-O") (ux - f3I u)o -! h 1lt,o +μI is the error
of approximation of condition (50) by the difference condition (51) on a
solution u. Developing Taylor's series for u about the node (0, tj +! r)
and denoting by ii 0 the value of the function v at this node, we can write
down in the preceding notations u' = ou/ OX and ii = ou/ ot

• • l. 2 h u 0 + l. 2 h u" 0 + O(h^2 + r^2 ).

Inserting here u~ = (3 1 110 - P, 1 and '11~ = u. 0 - 10 both recovered from the
equation

we deduce that

(53)

vI=O(h^2 +r),

VI= O(h^2 + r^2 ), ~-1 v - 2.