Page 170 - Computational Fluid Dynamics for Engineers

5.6 Finite-Volume Methods 157

AE
*i+l/2 - -\( tl/2 ' AE U t2 + AE W/2 ~ ^ " + 3 / 2 ) (5-5.26)
Case 4. For the fifth-order accurate upwind method

AE
fc+1/2 = - ^ ( - 2 ^ + 3 / 2 + 1 1 ^ + 3 / 2 " ^ Ul2 - 3 ^ + 3 / 2
2AE llAE 5 5 27
+ ^/2 - i z,2+e^r+i/2+3^r-i ) ( - - )
+
/2
The flux-difference-splitting method discussed by Osher [6] and Roe [7] is
based on Godenov and Riemann solver as discussed in detail in [1]. Unfor-
tunately this method is beyond the scope of this text, and only a very brief
description is given below.
If we use the exact formula for |^|z+i — \E\i in Eq. (5.5.17)

rQi+i \QE\ rQi+i
A d
d
i+1-\E\i= \^\ Q = \ \ Q, (5-5.28)
Equation (5.5.15) can be written as
/"Wi + l
T T I rQi+i rQi
pOi
Ei+1 _ E._ i)n
Qn+l_ Qn = _L_ {Ei+i n + +
2^ ~ ^ - i ) 2 / \A\dQ- \A\dQ
jQi JQi-1
Then the numerical flux E i+1/ 2 becomes
= \ | [E(Qi+i) + E(Qi)] - J Z+1 \A\ dQ 1 (5.5.29)
E z+i/ 2

5.6 Finite-Volume Methods

While finite-difference methods are based on a discretization of the differential
form of the conservation equations, the finite-volume methods are based on a
discretization of the integral forms of the conservation equations. To examine the
numerical solution of the conservation equations with this approach, consider
the generic form of the conservation integral equation given by Eq. (2.2.24),
that is,

d_ jffudQ^ ffF-dS= fffQ dQ+ flQ dS (2.2.24)
dt v s
Q s n
where, according to the usual sign convention for control volumes, S is perpen-
dicular to the control surface in a direction away from the control volume.
In order to discretize this equation, as with the finite-difference methods, it
is necessary to divide the physical space into a discrete network of cells. Two