1549301742-The_Theory_of_Difference_Schemes__Samarskii

460 Homogeneous Difference Schemes for Time-Dependent Equations

and the boundary conditions

(3) u(O,t)=u 1 (t), u(l,t)=u 2 (t), O<t<T,

under the assmnptions that the coefficient k(x, t) is bounded from below

and from above:

(4) 0 < c 1 < k(x, t) < c 2 , (x, t) E Dr,

where c 1 and c 2 are constants.

Also, we take for granted that problem ( 1)-(3) possesses a umque

solution with all necessary derivatives.

2. Homogeneous difference schemes with weights. In a common setting it
   seems natural to expect that a difference scheme capable of describing this
   or that nonstationary process would be suitable for the relevant stationary
   process, that is, for fJu/fJt 0 we should have at our disposal a difference
   scheme from a family of homogeneous conservative schemes, whose use
   permits us to solve the equation Lu+ f = 0.
   One way of covering this for the heat conduction equation is to con-
   struct a homogeneous conservative scheme by n1eans of the integro-interpo-
   lation method. To make our exposition more transparent, we may assume
   that the coefficient of heat conductivity k = k( x) is independent oft. The
   general case k = k(x, t) will appear on this basis in Section 8 without any
   difficulties.
   We proceed as usual. This amounts to introducing the following grids:
   an equidistant grid on the segment 0 < x < l with step h

wh={x;=ih, i=O,l, ... ,N, h=l/N};

a grid on the segment 0 < t < T with step T

.

W 7 ={tj=jr, j=O,l, ... ,No, 7:=T/No};

a grid in the rectangle Dr

and to forming