1549301742-The_Theory_of_Difference_Schemes__Samarskii

The alternating direction method 545

which is stable for T < 0.5 h^2 / p. If equation ( 1) contains the variable

coefficients, that is, it acquires the form

then

p

Aau = (aa Y;;a)Xa' A= LA('{' 0 < ae> < C2,

a=l

and the explicit schen1e (3) is stable for r < 0.5 h~ /(p c 2 ).

From here it seems clear that the admissible step in the explicit sche1ne

is yet to be refined along with increasing the maxinmm value of the coef-

ficient of heat conductivity. As a matter of fact, the last requiren1ent is

unreal for the problerns with fastly and widely varying coefficients. Just for

this reason explicit schemes are of little use not only for rnultidimensional

proble1ns, but. also for one-dimensional ones (p = 1). On t.he ot.her hand,

the explicit schemes offer real advantages that the value f; = Yn+l on every

new layer tn+l = tn + T is found by the explicit formulas ( 3) with a finite

number of operations at every node of the grid wh, so that the an1ount

of arithmetic operations required in passing from one layer to another is

proportional to the total number of the grid nodes and so it is a quantity

of 0(1/hP).

Being concerned with the in1plicit scheme for (J = 1, we may set up

the problen1 for determination of yn+l:

Yn+^1 _ TA yn+ I = , y" , y n+^1 I 1'h = ,^0 y ( x, 0) = u 0 ( ) :r.

N u1nerical solution of this system containing 1 / hP equations requires, for
exan1ple, during the course of Gaussian elimination 0(1/h^3 P-^2 ) operations
in connection with a special structure of the matrix E - r A.
Some consensus of opinion is desirable in this tnatter, since a s1naller
nu1nber of operations is performed in the explicit schenw, but. it is stable
only for sufficiently small values of r. In turn, the implicit scherne being
absolutely stable requires rnuch more arithmetic operations.
What sche1nes are preferable for later use? Is it possible to bring to-
gether the best qualities of both schemes in line with established priorities?
In other words, the best scheme would be absolutely stable as the implicit
schemes and schould require in passing from one layer to another exactly Q
arithmetic operations. As in the case of the explicit schemes, Q would be
proportional to the total number of the grid nodes so that Q = 0( 1 / hP).