1549301742-The_Theory_of_Difference_Schemes__Samarskii

The quasilinear heat conduction equation 521

tj+l can be found, for exa1nple, by the elimination method in terms of the

values of the function yj on the current layer tj. Because the scheme is ab-

solute stable, the well-founded choice of the step Tis stipulated by accuracy

reasoning only. Unlike the preceding schen1e, tile schen1e b) is nonlinear

with respect to the value of the function yj +^1 , so there is a real need for

employing the iterative method. Still using its fra1nework, the iteration

process is governed by the rule

(39)

( s+l)

Y i - Yi

T

( s+l)

y i

((s)) Yi- Y i-1

(s+l) (s+l) l

• a; y h

(s)

+f(Y;)·

As a final result of such operations a revised scheme becomes linear

w1t. h respect to t e h value o t e f h f unct10n · s+l y and only t e h m1t1al · · · approx-

imation ~~ remains a.s yet unknown. One way of avoiding this obstacle is

IJ.

to accept y = y1. vVe note in passing that n1ost of the iterative n1ethocls

converge in practical in1plen1entations for rather broader classes of coeffi-

cients k and f after two-three iterations performed. Even if the process

in view is divergent, two iterations can result in improved accuracy of the

describing scheme. In trying to adapt the iteration scheme (35), (39) the

usual practice is connected with specifying the condition

(s+l) (s)

nrnx '/ I y i - Y i I < E ,

where either the total number of iterations is known in advance or a desir-

able accuracy E is beforehand prescribed.

Let us stress here that the iteration scheme (35), (39) requires the

double storage in comparison with the scheme a). This is caused by the

necessity of calculating and saving the values of the function stl in terms

• s
  of the values of two funct10ns y and y.
  One more difference lies in the fact that the transition fr01n the value
  iji to the value jtl is possible only after several iterations n1ade in schen1e
  (35), (39), while the value jtl imn1ediately follows in the algorithm of the
  scheme a).
  But it would be erroneous to think in light of the same properties of
  the indicated schen1es such as their absolute stability and the same order of