Page 83 - Computational Fluid Dynamics for Engineers

68 2. Conservation Equations

In the hyperbolic case, with a > 0, c < 0 and using Eq. (2.6.4), we obtain
from Eq. (2.6.3)

2
2
d ?/ <9 ?/ =9w = 9w
d
g
Lu = ^ ^ 2 " -£=2 + 7 ^ + — +fu = h (2.6.6)
2
<9x dy dx dy
Here the leading terms form the wave operator and
2
2
d u/dx 2 - d u/dy 2 = 0
is the wave equation, so that any hyperbolic equation in the plane is analogous
to the wave equation, with additional lower-order terms.
In the parabolic case, with a > 0, c = 0, e < 0 and using
S = - 4 , £ = R (2-6.7)

rather than Eq. (2.6.4), we rewrite Eq. (2.6.3) as

2
d u du =du
L l / d
= T^2 2 "" 7SE + ^H + fu = h (2.6.8)
dx dy dx
The leading terms here form the heat or diffusion operator, that is
2
d u/dx 2 - du/dy = 0
is the heat-conduction equation with u representing a nondimensional temper-
ature.
One of the basic differences between the various types of partial-differential
equations is in their "domains of dependence" and their "regions (or domains)
of influence." Suppose each Eq. (2.6.1) is to be solved in some region R of
x7/-space with boundary B. Then the domain of dependence of a point P in
R consists of all those points on B which are required in order to uniquely
determine the solution at the point in question (see Fig. 2.3). Conversely, the
"region of influence" of a point P consists of all those points in R at which the
solution is altered when a change in the solution at the point P occurs (see
Fig. 2.4). In general, elliptic equations have the property that the domain of
dependence of any point is a curve or surface completely enclosing the point,
as in Fig. 2.3a. In parabolic and hyperbolic equations this is not the case. The
extent of the domain of dependence of a point is determined by the intersection
of the so-called characteristic curves through that point with the "physical"
boundary B. The characteristics also form part of the boundary of the domains
of influence, see Fig. 2.4. The total number of characteristics is equal to the
number of dependent variables. For a parabolic system they all coincide with a
line normal to the timelike direction (Fig. 2.4b), whereas they are distinct for a
hyperbolic system. An elliptic equation has no real characteristics. For further
details, see Hirsch [11].