Page 168 - Computational Fluid Dynamics for Engineers

5.5 Upwind Methods 155

+
E(Q) = AQ = E + E~ (5.5.13a)
\E\ = E + - E~ = \A\Q (5.5.13b)
§ f - Ml (5.5.13c)

In terms of the flux-vector-splitting, the upwind method for Eq. (5.1.2) can
be written in a form similar to Eq. (5.5.1), that is

Q n + 1 _ n = _ T{E+ _ ^ n _ T{E- +i _ Er)n { 5 5 U )
Q
or with the relations given in Eq. (5.5.13),

Qn+1 _ Qn = - I ( £ m - Ei^r + W (5.5.15)

where ipf is the numerical dissipation given by

C = ^{\E\ i+1 - 2\EU + |£?|i_i) n (5.5.16)

To show that Eq. (5.5.15) is first-order accurate in space and time, we apply
Taylor series expansion to Eq. (5.5.13b) and obtain

which can also be written as

2
= ^ A Q + 0{Ax )
\E\ i+l-\E\ i i
or
- \E\i = - Qi) + - Q {) (5.5.17)
\E\ i+l \A\ i+1/2(Qi +1 0(Ax)(Q i+1
In terms of Eq. (5.5.17), Eq. (5.5.15) can be written as

Qn +l_ Qn = _T_ {E. +i_ Ei_ i)n

+ \U\ i+i/2{Qi +i - Qi) - l4i_i/ 2 (Qi - Qi-i)} n (5-5.18)

In terms of the numerical flux E i+i/ 2 defined by

Ei+i/2 = \{\E{Qi+i) + E(Qi)] - \A\ i+l/2(Q i+1 - Qi)}

= \{[E(Q i+i) + E(Qi)] - (A+ +1/2 - A7 +1/2)AQ i+1/2} (5.5.19)

= \{[E(Q i+1) + E(Q t)] ~ (AE&i /2 ~ AE~ +1/2)}

Equation (5.5.18) can be written in a compact form,