United States
Environmental Protection
Agency
National Exposure
Research Laboratory
Research Triangle Park, NC 27711
Research and Development
EPA/60Q/SR-97/142 March 1998
on
/L Talat Odman
Errors introduced by advection
schemes are considered to be an im-
portant source of inaccuracy in air qual-
ity models. Recently, significant re-
search efforts focused on construction
of advection schemes that yield highly
accurate and oscillation-free solutions
on a fixed grid system. Other require-
ments are also imposed on pollutant
advection schemes such as mass con-
servation and positive solutions.
conservation is necessary to account
accurately for pollutant mass balance.
Preservation of positive concentrations
is important because of the nonlinearity
of atmospheric chemistry. There is no
scheme, at present, that fully satisfies
all of these requirements. For example,
high accuracy usually comes at the ex-
pense of spurious oscillations, espe-
cially near steep gradients. The objec-
tive of this study is to identify advec-
tion schemes with more desirable prop-
erties for air quality modeling.
The performances of the following
eight schemes are evaluated:
(1) Smolarkiewicz' scheme, (2) piecewise
parabolic method, (3) Bott's scheme,
(4) Yamartino's scheme, (5) flux-cor-
transport, (6) the semi-Lagrangian
method, (7) chapeau function scheme,
and (8) accurate space derivative
scheme. The evaluation cases are se-
lected from idealized problems with
known analytic solutions. Problems rel-
evant to air quality modeling are pre-
ferred such as one-dimensional advec-
tion with uniform velocity, rotation of a
cone-shaped puff, skew advection of a
point-source plume, shear flow tests
and rotation of chemically reactive
puffs. Quantitative measures are used
in the evaluation so that properties of
different schemes can be compared
readily. Most schemes performed con-
sistently in all the tests, but a few failed
in some stressful tests. The differences
between performances were more pro-
nounced in some tests, and the rank-
ing differed from test to test. However,
important properties of the schemes
were identified as a result of the diver-
sity of the tests. Bott's, Yamartino's and
accurate derivative schemes are in
general more accurate than the others.
Bott's and Yamartino's schemes are
computationally demanding. Short-
wavelength performance of Yamartino's
scheme and its essentially oscillation
behavior compared to Bott's scheme are
noteworthy.
This Project Summary was developed
by EPA's National Exposure Research
Laboratory, Research Triangle Park, NC,
to announce key findings of the re-
search project that is fully documented
in a separate report of the same title
Project Report ordering informa-
tion at back).
Introduction
Modeling the fate of pollutants on urban
and regional scales is a major research
area for the U.S. EPA. Simulation of
pollutant transport by the wind has been
one of the focal research topics of the air
quality modeling. We assume that the
transport of pollutants in the atmospheric
turbulent flow field can be described by
means of differential equations and ap-
propriate initial and boundary conditions.
In Eulerian air quality models, the trans-
port process is solved using appropriate
numerical algorithms. These numerical
algorithms for the advection processes
must satisfy several properties that are
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essential for making useful air quality simu-
lations. As with all numerical methods,
the numerical schemes for solving the
transport equation must meet the conver-
gence condition and correctly model the
conservative, transportive, dissipative, and
dispersive properties of the governing
equation.
Due to the presence of pollutant sources
and highly nonlinear chemical interactions
between the concentration fields of differ-
ent species, it is necessary to consider
schemes with special properties: mass
conservation, small numerical diffusion and
small phase errors. Advection schemes
with different properties introduce differ-
ent errors, all of which are sources of
uncertainty in air quality model predictions.
It is critical to identify which of the above-
mentioned properties a scheme possesses
before recommending its use. Since an
advection scheme with all the desired prop-
erties is not currently available, the issue
becomes identifying the scheme with the
most desirable properties and efficiency.
This report summarizes the work that
was done under a cooperative agreement
between the U.S. EPA and the MCNC-
North Carolina Supercomputing Center.
The objectives of the program were to
study important numerical characteristics
of various advection schemes, to test fea-
sibility of their use in air quality simula-
tions, and to show that the algorithms are
capable of following accurately and effi-
ciently pollutant transport. In this report,
we describe desirable characteristics of
eight one-dimensional numerical advec-
tion schemes currently used in air quality
models. We evaluated these schemes by
comparing a few quantitative measures
obtained through several robust test cases
relevant to air quality simulation. The
numerical algorithm studied during the pe-
riod of the project were incorporated into
the pool of optional advection modules of
the EPA's Models-3 Community Multiscale
Air Quality (CMAQ) modeling system. The
research findings from this project should
be applicable to other regional/urban air
quality models.
Advection Algorithms
There are many different ways of clas-
sifying advection schemes. A common way
is to classify the schemes based on the
method used in their formulation. Since a
wide variety of methods were used, any
classification may fall short of being com-
plete. The following is a fairly comprehen-
sive list: (1) finite difference schemes, (2) fi-
nite volume schemes, (3) flux corrected
schemes, (4) Lagrangian schemes, (5) fi-
nite element schemes, and (6) spectral
schemes. Current trends in advection
scheme development show a merging of
the methods to take advantage of each
approach's most desirable properties. For
example, the Characteristic-Galerkin
method combines the best of the finite
element and Lagrangian methods. Flux
corrections are being used in the frame-
work of finite element and spectral
schemes. Also, the classical finite differ-
ence schemes are being abandoned in
favor of modern finite volume schemes.
Eight advection schemes studied are listed
below:
The Smolarkiewicz scheme (SMO) is
based on the first-order accurate upstream
or "donor cell" method. To increase accu-
racy, Smolarkiewicz reversed the effect of
this artificial diffusion by defining an
antidiffusion velocity. Though diffusion is
physically irreversible, it is numerically pos-
sible to apply an antidiffusive step and
partially recover what has been lost to
diffusion; this resembles reversing a film
that shows diffusion. It is customary to
express the algorithm as a multistep
scheme.
Bott scheme (BOT) uses normalized
advective fluxes represented in polyno-
mial form in an attempt to reduce the
phase-speed errors. Negative values of
the transported quantity are suppressed
by nonlinearly limiting the normalized
fluxes. Recently, a monotonic version of
the scheme was developed and the time-
splitting errors associated with the use of
one-dimensional operators in multidimen-
sional applications were reduced.
In the piecewise parabolic method
(PPM), the subgrid distribution of the ad-
vected quantity is represented by a pa-
rabola in each grid interval. PPM not only
provides a local fit of the data, but is
monotonic and uses a special steepening
procedure in the vicinity of sharp gradi-
ents. This ensures that positive quantities
will remain positive and that sharp gradi-
ents will be handled correctly without the
generation of spurious oscillations.
Yamartino scheme (YAM) uses piece-
wise cubic interpolands as a starting point
for his higher-order scheme. In this
scheme, the coefficients of a cell-centered
cubic polynomial are constrained from the
point of view of maintaining high-accuracy
and low-diffusion characteristics while
avoiding undesirable byproducts associ-
ated with higher-order schemes but ab-
sent in low-order schemes. In addition, a
filter is used for filling in undesired short-
wavelength minima. One advantage of
Yamartino's scheme is that it was de-
signed to follow short-wavelength features.
Flux-corrected transport (FCT) is a tech-
nique developed by Boris and Book. It
constructs the net transportive flux as a
weighted average of a flux computed by a
low-order scheme and a flux computed by
a higher-order scheme. The weighting is
done in a manner that ensures that the
higher-order flux is used to the greatest
extent possible without introducing the
overshoots and undershoots. Zalesak gen-
eralized FCT to multidimensions.
In a semi-Lagrangian method (SLT), one
estimates the backward trajectory of a par-
ticle that arrives at a certain grid point.
Since the origin of a particle does not
always coincide with a grid point, an inter-
polation scheme is necessary to estimate
the original concentration. Once estimated,
this concentration is assigned to the grid
point of arrival. One advantage of the
scheme is that it is not subject to the
Courant stability condition, so large time
steps can be used.
The accurate space derivative (ASD)
scheme is based on a pseudo-spectral
method. This scheme is highly accurate;
however, it needs to be coupled with a
nonlinear filter (such as the Forester filter)
to suppress the spurious oscillations that
may exist in the solution. A disadvantage
of the scheme is the requirement of peri-
odic boundary conditions inherent to all
spectral schemes.
The chapeau function method (HAT) is
a classical weighted-residual finite element
method. The solution is expanded in piece-
wise basis functions that look like hats
("chapeau" is French for hat) in one-di-
mensional space: Then the residual is as-
sumed to be orthogonal to the weighting
functions, which may be the basis func-
tions themselves, as is usually the case in
the Bubnov-Galerkin methods.
Results and Discussion
Eight advection schemes (not counting
the variations of Bott Scheme) were com-
pared using test cases ranging from very
simple one-dimensional advection to more
robust two-dimensional shear and chemi-
cally reactive flows. Most schemes per-
formed consistently in all the tests, but a
few failed in some stressful tests. The
differences between performances were
more pronounced in some tests, and the
ranking differed from test to test. How-
ever, important properties of the schemes
were identified as a result of the diversity
of the tests. This preliminary testing of the
schemes resulted in the following find-
ings:
• ASD has very high accuracy but is
not strictly mass conservative. It is
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not monotonic but usually yields a
positive result.
BOT is highly accurate and mass con-
servative. It may create ripples and
overshoots, but it is positive-definite.
FCT is fairly accurate and mass con-
servative. It does not create any
ripples or overshoots, but does so at
the expense of diffusion to the back-
ground.
HAT has fair accuracy and is mass
conservative. However, it may lead to
ripples that can grow and cause in-
stabilities.
PPM also has good accuracy and is
strictly mass conservative and mono-
tonic. While the diffusion to the back-
ground is very small, peak clipping
can be significant
SLT has poor accuracy and severe
mass conservation problems. The only
positive feature of SLT is that it does
not lead to ripples.
SMO has relatively low accuracy. It is
mass conservative but it may lead to
ripples in concentration fields.
YAM has very high accuracy even for
the shortest wavelengths, and is mass
conservative. It can lead to overshoots
under certain situations.
In addition to the properties discussed
above, the computational performances of
the schemes were also considered in the
ranking. The CPU times for all the tests
were averaged and normalized with re-
spect to SMO. The only scheme that is
less CPU intensive than SMO is BOT;
other schemes require more CPU time.
ASD is the most CPU-intensive scheme:
it requires more than four times more CPU
than SMO does.
M. Talat Odman is with the MCNC-North Carolina Supercomputing Center,
Research Triangle Park, NC 27709-2889.
Daewon W. Byun (on assignment from the National Oceanic and Atmospheric
Administration, U.S. Department of Commerce) is the EPA Project Officer (see
below)
The complete report, entitled "Research on Numerical Transport Algorithms for Air
Quality Simulation Models," (Order No. PB98-127327; Cost: $21.50, 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:
National Exposure Research Laboratory
U.S. Environmental Protection Agency
Research Triangle Park, NC 27711
United States
Environmental Protection Agency
Center for Environmental Research Information
Cincinnati, OH 45268
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EPA/600/SR-97/142
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