Page 185 - Computational Fluid Dynamics for Engineers

172 5. Numerical Methods for Model Hyperbolic Equations

Here, for simplicity, we have dropped the subscript i and superscript n, with the
understanding that all derivatives are being evaluated at point i and at time n.
Using a similar procedure, the following expressions can be obtained,
3
3
d u od u „ f A A N
C
>
W = - 3^ + 0{At Ax) (5.8.5a)
3
dh ,d U
(5.8.5b)
2
dt dx
3
3
d u d u
(5.8.5c)
Substituting Eqs. (5.8.5) into Eq. (5.8.4), we get
3
3
„2 d u At 3d u 3d u ^. Ax . '
dt 2 dx 2 2 C 0 ^ 3 + C 0 x - 3 + O ^ A x )
3
3
Ax ,d u 2d u ^. Ax A .
3+
C
~~2~ '0^- 0x- °^ Ax) (5.8.6)
Substituting Eqs. (5.8.6) and (5.8.5a) into Eq. (5.8.3), and after rearranging,
we obtain
2
du du cAx d u c{Axf
(3a 2a' 1' u
dt dx ^ { 1 - a ) d x - 2 6~ dx 3
2
2
+ o (Atf,(At) (Ax),(At)(Ax) ,(Ax) i (5.8.7)
where a is the Courant number defined by Eq. (5.2.2),
At_
a = c (5.8.8)
Ax
The above equation known as the modified equation, is the partial-differential
equation that is actually solved when a finite-difference method is applied to
Eq. (5.1.1). It is important to note that when Eq. (5.8.1) is being used to obtain
a numerical solution of Eq. (5.1.1), actually the difference equation is solving
Eq. (5.8.7) rather than Eq. (5.1.1).
The modified equation (5.8.7) can provide useful information on the behavior
of the numerical solution of the difference equation. To discuss this point further,
consider the viscous term in the one-dimensional incompressible Navier-Stokes
equation.
2
d 4 d u
fc('-»=8"^ ( 5 8 9 )
2
Here the coefficient of d u/dx 2 represents the dissipative aspect of the phys-
2
ical viscosity /i on the flow. The term d u/dx 2 in Eq. (5.8.7) also acts as a
dissipative term, much like the viscous terms in the Navier-Stokes equations.
Unlike the viscous terms in the Navier-Stokes equations, however, this term is