1549301742-The_Theory_of_Difference_Schemes__Samarskii

Conservative schemes^149

From (3) and (4) we find that an = ~ (5k 1 - k 2 ), an+ 1 = ~ (k 1 + 3k 2 ),

bn = ~ (3k 1 + k 2 ) and bn+l = ~ ( 5k 2 - k 1 ). Solving equations (7) with respect

to cv and f3 and recalling the expressions xn = (-8h and xn+l = (+(l-8)h,

we determine the values ascribed to the parameters

f3=μcv,

cv- ~~~~~~~~~~~~~~^1

• μ + ( 1 - μ) ( + h ( >. - e - ( 1 - e) μ) '
  (8)
  5x - 1

  
  

  . =.
  3x+ 1

  
  

  
  

The passage to the limit as h ---+ 0 yields

where

(9)

lim cv = cv 0 ,

h--+0

lim f3 = (3 0 ,

h--+0

Via the linear interpolation we extend the functions (6) on the whole seg-

ment 0 < x < 1. Under such an approach we have at our disposal a new

function y(x, h), x E [O, 1], which coincides with Yi at the grid nodes xi= ih

and possesses the limiting function as h---+ 0:

( 10) u(x) = lin1 y(x, h) = -{

1 - cv 0 x,

h--+O (3 0 (1-x), (<x<l.

Comparison of u( x) with the exact solution u( x) specified by ( 5), where the
coefficients eta, /Jo ilre determined by formula ( 9), shows that u( x) = u( x)
for cv 0 = cv 0 and (3 0 = (3 0. But it is possible only if x = 1 or k 1 = k 2.
This provides support for the view that, as h ---+ 0, solution (6) of the
difference problem (2), (5) approaches the function u(x) other than the
exact solution u(x) of problem (1) in the case k 1 f:. k 2. Due to this fact
scheme (2) is divergent.
Before stating the main results, it will be sensible to clarify a phys-
ical sense of the function u(x), which solves problem (1) subject to the
conditions [u] = 0 and [k u'] = -cv 0 (μ - x) k 2 = q at the point x = (.
Here q stands for the capacity of a point heat source (sink) at the point
x = (. Being dependent on x, the quantity q varies very widely. Specifi-
cally, q ---+ ±oo as x---+ 5 ± 0. Thus, the physical reason for the convergence
of scheme (2) is that the heat balance (the conservation law of heat) is