United States
Environmental Protection Agency
Agency
Atmospheric Sciences Research
Laboratory
Research Triangle Park NC 27711
Research and Development
EPA/600/S3-85/037 June 1985
&EPA Project Summary
A Regional-Scale (1000 km)
Model of Photochemical Air
Pollution: Part 3. Tests of the
Numerical Algorithms
Robert G. Lamb and Gerard F. Laniak
The Regional Oxidant Model devel-
oped in two previous reports is applied
to a series of test problems whose exact
solutions are known. The predicted
concentrations are compared with the
true values to obtain a measure of the
accuracy of the numerical algorithms
that comprise the model's governing
equations. Some of the problems test
only the model's chemical kinetics algo-
rithm, others test the kinetics and
transport/diffusion algorithms jointly,
and a few test all three of the model's
basic algorithms together—kinetics,
transport/diffusion, and vertical fluxes.
It is found that the kinetics algorithm
produces exact solutions of the chemi-
cal rate equations over the full range of
species concentrations that are likely to'
be encountered in applications. A modi-
fied version of the algorithm yields con-
centration estimates that are within
±5% of the correct values in one-half to
one-third of the computer time needed
for exact solutions.
In simulations of the advection of
clouds of chemically reactive com-
pounds, the kinetics and transport/dif-
fusion algorithms jointly reproduce the
correct shapes and motions of clouds,
and they predict the peak concentration
in the clouds to within 10% of the true
value over 48-h simulation periods.
In applications to continuous finite
line sources in steady, spatially variable
flows, the combined algorithms repro-
duced plumes with negligible spreading
due to pseudo diffusion. In the case of
ozone, the predicted plume centerline
concentration was within 5% of the true
value in a plume five grid cells wide and
within 15% of the true value in a plume
two grid cells wide. Corresponding
errors in the predicted CO concentra-
tions were about 50% larger. In general,
it was found that ozone is among the
species most accurately simulated,
while compounds such as nitrous acid,
nitric acid, alkyl nitrate, and related
nitrogen-containing species are the
most poorly simulated. The predicted
concentrations of free radical species
are of intermediate accuracy. Evidence
was also found that errors in plume
concentration can be amplified when a
plume crosses a second source. The
zone of enhanced error tends to be con-
fined to the vicinity of the second
source.
This Project Summary was developed
by EPA's Atmospheric Sciences Re-
search Laboratory, Research Triangle
Park, NC, to announce key findings of
the research project that is fully docu-
mented in a separate report of the same
title (see Project Report ordering in-
formation at back).
Introduction
In two previous reports (A Regional
Scale (1000 km) Model of Photochemical
Air Pollution: Part 1. Theoretical Formula-
tion, EPA-600/3-83-035, May 1983; and
Part 2. Input Processor Network Design,
EPA-600/3-84-085, August 1984), a
model was developed for simulating the
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fate of photochemical air pollutants over
large time and space domains. The theo-
retical basis of this model is a set of dif-
ferential equations that describe the
chemical and physical processes that
control the fate of substances in the
atmosphere. Since the general forms of
the solutions of these equations are not
known, an operational model must be
based on analogues of these equations
that are amenable to computer solution.
These analogues are referred to as numer-
ical algorithms. Therefore, for the model
to produce accurate concentration pre-
dictions, the solutions of the numerical
algorithms must be accurate facsimiles of
the corresponding solutions of the differ-
ential equations. In this study, we inves-
tigate whether the algorithms that form
the basis of the Regional Oxidant Model
(ROM) satisfy this condition. We consider
this task to be a necessary prerequisite to
any attempt to infer the model's predic-
tive capability through comparisons of
predicted and observed concentrations.
Procedure
The ROM's predictive equations are
composed of three basic numerical algo-
rithms: chemical kinetics, transport/dif-
fusion, and vertical fluxes. To obtain
quantitative measurements of the per-
formance of these algorithms both indi-
vidually and jointly, we apply the ROM to
a series of hypothetical problems whose
exact solutions are known. For example,
to test the chemical kinetics algorithm
alone, we apply the ROM to a situation in
which the atmosphere is at rest; there is
no diffusion, vertical fluxes, material
deposition, or sources. In this case the full
set of equations reduces to a system of
nonlinear first-order ordinary differential
equations whose exact solutions can be
calculated by using the well-tested Gear
routine. By examining situations charac-
terized by various combinations of initial
concentrations, we can determine wheth-
er the accuracy of the algorithm is con-
stant over the range of species concentra-
tions that the model will be called upon to
treat in actual applications.
Similarly, to test the chemical kinetics
and transport/diffusion algorithms joint-
ly, we apply the model to a situation in
which the horizontal flow speed is finite,
but horizontal diffusion, vertical fluxes,
and deposition are all zero. In this in-
stance, a translation of coordinate sys-
tems to a moving frame reduces the full
set of equations to the same set of ordi-
nary differential equations described
above. In this case, the accuracy of the
combined kinetics and transport/diffu-
sion algorithms is judged by comparing
the exact solutions with the values pre-
dicted by the model along the moving ref-
erence frame.
Tests of the vertical flux algorithms
were reported in Part 1, Section 9 of the
previous project reports1. We test it in
combination with the chemical kinetics
algorithm in this study only in the case in
which the vertical flux is infinite. We
believe that the performance exhibited in
this single extreme case is indicative of
joint performance of these two algorithms
in general, because the vertical flux dif-
ferential equations are of the same type
as the chemical kinetics equations, but
they are linear in form.
Although the combined set of condi-
tions under which we test the model algo-
rithms constitutes only a limited part of
the parameter space in which the model
would be applied in actual applications,
these tests provide a minimum standard
for judging the model's predictive capabil-
ity. Indeed, if the model is unableto accu-
rately simulate these rather elementary
situations, there are no grounds for
expecting it to perform well in general
applications.
Conclusions
The chemical kinetics algorithm was
found to provide exact solutions of the
corresponding differential equations over
a wide range of species concentrations.
In fact, the algorithm's accuracy is unnec-
essarily high because none of the input
data used in the model nor any of the
model's other components are equally
accurate. For this reason, we sacrificed
some accuracy in exchange for higher
computing speeds. The modified algo-
rithm yields concentration estimates with-
in ±5% of the true values in about one-
tenth of the time required to calculate the
exact solutions with the Gear routine.
Tests of the combined kinetics a
transport/diffusion algorithms showed
that plumes were underestimated by only
10% or less over a 48-h simulation period.
Widening of the concentration distribu-
tions due to pseudo diffusion was negli-
gible, and errors in the predicted shapes
and transport speeds of clouds and
plumes were nil. These results are par-
ticularly significant, since the transport/
diffiusion algorithm used in the ROM is a
type that does not maintain positive defi-
nite concentrations. When negative con-
centrations are generated, they are mere-
ly reset to zero or clamped, before the
chemical kinetics algorithm operates on
them.
Not all species were simulated equally
well. Ozone is among the compounds
most accurately predicted while nitrous
and nitric acid were among those most
poorly simulated. For any given species,
the error level is inversely proportional to
the width of the cloud or plume and there
is a slight enhancement of the error in the
concentrations of some species once they
cross a second source. Overall, our tests
showed that the algorithms on which the
ROM is based are quite accurate descrip-
tions of the physical and chemical pro-
cesses that the model is intended to treat.
The EPA author, Robert G. Lamb (also the EPA Project Officer, see below), is with
Atmospheric Sciences Research Laboratory, Research Triangle Park, NC
27711; and Gerard F. Laniak is with Program Resources, Inc., Annapolis, MD
21401.
The complete report, entitled "A Regional-Scale (1000 km) Model of Photo-
chemical Air Pollution: Part 3—Tests of the Numerical Algorithms," (Order No.
PB 85-203 818/AS; Cost: $23.50, subject to change) will be available only
from:
National Technical Information Service
5285 Port Royal Road
Springfield, VA 22161
Telephone: 703-487-4650
The EPA Project Officer can be contacted at:
Atmospheric Sciences Research Laboratory
U.S. Environmental Protection Agency
Research Triangle Park, NC 27711
U.S. GOVERNMENT PRINTING OFFICE: 1985-559-016/2
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