1549301742-The_Theory_of_Difference_Schemes__Samarskii

324 Difference Sche1nes with Constant Coefficients

But it is absolutely unstable: this scheme becon1es unstable for any way of

tending T and h to zero.

As further developtnents occur, we rewrite equation (59) as

..
(60)

j +I j-I

Yi - Yi

2T

Yi-I J -^2 Yi J + Yi+1 J

h2

Upon substituting the sum y/+I + y/-^1 for 2 y/ in the right-hand side of

equation (60) the resulting three-layer "rho1nbus" schen1e known as the

Du-Fort-Frankel schen1e

(61)

j +I j -I

Yi - Yi

2T

j j+I j-I j

Yi-I - Yi - Yi + Yi+1

h2

becomes explicit with respect to y/+^1 and appears to be absolutely stable

for any h and T. In giving an alternative form of the "rhombus" scheme

(62)

with Ytt = (y/+I - 2y/ + y/-^1 ) r-^2 we may attempt the right-hand side

of equation (61) in the form

Yi-I - Yi - Yi+ Yi+1

h2

Yi-I - 2 Yi+ Yi+1

h2

T2

= Yxx - h 2 Ytt ·

Yi - 2 Yi +Yi

h2

Substitution of the last expression into (61) yields (62). As a matter of fact,
the "rhombus" scheme is some modification of the Richardson scheme with
the extra member on the left-hand side of (61) r^2 h-^2 Yrt, which assures
us of its stability. The proof of stability of sche1ne (62) is an immediate
implication of the general theory of Chapter 6 and is omitted here. The
error of approximation for scheme (62) is

2 2 2

1/J = Au-u 1 - ~ 2 uft = u"-i"t- ~ 2 ii+O(h^2 +r^2 ) = - ~ 2 ii+O(h^2 +r^2 ).

From such reasoning it seetns clear that the "rhombus" scheme provides
the conditional approximation