Page 169 - Computational Fluid Dynamics for Engineers

156 5. Numerical Methods for Model Hyperbolic Equations

n +l
Q i -Q? = " ^ ( ^ 1 / 2 " ^ - 1 / 2 ) (5-5-20)
where
^<2m/2 = Qi+i ~ Qi (5.5.21a)
= ± (5.5.21b)
AEf +l/2 A (Q i+1/2)AQ i+l/2

\
Oi+i/2 = Ql+ + Qi (5.5.21c)
Thus, in an upwind method using the flux-vector-splitting method, the deriva-
tive of the convective flux in Eq. (5.1.2) is approximated by

E
dE ^ Ei+1/2 ~ i-i/2 _ Ej+i/2 ~ E i-i/2
(5.5.22)
dx ~ x i+1/2 -Xi-1/2 A x
Equation (5.5.19) can be generalized and expressed in the form

^z+i/2 = \{[E{Qi+i) + E{Qi)] - 0 i + 1 / 2 } , (5.5.23)

where ^+1/2 denotes the dissipation term which can be of first-order, second-
order or higher-order. A summary of dissipation terms of different orders are
given below.

Case 1. If there is no numerical dissipation, that is

&+1/2 = 0, (5.5.24)

Equation (5.5.22) becomes
dE ^ Ei+i — E{-i
~dx 2Ax
The corresponding numerical method
Qn +l_ Qn = _^t_ {E. +i_ Ei_ i)

is unstable.

Case 2. For a first-order accurate upwind method (described by Eq. (5.5.19)),

= (5.5.25)
4> l+l/2 AE+ +1/2-AE7 +1/2
First-order upwind methods introduce significant amounts of implicit numerical
diffusion into the solution. Higher-order upwind methods give more accurate
solutions.

Case 3. For the third-order accurate upwind method,