1549301742-The_Theory_of_Difference_Schemes__Samarskii

484 Homogeneous Difference Schemes for Time-Dependent Equations

thereby completing the staten1ent of the problem under consideration:

(68)

au

m=Lu+f(x,t), O<x<l, t>O,

c 0 au = k au for

at ax

x = 0 , u(l,t)=O, u(x, 0) = u 0 (x).

The design of a homogeneous difference scheme necessitates approxi-

mating the boundary condition at the point x = 0.

The first step during the course of the integro-interpolation method is

to rely on the balance equation, say in the rectangle { 0 < x < x l/ 2 = 0. 5 h,

tj < t < tJ+ 1 }, leading to

'"I /2

j [u(x,tj+i)-u(x,tj)] dx

0

tj +l

.I [w(x 112 , t) - w(O, t)] dt +

tj

tj+I x!/2

j j f(x,t)dxdt,

t J. 0

au

where w(x, t) = k ax. The next step is to substitute here

(

w(O,t)= k~ au) =Co~(O,t) au

ux x=O ui

and take into account that.

tj +1

.I Co~~ (0, t) dt =Co ( u(O, tj+i) - u(O, tJ)) =Cu T u 1 , 0.

tj

Then upon replacing the integrals in x by the simplest expressions 0.5 h u 0

and 0.5 h j~ and the integrals int by the expressions rw;~; and T f~a), the
difference boundary condition is taken to be

(69) C (

(^0) ) 0 5 h J,( (^0) )
Yt.O = Cl) Y:i:.O +. 0 , C = C^0 + 0.5 h.