ME 540

**Numerical Solutions Applied To Heat
Transfer And Fluid Mechanics Problems**

# Quick and Quickest

This lecture is based upon the paper by B. P. Leonard,
deriving the QUICK and QUICKEST methods. Note that we have already
derived these methods via a different path in previous
lectures. Take a look at Leonard’s
Equation 23. Renaming the variables to
match my notation, we have:

_{}

or

_{}

Look familiar?

Notice that QUICK is one of those instances
where you get something by thinking about practical bounds on the wave number
in the stability analysis.

If you are having trouble with Equations 49 through 51,
remember that they work from the assumption that the density is a quadratic
function of x. For example, if I need to
know the density near the face between volumes i and
i+1, I work with the first three terms of a Taylor
series expansion about the face.

_{}

To simplify notation Leonard introduced a new space variable _{}. A positive sign of
this new variable corresponds to the upstream direction. He’s avoiding use of _{} in this context, since
that denotes the length of the spatial mesh.

The two derivatives in the expression are approximated by finite difference,
giving:

_{}

Assume that a spatial density distribution is advected without change in
shape. The time history of density at the right face can be obtained using the
density profile at the beginning of the time step upstream of the face. The relationship between time and space
variables in these two ways of viewing density is:

_{}

So the time integral converts to a space integral as follows:

_{}

or in terms of the material Courant number c, we
have

_{}

## Error Cancellation in the Presence of Source Terms

Consider a mass conservation equation that might result from multi-phase or
reacting flow:

_{}

If the mass source term S(x,t)
is small compared to the mass flux term then the Quickest difference formula is
a reasonable approximation. However, in
many such problems a quasi-equilibrium exists with:

_{}

Hence, the substitutions imbedded in Quickest:

_{}

are no longer valid. Use of a vanilla Quickest
difference operator for flux terms does not give higher order accuracy. Be cautious of any numerical solution of flow
equations that advertises both the Quickest difference
method, and the ability to model multi-phase and/or reacting flows.

**Maintained by John Mahaffy : jhm@psu.edu**