1549301742-The_Theory_of_Difference_Schemes__Samarskii

Conservative difference schen1es of nonstationary gas dynamics 539

, , k+l k+l k+L k+l

By insertmg now b vs = -cr 1 Tb gss and b 17 = 0.5TO v 8 we deduce

with regard to

that

(53)

k

(a+cr 1 )017

qk = -~--k~--'

(a+ cr 1 ) 1) - cr 117

provided the condition (a+ cr 1 ) ~ - cr 117 > 0 holds. The meaning of this is

that

k CT I

1) > 17 for all k = 0, 1, 2, , ..

a+ cr1

(54)

was supposed before proceeding to further derivations.

When the pressure is prescribed for i = 0 and i = N, the boundary

cond1t10ns '. f or u i: k+l g are certamly ' h omogeneous:

(55)

After scrutinising the canonical form of equation (53) with respect to

k+l

b g ( s.,)

A(P) y(P) = B(P, Q) y(Q) + F(P)

QEIII'(P)

we can be pretty sure that

A(P) > 0, B(P, Q) > 0, D(P) = A(P) - B(P, Q) = 1.

QEIII'(P)

This serves to motivate the validity of the maxi1num principle with regard
to equation (58) supplied by the homogeneous boundary conditions ( 55).
By utilizing this fact it is plain to show that the estimate holds:

(56)

whence it follows that the iterations converge under either of the following
conditions:

lqk I < q < 1 for all k=0,1,2, .,

(57) k
11 7 -^171
< q' b=

CT I

k: a+ cr 1

17 - b 17