1549301742-The_Theory_of_Difference_Schemes__Samarskii

Homogeneous difference sche1nes for the heat conduction 469

4. The equation with a discontinuous coefficient of heat conductivity. In
   the general setting the convergence of the weighted homogeneous scheme
   (7) is of our initial concern. A case in point is that the coefficient k( x) has
   a discontinuity of the first kind on the straight line x = ~ in the plane ( x, t).
   Some consensus of opinion is that the usual condition of conjugation

(23) [u] = 0, [ k ~~] = 0 for x -- " s: ) t > 0)

is fulfilled on every discontinuity line. In the physical language, this is a

way of saying that the temperature u(x, t) and the heat flow (-ku') are

continuous.

By relating the functions k(x), f(x, t) and a solution u(x, t) to be

sufficiently smooth everywhere except this discontinuous line we proceed to

evaluate the residual (the error of approximation)

'ljJ = A ( (! it + ( 1 - (!) u) + 'P - Ut

u(a) = CJ u + (1 - CJ) u.

Let now~= xn + flh, xn = nh, 0 <fl < 1, n > 1. As far as a three-point

operator L is concerned, we might have

On this basis it remains to calculate V'i for i = n and i = n + 1. Also, it

will be sensible to introduce

(25) h of, 'Pn -- Wn+I -(a) - wn -(a) + h 'Pn - / l ut,n' wi =a. ux,i.

Since

(a. ux)i = (k u')i-i/ 2 + O(h^2 )

fork E c<^2 l[x; 1 , :.!'.;]and u. E C(^3 l[J:; 1 , xi], we obtain

( 26) w,, - = (k 11 I L-1/2 + O(h ~ )

= (k u')teft - (fl+ 0.5) h (k u');eft + O(h^2 ),

where vleft = v(~ - 0, t). By the same token,

( ) 27 h · of, 'Pn+I -- Wn+2 -(er) - Wn+I -(er) + h · l.f!n+I - h ut,n+I'

(28) 1Vn+ 2 =(a. tlx)n+ 2 = (k u')nght + (1.5 - fl) h (k u')'.·ight + O(h^2 ),