United States
Environmental Protection
Agency
Atmospheric Sciences Research
Laboratory
Research Triangle Park NC 27711
Research and Development
EPA-600/S3-84-112 Jan. 1985
&EPA Project Summary
Evaluation of Two Numerical
Integrator Schemes
A. A. N. Patrinos and M. J. Leach
This report compares the performance
of two numerical integrator schemes
that have been used in air pollution
simulations. The schemes are the inte-
grator used in the Northeast Regional
Oxidant Study and a version of the
pseudospectral integrator; both schemes
have been considered candidates for
the Regional Acid Deposition Model.
The two schemes are discussed in some
detail, along with their principal advan-
tages and drawbacks. An objective
methodology is then developed to pro-
vide quantitative measures of the
schemes' performances on accuracy,
conservation, and efficiency. Represen-
tative results are given from two test
simulations: the advection-diffusion of
a single puff in a uniform wind field and
the continuous plume in a uniform wind
field. Although less efficient, the
Northeast Regional Oxidant Study in-
tegrator demonstrated superior consis-
tency in accuracy and conservation over
the pseudospectral scheme.
The 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
An important focus of the National Acid
Precipitation Assessment Program is the
development of the Regional Acid Depo-
sition Model (RADM). This model is
expected to integrate state-of-the-art
knowledge of the important chemical
reactions leading to acid deposition pre-
cursors with state-of-the-art routines
describing atmospheric processes The
end product will be a Eulerian grid model
capable of performing assessment calcu-
lations and alternative scenario simula-
tions. The choice of a suitable numerical
scheme for the transport simulations of
RADM is a critical one, and a decision
was made to compare two candidates for
that role on the basis of an objective
evaluation methodology. The schemes
were the NEROS (Northeast Regional
Oxidant Study) integrator and a version of
the pseudospectral integrator. The NEROS
scheme may be described as a Lagrangian,
upwind differencing scheme employing
bi-directional qumtic polynomials. The
pseudospectral integrator employs the
traditional, finite Fourier series expansion
coupled with a coordmatetransformation
to eliminate the need for periodic bound-
ary conditions.
Evaluation Procedure
The evaluation methodology concen-
trated on three criteria; accuracy, conser-
vation, and efficiency. Accuracy was
judged by three numerical measures
borrowed from the nomenclature of ap-
plied mathematics, LL L2, and L«. The
first two are measures of the global
performance of a scheme, the third is
important where the forecasting of peak
values is crucial In selecting the test
cases for comparison, an important re-
quirement was the availability of the
exact solution in order to effectively
assess accuracy Two test cases were
formulated, one involved the advection-
diffusion of a "spline" puff and the other
the constant point source in a uniform
wind, particularly the early transient part
prior to the development of the steady
state plume
-------
Within the framework of the two cases
mentioned above, the schemes were
tested in a variety of parametric changes,
including wind speed, diffusivity, time
step, and position of point source. Every
effort was made in creating fair compari-
son circumstances. Various filtering
algorithms for the pseudospectral inte-
grator were also attempted.
Conclusions
The importance of an objective evalu-
ation methodology is emphasized. This
should include a variety of test cases such
as "solid body" rotation, transport across
boundaries, and point and areal sources.
Puffs should be of general shape, and
symmetric flow characteristics should be
avoided It is important, however, that the
exact solution be available in analytical
closed-form terms or in simple and veri-
fiable numerical form.
The NEROS integrator demonstrated a
more consistent performance than the
various versions of the pseudospectral
integrator. This consistency was main-
tained throughout most parametric
changes and particularly for simulations
involving transport across boundaries.
Although more than twice as expensive,
the NEROS integrator demonstrated
superior accuracy and conservation
properties. The performance of the filter-
ing algorithms for the pseudospectral
integrator was erratic; only high diffusion
appears to favorthis integrator, but atthe
price of efficiency, since computational
stability restrictions decrease the allowed
time step.
RADM is expected to be dominated by
the chemical reaction simulations in
terms of computer time demands. It is
essential to keep this in perspective in
attempting to optimize the advection-
diffusion integrator. The "best" integrator
may in fact be the one which allows better
optimization of the chemical reaction
simulations while maintaining reasonable
standards of accuracy and conservation
for the transport terms
A.A.N. Patrinos andM. J. Leach are with Brookhaven National Laboratory, Upton,
NY 11973.
F. Binkowski is the EPA Project Officer (see below).
The complete report, entitled "Evaluation of Two Numerical Integrator Schemes,"
(Order No. PB 85-138 196; Cost: $8.50, 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 PFIINTING OFFICE, 1985 — 559-016/7887
United States
Environmental Protection
Agency
Center for Environmental Research
Information
Cincinnati OH 45268
BULK RATE
POSTAGE & FEES PAIC
EPA
PERMIT No G-35
Official Business
Penalty for Private Use $300
U i>
------- |