Page 289 - Numerical Methods for Chemical Engineering

278 6 Boundary value problems

Now, if Q were either within the interior of the region of influence or somewhere outside
of it, then small changes in the first derivatives would correspond to small changes in the
second derivatives. If Q is on a characteristic line, however, even an infinitesimal change
in the first derivatives may have a finite effect upon the second derivatives, requiring the
matrix of (6.107) to be singular,
 
dt dz
2
det  dt dz  = 0 = c(dt) + b(dz)(dt) + a(dz) 2 (6.108)
a b c
2
Dividing by (dt) , we obtain the slope of a characteristic line,
√
dz b ± b − 4ac
2
= (6.109)
dt 2a
2
The sign of b − 4ac controls the number of real characteristic lines. For the 1-D time-
varying convection equation, we find as expected,

2
dz v ± v − 4(1)(0) v ± v
= = = v, 0 (6.110)
dt 2(1) 2
Thus, we distinguish the three characteristic types of second-order PDEs.
2
Hyperbolic equations, b − 4ac > 0
Two real characteristic lines exist; therefore, there exist distinct regions of space-time that
are influenced, or not influenced, by each point P. Examples of hyperbolic equations are, of
course, the 1-D convection equation, and also the wave equation
2
2
∂ ϕ ∂ ϕ
− = 0 (6.111)
∂t 2 ∂z 2
which has eigenfunctions of the form of traveling waves,
2 2
∂ ∂ i(kz−ωt) 2 2 i(kz−ωt)
− e = (k − ω )e (6.112)
∂t 2 ∂z 2
In hyperbolic equations, information about the field is transmitted as traveling waves, and
the effects of numerical error usually appear as oscillations.

2
Elliptic equations, b − 4ac < 0
Here, there exist no real characteristic lines and no distinction can be made between regions
of influence and no influence. In such problems, the value of the field at each point affects the
value of the field at all other points. As an example, consider the 2-D steady-state diffusion
equation
2
2
∂ ϕ ∂ ϕ
+ = 0 (6.113)
∂x 2 ∂y 2
2
that, with a = 0, b = 0, c = 1 has b − 4ac =−4.
In elliptic problems, information about the field is transmitted diffusively in all directions,
and thus numerical error is “smoothed out.”