1549301742-The_Theory_of_Difference_Schemes__Samarskii

354 Difference Schemes with Constant Coefficients

with o:k and /]k arising from the initial conditions

y^0 = 'Uo'

or

Y1 _Yo

T

y I = Uo -

T

srn 'Pk

No wishing to load the book down with more a detailed derivation of pos-

sible estimates we quote here only the final result

II Yj II < M (II Y

0

II+ II Y~ II) for ()" >

1

4

h4

16 T^2

Careful explorations of this sort will appear in Section 6 in trying to estab-

lish stability of schemes for the equation of vibrations of a string.

5.5 THE TRANSFER EQUATION

1. Explicit schemes for the Cauchy problem. The first-order equation

au au

at + a ax =^0 '

known as the transfer equation, describes, for instance, the behavior of
the density p = p(x,t) of incompressible liquid moving along the Ox-axis
with velocity v:
op op
at + v ox =^0 ·
Here we treat it as a model one. However, the arguments about this mat-
ter can result in the design of interesting experiments, whose aims and
scope are to test and improve adrnisssible scbernes for rather complicated
equations of acoustics, kinematic integroclifferential equations of neutron
transfer, nonlinear equations of gas dynamics, etc. Because of the enor-
mous range and variety of problems dealt with by mathematical physics,
the contents of this section would be of the methodological merit.