292
5. Finite-Volume Methods
Here 1/f1 and 1/fz represent the concentration of each chemical species, and r is
the rate at which they react, transforming 1/f1 into 1/fz. Suppose there is a nonzero
background concentration of 1/f1 throughout the domain and that 1/fz drops very
rapidly to zero outside some localized "plume," If leapfrog-time centered-space
differencing is used to simulate the downwind transport of the plurne, small dispersive ripples will appear at the edge of the plurne, and regions will develop
where 1/fz < 0, 1/f1 > O. In the absence of the chemical reactions, these negative
regions would remain small and relatively insignificant. However, at any point
where 1/fz < 0 and 1/f1 > 0, the chemical reaction terms in (5.57) drive 1/fz more
negative while simultaneously increasing 1/11, thereby amplifying the undershoot
and ultimately destabilizing the numerical integration. The difficulties associated
with the generation of false negatives in the simulation of physical fields that
should never become negative are sufficiently serious that several positive definite
advection schemes have been specifically proposed to avoid this problem.
5.8 Schemes for Positive Definite Advection
A positive definite advection scheme is a method that never generates a negative value from nonnegative initial data.!" Any monotone scheme is positive definite, but there are no other simple relationships between the sets of methods that
are positive definite and those that are monotonicity-preserving or TVD. TVD
schemes need not be positive definite , and positive definite schemes need not be
TVD.
Early attempts to construct positive definite advection schemes involved "filling algorithms," in which the solution obtained after each integration step was
corrected by filling in any negative values. In order to conserve the total mass of
the advected species, negatives cannot simply be set to zero; compensating mass
must removed from positive regions. There are a variety of filling algorithms designed for this purpose . Some filling algorithms attempt to fill local negative regions from adjacent positive areas (Mahlman and Sinclair 1977). This may be a
physically satisfying way to remove dispersive undershoots, but it requires a great
deal of logical testing that cannot be performed efficiently on vector computers .
In other approaches the compensating mass is removed from the entire field by
multiplying the value at every grid point by the ratio of the total original mass
to the total nonnegative mass. Multiplicative compensation is computationally
efficient, but it preferentially damps the regions of highest tracer concentration.
Other filling algorithms are reviewed by Rood (1987). Although empirical testing
has shown the value of filling algorithms, the theoretical basis for these schemes
is largely undeveloped.
IONegative definite schemes may be similarly defined as any method that never generates positive
values from nonpositive initial data, Any positive definite scheme can be trivially converted to a negative definite method.
5. Finite-Volume Methods
Here 1/f1 and 1/fz represent the concentration of each chemical species, and r is
the rate at which they react, transforming 1/f1 into 1/fz. Suppose there is a nonzero
background concentration of 1/f1 throughout the domain and that 1/fz drops very
rapidly to zero outside some localized "plume," If leapfrog-time centered-space
differencing is used to simulate the downwind transport of the plurne, small dispersive ripples will appear at the edge of the plurne, and regions will develop
where 1/fz < 0, 1/f1 > O. In the absence of the chemical reactions, these negative
regions would remain small and relatively insignificant. However, at any point
where 1/fz < 0 and 1/f1 > 0, the chemical reaction terms in (5.57) drive 1/fz more
negative while simultaneously increasing 1/11, thereby amplifying the undershoot
and ultimately destabilizing the numerical integration. The difficulties associated
with the generation of false negatives in the simulation of physical fields that
should never become negative are sufficiently serious that several positive definite
advection schemes have been specifically proposed to avoid this problem.
5.8 Schemes for Positive Definite Advection
A positive definite advection scheme is a method that never generates a negative value from nonnegative initial data.!" Any monotone scheme is positive definite, but there are no other simple relationships between the sets of methods that
are positive definite and those that are monotonicity-preserving or TVD. TVD
schemes need not be positive definite , and positive definite schemes need not be
TVD.
Early attempts to construct positive definite advection schemes involved "filling algorithms," in which the solution obtained after each integration step was
corrected by filling in any negative values. In order to conserve the total mass of
the advected species, negatives cannot simply be set to zero; compensating mass
must removed from positive regions. There are a variety of filling algorithms designed for this purpose . Some filling algorithms attempt to fill local negative regions from adjacent positive areas (Mahlman and Sinclair 1977). This may be a
physically satisfying way to remove dispersive undershoots, but it requires a great
deal of logical testing that cannot be performed efficiently on vector computers .
In other approaches the compensating mass is removed from the entire field by
multiplying the value at every grid point by the ratio of the total original mass
to the total nonnegative mass. Multiplicative compensation is computationally
efficient, but it preferentially damps the regions of highest tracer concentration.
Other filling algorithms are reviewed by Rood (1987). Although empirical testing
has shown the value of filling algorithms, the theoretical basis for these schemes
is largely undeveloped.
IONegative definite schemes may be similarly defined as any method that never generates positive
values from nonpositive initial data, Any positive definite scheme can be trivially converted to a negative definite method.