1549301742-The_Theory_of_Difference_Schemes__Samarskii

Hornogeneous difference schemes for the heat conduction 491

of the general form in the rectangle Dr = { 0 < x < 1, 0 < t < T}:

(78)

01l

c( x, t) 8t = Lu+ f( x, t) ,

u(O, t) = u 1 (t), u(l, t) = u 2 (t), a(x, 0) = u 0 (x),

Lu=-,=;-o ( k(x,t)c;-ou) +r(x,t)-, ou -q(x,t)u,

ox ux dx

A similar problen1 has been solved in Chapter 3, Section 5 for the stationary

equation Lu+ f = 0 through the use of monotone schemes of second-ordet

accuracy attainable for any step h and the function r( x).

In order to construct a monotone scherne for problem (78) fol' which

the maximun1 principle would be valid for any hand r, we involve in subse-

quent considerations the equation of the same type, but with the pei-turbed

operatoi-L:

(79)

OU -

c(x,t)Ft=Lu+f, Lu=u--^0 ( k-Ou ) +r--qu, OU

ox ox ox

u=(l+R)-^1 , R = 0.5 h lrl/k.

As usual, the operator Lis apptoxiniated for fixed t = i = tj+l/ 2 by the

difference operator

;i. A' Y -- u ( a Yx ) x + b+ a (+l) Yx + 1-J a Yx -d Y,

whete (for n10re detail see Chaptei- 3, Section 5)

a= A [k(x +sh, i)], d = F[q(x +sh, i)], b± = F[r±(x +sh, i)],

r±=r±/k, r+=0,5(r+lrl)>O, r- =0,5(r-lrl)<O.

Here the same pattern functionals A and F are adopted in achieving much
progress as was done in Chapter 3, Section 2, n1aking it possible to generate
an approximation of order 2.