5.8 Schemesfor Positive Definite Advection
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method. In particular, the second step (5.60) is a numerical approximation to
in which
c= I
if 1/1 > 0;
0,
if 1/1 = o.
Although the second step utilizes upstream differencing, the ifJn+ I are highly
nonlinear functions of the ifJ*, and the second step is not monotone. The second
step will, nevertheless, be positive definite provided that
which guarantees that even when both antidiffusive velocities are directed out of a
particular grid cell, the antidiffusive fluxes will be too weak to generate a negative
value. The preceding condition is satisfied whenever the initial ifJ are nonnegative
and the maximum Courant number associated with the physical velocity field
satisfies
CH I /),.t I < 1
I
tu
-
for all j.
Then
Since the first step is monotone, all the ifJ* are nonnegative and
The Smolarkiewicz scheme can easily be extended to multidimensional problems
(Smolarkiewicz 1984) and can be made monotonicity-preserving by applying Iimiters in the antidiffusion step (Smolarkiewicz and Grabowski 1990).
One consequence of the nonlinear dependence of the antidiffusive velocities on
ifJ* is that the solution obtained using the Smolarkiewicz scheme will change if a
spatially uniform background field is added to the initial tracer concentration. This