Page 166 - Computational Fluid Dynamics for Engineers

5.5 Upwind Methods 153

As we also discussed in the same section, the characteristic lines play an
important role in the development of the numerical methods and, for conve-
nience of explanation, we consider the linear convection equation, Eq. (5.1.1).
In this case, information propagates along the characteristics curves specified by
dt/dx = 1/c [see Eq. (5.1.9)] from either the right or the left side of the solution
point depending on whether c < 0 or c > 0, respectively. The propagation of
information of this type is referred to as upwind propagation since the informa-
tion comes from the direction from which the convection velocity comes, that
is, the upwind direction. Finite difference methods that account for the upwind
influence are called upwind methods.
To describe an upwind method for Eq. (5.1.1), we use a one-sided space
differencing in the characteristic direction, and this results in,

< + 1 -i - "i+i
= _ c + ^ ^ i _ c - _ l ± i L (5.5.1)
t t X{ X{—\ Xi-\-\
where

c + = max(c, 0) = —-— and c~ = min(c, 0) = —-— (5.5.2)

For a uniform grid in space and time, Eq. (5.5.1) becomes

5 5
c
< + 1 = «? - c + ^ « - uU) - " ^ « + i - «") ( - -3)
Note that if central differences are used for u x in Eq. (5.1.1), the preferred
pathes of information are ignored and the explicit scheme becomes unstable as
will be discussed in Section 5.7. However, the explicit upwind method for Eq.
(5.1.1) is stable provided
At
0 < < 1 (5.5.4)
Ax
Equation (5.5.3) can be written as

71+1 TC
u, i i 2 ( ^ + i - i i ? _ i ) + ^ ? (5.5.5)
where r = At/Ax and ip™ is called numerical dissipation given by

^ ? = ^ M ( u ? + i - 2 u ? + u?-i) (5-5.6)

The extension of the above approach for the linear convection equation to
nonlinear partial differential equations PDEs, large systems of PDEs and multi-
dimensional problems can become rather complicated. The flux-vector-splitting
method of Steger and Warming [5] and the flux-difference-splitting method dis-
cussed by Osher [6] and Roe [7] provide a systematic approach to identify the
direction of information propagation of each spatial derivative, and allow the