Zienkiewicz O.C., Taylor R.L. Vol. 3. The finite - tiera.ru

58 Convection dominated problems
This in general may still be a function of , thus destroying the linearity of the
problem.
Before proceeding further, it is of interest to discuss the general behaviour of
Eq. (2.1) in the absence of source and di€usion terms. We note that the matrices A i
can be represented as
A i ˆ X i iX ÿ1
i
by a standard eigenvalue analysis in which i is a diagonal matrix.
If the matrices X i are such that
…2:139†
X i ˆ X …2:140†
which is always the case in a single dimension, then Eq. (2.137) can be written (in the
absence of di€usion or source terms) as
@
@t ‡ X iX ÿ1 @
ˆ 0 …2:141†
@xi Premultiplying by X ÿ1 and introducing new variables (called Riemann invariants)
such that
f ˆ X ÿ1
…2:142†
we can write the above as a set of decoupled equations in components of f and
corresponding of :
@
@t ‡ @
i ˆ 0 …2:143†
@xi each of which represents a wave-type equation of the form that we have previously
discussed. A typical example of the above results from a one-dimensional elastic
dynamics problem describing stress waves in a bar in terms of stresses … † and
velocities …v† as
@ @v
ÿ E ˆ 0
@t @x
@v 1 @
ÿ ˆ 0
@t @x
This can be written in the standard form of Eq. (2.1) with
ˆ
v
F ˆ Ev
=
The two variables of Eq. (2.142) become
p
where c ˆ E=
1 ˆ ÿ cv 2 ˆ ‡ cv
and the equations corresponding to (2.143) are
@ 1
@t ‡ c @ 1
ˆ 0
@x
@ 2
@t ÿ c @ 2
ˆ 0
@x
representing respectively two waves moving with velocities c.