1549301742-The_Theory_of_Difference_Schemes__Samarskii

Asymptotic stability 331

for the symmetric scheme with (} = ~ under the constraint T < 2 ( c5 b. )-^112.

We bring together the results obtained and state that

1

for

1+T0

(} = 1 and any T > 0,

p= 1 - T^0 for (} =^0 and

2

T < f 0 =

c5 + ,6.

,

1-1.To

(^2) for (} - 1. and T < T 0 =^2
l+~rc5

• 
  

  (^2) J[t;.
  In each such case the estimate II yJ II < pJ II y^0 II is certainly true. Moreover,
  pj --+ 0 occurs as tj = j T --+ oo. As we will see a little later, it will be
  convenient to represent p by
  T zp = μ 1 - log -
  p
  It is straightforward to verify that for (} = ~
  (^1) + (^1 2) μ1 ( μl (^3) μ1 5 )
  T zp = μ 1 - log l = - - + - + · · ·
  1 - - 2 μ 1 12 80
  It follows fron1 the foregoing that
  e -Ii t. J
  The regular behavior for the symmetric scheme with (} = ~ can be
  described as follows:

  
  


• 
  

  Y(x i> t j ) ~ c1 e-li tj+!l t.i ,". Y1(x.,), '

  
  

where

(

T2 03 T4 05 )

(3=- -+-+ 12 80 ··· =0(T^2 ) ,

In dealing with the pure implicit scheme we might have

(^1) -Tli
p = 1 +TD > e , p J. = e -lit·+f3t· J J