Transactions on Ecology and the Environment vol 7, © 1995 WIT Press, www.witpress.com, ISSN 1743-3541
Application of the random choice method to the
estuarine mass transport
A. Jazcilevich-Diamanf, E. Gomez-Reyes*, A. Valle-Levinsorf,
V. Fuentes-Gecf
"Division de Estudios de Posgrado, Facultad de Ingenieria,
Universidad National Autonoma de Mexico, Mexico, D.F. 04510,
Mexico
^Departamento de Ingenieria de Procesos e Hidraulica,
Universidad Autonoma Metropolitana, Av. Michoacany la
Purisima, Col Vicentina Iztapalapa, Mexico, D.F. 09340, Mexico
'Center for Coastal Physical Oceanography, Old Dominion
4 2312P,
Abstract
Performance comparisons of the Random Choice Method (RCM) with
conventional advection schemes to compute simple cases of the Mass Transport
Equation in estuaries are conducted.
Test cases treated here include
one-dimensional front propagation and advection of a rotary pulse. Both cases
represent extreme conditions of the mass transport in estuaries, i.e., absence of
physical diffusion. The RCM is capable of producing numerical solutions that are
exactly equal to the analytical solutions, except for a displacement of coordinates
which on the average is zero. All other numerical schemes compared introduce
numerical diffusion and/or dispersion. The RCM therefore offers a very good
alternative for modeling mass transport in estuaries.
1 Introduction
The recent interest in hydrodynamic modeling as a tool to study water quality
problems in estuaries has led to the use of equations that simulate the transport of
dissolved or suspended matter in water. The advection component of this equation
is often an important source of error because of the generation of numerical
diffusion and dispersive waves. Hence, devising an accurate scheme to solve the
transport equation continues to be an active area of research.
Transactions on Ecology and the Environment vol 7, © 1995 WIT Press, www.witpress.com, ISSN 1743-3541
254
Water Pollution
Scientists from research fields other than hydrodynamics have applied the
Random Choice Method (RCM) to simulate propagating fronts without
introducing numerical diffusion and with efficient computing time. The RCM
seems to be a very good alternative for the computation of advective processes.
The present study compares performances of the RCM with conventional schemes
(Centered Differences, Upwind, Moments and Smolarkiewicz) to compute the
advective component of the mass transport in estuaries. The following section
briefly describes the RCM
Performances for one and two-dimensional cases are
shown in the subsequent section. Conclusions are presented in thefinalsection.
2 Random Choice Method
The RCM wasfirstintroduced by Glimm [1] as part of a technique to prove the
existence of solutions for hyperbolic equations. Chorin [2] developed RCM into
a numerical method. Further development and successful applications of the
method in researchfieldsother than hydrodynamics have been documented. For
instance, in gas dynamics by Sod [3] & Colella [4]; in petroleum engineering by
Concus & Proskurowski [5]; in hydraulics by Marshall & Menendez [6]; in air
pollution by Jazcilevich-Diamant & Fuentes-Gea [7].
In the RCM, the solution of the Mass Transport Equation is constructed as a
superposition of local analytical solutions of Riemann problems and sampling
techniques. At each time step nAt, where At is the time interval, the concentration
field c^At) at the point x = iAx, isfirstapproximated by a piecewise constant
function, i.e., a sequence of Riemann problems,
(1)
Then c^w ^ advanced in time using the exact solution of the Riemann problem
along the characteristics defined by the corresponding velocity field, provided
that the Courant-Friedrichs-Lewy condition is satisfied to avoid intersection of
waves propagating from adjacent discontinuities. Finally, new values for the
concentration are sampled from the advected field. This sampling is carried out
through an equidistributed random variable £ in the interval [-1/2, 1/2], i.e., £
has a probability density function that takes the value of 1 in [-1/2, 1/2] and 0
elsewhere. Sampling is performed at each interval [-Ax/2, Ax/2], obtaining a
single value on the left c^ and right c^ of the solution of the Riemann problem,
t- Ax/2+
— c
Transactions on Ecology and the Environment vol 7, © 1995 WIT Press, www.witpress.com, ISSN 1743-3541
Water Pollution
255
Concentration values at time (n+\)At are assigned from c^ and c^ as follows,
L,T
far
*
£ > 0
/s*. \
Because the solution is either c^ or c& the region in which the concentration is
different from zero remains sharply defined if it is sharply defined initially. That is,
if the concentration at the initial time is a step function, it remains so at all times.
The method has therefore no numerical diffusion.
3 Performance comparisons
Performance comparisons of the RCM with conventional schemes (Centered
Differences, Upwind, Moments and Smolarkiewicz) to compute simple cases of
advective mass transport with known analytical solution, i.e., the one-dimensional
front propagation and the advection of a rotary pulse of a conservative water
quality constituent, are developed here. Since the analytical solution is known for
these cases, the numerical solution can be evaluated critically. Details of such
assessments are presented next.
3.1 Front simulation
In this case, the analysis is based on the one-dimensional numerical solution of a
conservative constituent of concentration c, such as salinity, and advected with
constant velocity u in the x direction,
6c
ck
^
— + u-— = 0
Varying velocityfields,multidimensions and nonrectangular coordinate systems
all increase the difficulty in modeling the transport equation. A simple onedimensional case has been chosen because if an algorithm can not model simple
cases correctly (such as eqn 4), then it is of little use in more complex
situations.
The analytical solution of Equation (4) for the case of a propagating front is
simply the initial distribution moving in the direction of the velocity field while
preserving its shape. This case represents the behavior of an idealized river
discharge along the major axis of an estuary. Figure 1 displays the numerical
solution of Equation (4) using RCM and other conventional advection schemes.
The figure shows that the Centered Differences scheme [8] produces small
dispersive waves, while the Upwind scheme [8] does not generate ripples but
Transactions on Ecology and the Environment vol 7, © 1995 WIT Press, www.witpress.com, ISSN 1743-3541
Water Pollution
Salinity (ppt)
256
30
25
20
15
10
5
0
-5
0. DO
Analytical
/ RCM
*
0.25 0.50 0.75
30
25
20 " SrnolarkiewicA Analytical
15
10
5
V
0
0.00
0.25 0.50 0.75
Figure 1: Comparison of RCM
a front propagation.
30
25
20
15
10
5
0
-5
1.00 0.30
30
25
20
15
10
5
0
Analytical l Moments
V
0.25 050
0.75
1.00
^xT\ ^Analytical
UpwindX
\ Centered
\\Differences
1.00 0.00 0.25 0.50 0.75
Relat ive Distance
1.00
with conventional advection schemes to simulate
dissipates the shape of the initial front. The Smolarkiewicz scheme [9] yields a
good solution, although some dispersion is still present and does not reproduce the
theoretical shape. The Moments Method [10] reproduces the analytical solution
almost perfectly. The solution by the RCM is the best fit to the analytical solution
as it preserves the shape of the front.
In the above comparison, the RCM perfectly matches the analytical solution
(i.e., ERROR=0), although in general the method yields a solution that fluctuates
about the true position. This displacement of coordinates is of O(Ax), which is
generated by the random sampling of the solution of the Riemann problem; the
coordinate displacement is zero on the average. Accordingly, the RCM has only
first-order accuracy but infinite resolution. Moreover, considering the
computational time efficiency (Table 1), the RCM is by far the best alternative to
model the one-dimensional component of the transport equation.
Transactions on Ecology and the Environment vol 7, © 1995 WIT Press, www.witpress.com, ISSN 1743-3541
Water Pollution
257
Table 1. CPU times for the front simulation
(normalized by the Centered Differences).
SCHEME
CPU
Centered Differences
LOO
RCM
1.30
Upwind
1.10
Smolarkiewicz
1.30
Moments
8.00
3.2 Rotating pulse simulation
Performance of the RCM that provides solutions for varying velocity fields and
multidimensional cases without introducing numerical dispersion, is illustrated in
Figure 2.
3/4 Cycle
Figure 2: RCM
solution for a rotary pulse.
Transactions on Ecology and the Environment vol 7, © 1995 WIT Press, www.witpress.com, ISSN 1743-3541
258
Water Pollution
The advection equation solved in the case of the rotary pulse is:
a,
aw
aw
a/
a*
ay
where the advected velocityfield(w,v) represents a steady anticyclonic gyre,
u —
—
and
v—
—
/^\
{'
This numerical solution shows that initial maximum value and shape of the
pulse concentration are integrally preserved through the entire cycle, as is the
case for the analytical solution. The position of the pulse, however, fluctuates
about the true position as much as an amount of O(Ax). On the average, this
fluctuation is zero.
4 Conclusions
Test cases treated here represent extreme conditions of the mass transport in
estuaries (i.e., absence of physical diffusion). They provide outermost events
for assessment of any numerical scheme to simulate the advective transport of
mass. The RCM is capable of reproducing numerical solutions for these cases
that are exactly equal to the analytical solutions, except for a displacement of
coordinates which on the average is zero. The RCM offers therefore a very
good alternative for modeling mass transport in estuaries.
Acknowledgement
This work was partially supported by the DGAPA, UN AM.
References
1.
Glimm, J. Solutions in the large for nonlinear hyperbolic systems of
conservation laws. Comm. ofPureAppl Math., 1965, 18, 697-715.
2. Chorin, A J Random choice solution of hyperbolic systems. Journal of
Computational Phsycis, 1976, 22, 517-533.
3.
Sod, G.A. A numerical study of a converging cylindrical shock. Journal of
f, 1977, 83, 785-794.
Transactions on Ecology and the Environment vol 7, © 1995 WIT Press, www.witpress.com, ISSN 1743-3541
Water Pollution
259
4. Colella, P. Glimm's method for gas dynamics. SIAM Journal of Science
Statitics Computational, 1982, 3(1), 76-109.
5. Concus, P. & Proskurowski, W Numerical solution of a nonlinear hyperbolic
equation by a random choice method. Journal of Computational Physics,
1979,30, 153-166.
6. Marshall, G. & Menendez, A.N. Numerical treatment of nonconservation
forms of the equation of shallow water theory. Journal of Computational
1981,44, 167-188.
7. Jazcilevich-Diamant, A. & Fuentes-Gea, V. The random choice method in the
numerical solution of the atmospheric transport equation. Environmental
1994,23-31.
8. Roache, P.J. Computational Fluid Dynamics.
Albuquerque, New Mexico, 1976.
Hermosa Publishers,
9. Smolarkiewicz, P.K. & Grabowski, W W The multidimensional positive
definite advection transport algortihm: nonoscillatory option. Journal of
Computational Physics, 1990, 86, 355-375.
10. Eagan, B.A. & Mahoney, J.R. Numerical modeling of advection and diffusion
of urban area source pollutants. Journal of Apllied Meteorology, 1972, 21.