Page 284 - Numerical Methods for Chemical Engineering

Finite differences for a convection/diffusion equation 273

Upwind difference scheme (UDS)

We now show that it is possible to remove the oscillations by using a different approximation
for the first derivative in the convection term, an upwind difference, defined in this case for
v> 0 (flow from left to right) as
(UDS)
dϕ ϕ j − ϕ j−1
= + O[( z)] (6.82)
dz z
z j
We take here a one-sided difference, and thus this formula is not as accurate, for finite z,
as the central difference approximation. The UDS equation for grid point j is

ϕ j − ϕ j−1 ϕ j−1 − 2ϕ j + ϕ j+1
− v + = 0 (6.83)
z ( z) 2
2
Multiplying by − ( z) /L and defining Pe loc as before yields
− (Pe loc + 1)ϕ j−1 + (2 + Pe loc )ϕ j − ϕ j+1 = 0 (6.84)

Defining β ≡ Pe loc + 2 and γ ≡ Pe loc + 1, the UDS linear system is
     
β −1
ϕ 1 γϕ 0
 −γ β −1     


 ϕ 2   0 
−γ β −1    
 
 
 ϕ 3   0 
. .   =   (6.85)
 
.
−γ . .  .   . 
 . .  .
  .
   . 
.    
 
 . . β   ϕ N−1   0 
 −1 
−γ β ϕ N ϕ L
Note that for all Pe loc ≥ 0, β> 0, γ> 0, and that the matrix elements have the same signs
as they do in the CDS system when Pe loc < 2. As Figure 6.8 shows, the UDS solution
remains well-behaved for all values of Pe loc .
Even though the UDS equations use a less accurate approximation for the convection
derivative than the CDS equations, they are safer and better behaved. If too few grid points
are used with the CDS method, large oscillations appear that can be disastrous with nonlinear
source terms, but with the UDS method, the effect of too large a z is merely a loss in
accuracy.
Why does upwind differencing work?
To see why upwind differencing works, we relate the upwind and central difference first-
derivative approximations through the identity

ϕ j − ϕ j−1 ϕ j+1 − ϕ j−1 ϕ j − ϕ j−1 ϕ j+1 − ϕ j−1
= + −
z 2( z) z 2( z)
(UDS) (CDS)
dϕ dϕ 2ϕ j − 2ϕ j−1 ϕ j+1 − ϕ j−1
= + − (6.86)
dz dz 2( z) 2( z)
z j z j
(UDS) (CDS)
dϕ dϕ z −ϕ j−1 + 2ϕ j − ϕ j+1
= +
dz dz 2 ( z) 2
z j z j