1549301742-The_Theory_of_Difference_Schemes__Samarskii

378 Difference Schen1es with Constant Coefficients

with the weight O" = O", is more precise and reproduces much more better

the characteristic features of generalized solutions.

A stability condition established for scheme ( 44) such as

()" > ~--1 = ~(1-2-)

• 4 4,2 4 ,2

is certainly true for the scheme with the weight O" = O", if I< 1 or T < h/a,

that is, under the same condition as for the explicit scheme. In this context,

Let us stress here, that in an attempt to relax the "ripple" by introduc-
ing the viscosity, the distortion of the solution profile and accuracy losses
occurred.

5.7 SELECTED PROBLEMS

1. For the heat conduction equation the difference scheme is suggested:

' - 1 ·+1.

Y1i,i - 2 (vkx,i + Ykx,i)' z = 1, 2, ... , N - 1,

Yii = y~ = 0,

Prove its absolute stability, find the order of approximation and point out

the method for solving the problem.

2. Find the order of approximation for the difference scheme
   . + 1 ' • 2
   11; - 11; - 1 ( '+1. ) h.
   T -2 Ykx,i+Yk:r:,i -12Y1ix:r:,i'

i=l,2, ... ,N-1,

prove its absolute stability and investigate the stability of the elimination
method being used for determination of

i=l,2, ... ,N-1.