A comparison of numerical techniques for solution of atmospheric kinetic equations

‡On assignment to the National Exposure ResearchLaboratory, U.S. Environmental Protection Agency.

Atmospheric Environment Vol. 32, No. 9, pp. 1535—1553, 1998( 1998 Elsevier Science Ltd. All rights reserved

Printed in Great Britain1352—2310/98 $19.00#0.00PII: S1352–2310(97)00381–6

A COMPARISON OF NUMERICAL TECHNIQUESFOR SOLUTION OF ATMOSPHERIC KINETIC EQUATIONS

ROHIT MATHUR,* JEFFREY O. YOUNG,-‡KENNETH L. SCHERE-‡ and GERALD L. GIPSON°

*Environmental Programs, MCNC, P.O. Box 12889, Research Triangle Park, NC 27709-2889, U.S.A.;‡Atmospheric Sciences Modeling Division, Air Resources Laboratory, National Oceanic and AtmosphericAdministration, Research Triangle Park, NC 27711, U.S.A. and °National Exposure Research Laboratory,

U.S. Environmental Protection Agency, Research Triangle Park, NC 27711, U.S.A.

(First received 16 February 1997 and in final form 21 July 1997. Published April 1998)

Abstract—Numerical modeling of atmospheric chemistry is a computationally intensive problem. Theequations describing the interaction among various modeled chemical species are coupled, nonlinearordinary differential equations. Spatial dependencies in comprehensive three-dimensional air qualitymodels require the solution of this system at thousands of spatial points. Even with increasing computerpower, there is a need for efficient and accurate numerical solvers with expanded capabilities, since the nextgeneration of air quality simulation models needs to address the increasingly complex chemistry issuesemerging in new model applications.

Variants of the commonly used quasi steady-state approximation and the hybrid methods currently usedin several modeling systems are examined against a reference mechanism describing chemical interactionsrelated to tropospheric oxidant and acid formation. Additional modifications to the methods are incorpor-ated to yield more robust integration techniques. The chemistry solution methodology used in the regionalacid deposition model is also incorporated in this comparison as a base methodology for representing thereference chemical mechanism. The methods are tested against the Gear integration scheme for a variety oftest cases including traditional box-model calculations and detailed three-dimensional simulations, andtheir relative accuracies and efficiencies are investigated. Performance and implementation issues related tochemical integration schemes are examined in the context of the demands and needs of the chemistrycomponent of future comprehensive atmospheric chemistry/transport simulation models. ( 1998 ElsevierScience Ltd. All rights reserved.

Key word index: Atmospheric chemistry, stiff ODEs, numerical integration, air quality modeling,photochemical mechanism.

1. INTRODUCTION

The chemical process describing the fate of atmo-spheric pollutants is mathematically representedthrough a set of coupled, nonlinear ordinary differen-tial equations (ODEs). A primary source of difficultyin the numerical solution of these equations arisesfrom the fact that the chemistry in atmospheric sys-tems involves reactions whose characteristic timescales vary by orders of magnitude, resulting in ahighly ‘‘stiff’’ system of equations. Following McRaeet al. (1982), a stiff system may be described as asystem

dci

dt"f

i(c

j, t), i, j"1, 2, . . . , n (1)

with the following properties: (1) the eigenvalues ofthe Jacobian matrix of the system are negative, i.e.Re(j

i)(0, i"1, 2, . . . , n; and (2) the ratio of the

maximum to the minimum eigenvalue (max DRe (ji) D /

min D Re (ji) D"R) is large.

In solving ODEs numerically, stiffness affects stab-ility or amplification of errors from step to step; thus,to prevent overwhelming errors, stiffness must be con-sidered when selecting step sizes for a numerical sol-ver. The stiffness ratio for ODEs arising from typicaltropospheric chemical systems is usually of the orderof 1010, leading to a very stiff system of equations.Further, the spatial dependencies typical of three-dimensional (3-D) air quality models require solutionof these equations at each computational node of themodeled domain, resulting in an extremely computa-tionally intensive problem. While significant advanceshave been made in the numerical solution of stiffODEs, the search for faster and more accuratesolution methodologies is of considerable interest

1535

to atmospheric modelers, because 80—90% of thecomputational effort in comprehensive atmosphericchemistry/transport models is devoted to solving thekinetic equations.

A variety of methods have been proposed overthe past two decades to provide accurate and fastersolutions to chemical kinetic problems in the atmo-sphere. Explicit methods (e.g. Young and Boris, 1977;Hesstvedt et al., 1978) are the simplest and fastestones, and variants of these have commonly been usedin air quality simulation models (e.g. Carmichaelet al., 1991; Chang et al., 1987; Lamb, 1983; McRaeet al., 1982). Other methods such as hybrid methodsuse a combination of semi-implicit and explicit solu-tion techniques by dividing the species into stiff andnon-stiff categories; an implicit integration techniqueis then used for the stiff or fast-reacting species and anexplicit scheme is used for integrating the rate equa-tions of the non-stiff or slow-reacting species (Gongand Cho, 1993; Sun et al., 1994). A variety of tech-niques have also been developed to take advantage ofcharacteristics of particular chemical mechanisms (e.g.Hesstvedt et al., 1978; Sillman, 1991; Hertel et al.,1993). While implicit methods (Curtiss and Hirschfel-der, 1952; Gear, 1971) have been used in a limitedsense for atmospheric chemistry problems, only re-cently have attempts been made to include them indetailed 3-D atmospheric transport/chemistry calcu-lations (e.g. Jacobson and Turco, 1994). A number ofcomparative studies have been performed to examinethe tradeoffs between accuracy and computationalefficiency of such integration methods (Seinfeld et al.,1970; Odman et al., 1992; Gong and Cho, 1993; Saylorand Ford, 1995; Verwer et al., 1996; Sandu et al.,1996). While such comparisons are useful, they havegenerally focused on chemical evolution of variousspecies based either on idealized conditions or a set oflimited conditions. Sillman (1991) and Chock et al.(1994) have attempted to perform similar compari-sons over a wider range of conditions, but for totalintegration periods ranging from 6 min to a half anhour. Very little effort has been devoted to examiningthese solution methods in situ in a comprehensive 3-Dmodeling environment, in the presence of other inter-acting processes, which cause the actual conditionsthat the integration methods experience to vary dra-matically both spatially and temporally. Recently,Dabdub and Seinfeld (1995) have examined the im-pact of alternate chemical integration schemes in theCIT model by examining average errors, whileWinkler and Chock (1996) have examined the impactof replacing the chemical integration scheme in theUAM on predicted domain wide peak concentrationsfor a variety of species.

In this study, we extend previous investigations byperforming a set of traditional box-model calculationsas well as testing selected integration schemes ina comprehensive atmospheric transport/chemistrymodeling system. The integration methods we con-sidered represent techniques or variations of tech-

niques commonly used in existing air qualitymodeling systems and include a modified version ofthe QSSA scheme (MQSSA) originally proposed byHesstvedt et al. (1978), a modified version of thehybrid scheme of Young and Boris (1977) (MY & B),and the solution method employed in the regionalacid deposition model (RADM) (Chang et al., 1987).We compare our results against solutions obtainedfrom higher-order solution techniques such asLSODE (Hindmarsh, 1983) for the box-model casesand SMVGEAR (Jacobson and Turco, 1994) for thecomprehensive modeling case. Section 2 describes theintegration techniques that are the focus of this study.Section 3 presents details on the comparison meth-odology along with results obtained from the testproblems designed to evaluate the integration tech-niques. In Section 4, we summarize the major con-clusions of this study and offer suggestions for futureresearch.

2. DESCRIPTION OF SOLVERS

At any spatial point, the rate of change in concen-tration for each species resulting from the chemicalkinetics is described by equation (1). That equationcan then be recast into the following form:

LCi

Lt"P

i!¸

iC

i(2)

where Ci

represents the concentration of species i,and P

iand ¸

iC

irepresent the chemical production

and loss rates of species i, respectively. In general,Piand ¸

iare functions of other species concentrations

and hence exhibit variations on various time scales.The reciprocal of ¸

i(q

i) represents the characteristic

time of decay of species i, indicating how quicklyC

ireaches its equilibrium value.If P

iand ¸

iare constant, then equation (2) can be

solved analytically to give

Cti"

Pi

¸i

#CC0i!

Pi

¸iD ei(L

it) . (3)

2.1. ¹he modified QSSA (MQSSA) solver

The MQSSA solver is a variant of the quasi-steady-state approximation (QSSA) method of Hesstvedtet al. (1978). We have incorporated the QSSA schemewithin a predictor—corrector algorithm with an addi-tional chemical integration time-step-determinationstep. Further, no a priori assumptions are made re-garding steady-state approximations for any of themodeled species. The solver is based on conceptsoriginally proposed by Lamb (1983). The methodo-logy in its original form as proposed by Lamb (1983)was implemented in the regional oxidant model(ROM) (Lamb, 1983) and optimized for vector com-puter applications by Young et al. (1993). Startingwith that solver, we have developed the modified

1536 R. MATHUR et al.

QSSA (MQSSA) solver by introducing several algo-rithmic and implementation changes based on ourtesting of the original scheme. We first present a briefoverview of the original scheme proposed by Lamband then discuss the modifications introduced to yieldthe MQSSA scheme.

2.1.1. ¹ime-step determination. The integrationtime step is determined through a three-step proced-ure as follows:

1. Determine a composite concentration CK "CNO

#CNOÈ

#COÊ

.2. Chemical characteristics of species with concen-

trations greater than 1% of CK are used in the time-stepdetermination. This empirical step effectively screensout of the time-step determination those specieswhose concentrations are quite low relative to theprimary species and those radical concentrations thatadjust rapidly to the local chemical conditions. It isassumed that P

iand ¸

iare constant over the time step

*tnsuch that the concentration Cn`1

iat t

n`1(tn`1

"

tn#*t) is given by

Cn`1i

"

Pni

¸ni

#CCni!

Pni

¸niD e!(¸n

i*t

n) . (4)

The above assumptions are assumed valid if the con-centration of the species does not change significantly(by j) over the integration time step such that

D Cn`1i

!CniD)jCn

i. (5)

From equations (4) and (5) we get

1!e!(¸ni*t

n))

jCni

DCni!Pn

i/¸n

iD,F

i. (6)

The time-step criteria of equation (6) is satisfied auto-matically for F

i'1. For other cases, a sufficient con-

dition for equation (4) to be a close approximation tothe solution of equation (1) follows from equation (6)as

*tn"min(!ln(1!F

i)/¸n

i). (7)

3. For the sake of computational efficiency, min-imum and maximum allowable time steps are alsospecified. The values of these time-step bounds alongwith that of j then determine the degree of accuracyrequired. If *t

n(*t

.*/then *t

nis set to *t

.*/, and if

*tn'*t

.!9then *t

nis set to *t

.!9.

2.1.2. Solution procedure. Predictor: Based on thetime scale of their variation, the concentrations of thesystem are classified into three categories.

1. The concentration of a species is assumed to bein equilibrium with other species if its characteristictime of variation is considerably smaller than theintegration time step, so that at any instant, ifqi(*t

n/10 then

Ci"

Pni

¸ni

(8)

where C*i

is an intermediate concentration solution.

2. If qi<*t

n(q

i'100*t

n), the time variation of

the species concentration is slow. In this case it can beassumed that the concentration variation is linearwith time and is calculated by the Euler formula

C*i"Cn

i#(Pn

i!¸n

iCn

i) *t

n. (9)

3. For all other cases, i.e. *tn/10)q

i)100 *t

n

C*i"

Pni

¸ni

#CCni!

Pni

¸niD e!(¸n

i*t

n) . (10)

Corrector: Based on the values of C*i, the chemical

production (P*i) and loss (¸*

i) terms are calculated to

yield

PKi"

P*i#Pn

i2

(11)

Ki"¸*

i. (12)

Steps 1—3 are then repeated using the terms PKiand K

ias the chemical production and loss terms, respect-ively. The categorization of species through equations(8)—(10) is performed dynamically during the course ofintegration.

2.1.3. Additional modifications to the MQSSAsolver. Based on our testing of this methodology, wefound that the original time-step determination pro-cedure described in Section 2.1.1, while robust, wascomputationally intensive. Analysis of the techniquefor selected test cases indicated that the computedtime step was often close to either the specified min-imum or maximum time-step bounds that were im-posed for efficiency considerations. Consequently,much effort was spent in computations that were notfully utilized. In the MQSSA implementation we havereplaced the procedure with a methodology similar tothat proposed by Young and Boris (1977), so that thetime-step control is maintained as

*tn"min A

eCni

Pni!¸n

iCn

iB (13)

with an additional constraint such that *t.*/

(

*tn(*t

.!9. Here, e represents the maximum allow-

able percentage change in species concentration overthe time step and *t

.*/and *t

.!9represent specified

minimum and maximum time-step bounds, respec-tively. Additionally, in the MQSSA implementation,for each chemical step, we perform the species cat-egorization into fast, intermediate and slow once anduse the categorization for both the predictor andcorrector steps, as opposed to Lamb’s implementa-tion where the categorization was performed bothfor the predictor and corrector step. This furtherhelps improves the computational efficiency of thealgorithm.

2.2. ¹he RADM solver

The RADM solver represents the chemistry inte-gration methodology implemented in the Regionalacid deposition model (RADM) (Chang et al., 1987).

Solution of atmospheric kinetic equations 1537

Of the 58 modeled gas-phase chemicals, the RADMsolver treats 18 (radical species) as explicit steady-state species. In addition, the RADM solver uses thefollowing four lumped species to remove pairs ofreaction rates that are large due to rapid interconver-sions between species:

[NOx]"[NO]#[NO

2] (14)

[HOx]"[OH]#[HO

2] (15)

[PAO3]"[PAN]#[ACO

3] (16)

[N2N

5]"[N

2O

5]#[NO

3]. (17)

Time-step control is maintained through equation(13). Like the MQSSA solver, the RADM solver alsobounds the computed time step with an allowableminimum and maximum. H

2O

2and HO

xconcentra-

tions are stepped forward in time using the analyticalsolution described by equation (4), while all otherspecies concentrations are computed using the follow-ing trapezoidal discretization scheme:

Cn`1i

"

Cni(1!¸n

i*t

n/2)#Pn

i*t

n1#¸n

i*t

n/2

. (18)

OH concentrations are obtained by solving the fol-lowing equation (see the appendix for details):

d[OH]

dt"A[OH]2!B[OH]#C. (19)

If OH is assumed to be in steady state, the roots of theequation

A[OH]2!B[OH]#C"0 (20)

yield the OH concentration. Finally, concentrationsfor NO, HO

2, PAN, and N

2O

5are then computed

from the lumped entities (equations (14)—(17)).Further, it should be noted that the RADM solver

is constructed to take advantage of specific features ofthe chemical mechanism of Stockwell et al. (1990). Inparticular, special attention has been devoted to thesequence of calculations of the steady-state species.Once the concentrations of the steady-state species iscomputed, these are used in the calculation of thetransported species through equations (4) and (18).Additionally, all reaction rates related to HO

xtrans-

formations are recomputed using updated values ofother species.

2.3. Modified ½oung and Boris (M½&B) solver

A second-order predictor, iterated-correctorscheme was suggested by Young and Boris (1977) forsolving the chemical kinetics of reactive flow prob-lems. This method has further been used for solutionof tropospheric kinetic equations (e.g. McRae et al.,1982; Odman et al., 1992; Yamartino et al., 1992). Thespecies are separated into two groups according to therelative size of the species time-constant with respectto the integration time step. For q

i'*t, the species

is considered to be non-stiff or long-lived, while

for qi)*t, it is considered to be stiff or short-lived.

The non-stiff species are integrated according to thealgorithm

Predictor: c*i"cn

i#*t

n(Pn

i!¸n

icni) (21)

Corrector: cn`1i

"cni#

*tn

2

](Pni!¸n

icni#P*

i!¸*

ic*i) .

(22)

The stiff species are integrated using

Predictor: c*i"

cni(2qn

i!*t

n)#2Pn

iqni*t

n2qn

i#*t

n

(23)

Corrector: cn`1i

"

cni(qn

i#q*

i!*t

n)#(*t

n/2)(Pn

i#P*

i) (qn

i#q*

i)

qni#q*

i#*t

n

.

(24)

Further, following McRae et al. (1982) and Odmanet al. (1992), we have introduced the pseudo-steady-state approximation (equation (8)) for species withqi;*t, which we refer to as the ‘‘fast’’ group. In our

implementation, the separation of species into fast,stiff, and non-stiff groups is based on the criteriaqi(0.2*t, 0.2*t)q

i)5*t, and q

i'5*t, respect-

ively. These species categorizations are performeddynamically during the integration. In constructingthe modified Young and Boris (MY&B) solver, wehave attempted to use the integration formulae sug-gested by Young and Boris (1977) in a two-steppredictor—corrector mode similar to the MQSSAtechnique. In addition, we have added an automatictime-step control by implementing the procedure de-scribed by equation (13). Consequently, the basic al-gorithmic structures of the MQSSA and the MY&Bmethods are similar except in the specific integrationformulae used for different species classifications.

There are significant differences between theRADM solver and the MQSSA and MY&B solvers.As indicated earlier, the RADM solver has been care-fully constructed and optimized to compute solutionsof the chemical transformation equations represent-ing the Stockwell et al. (1990) mechanism. The algo-rithm is designed such that the concentrationsof fast-reacting species are computed first. On theother hand, our implementation of the MQSSA andMY&B solvers is more general; no particular atten-tion is paid to the sequence in which the concentra-tions of species is computed. These considerations areimportant from the perspective of future designs of‘‘general’’ solution techniques where different mecha-nisms can be easily replaced. Another significant dif-ference between the implementations is the treatmentof radical species. The RADM solver treats all radicalspecies as steady state, whereas no a priori assump-tions regarding steady state are made in the MQSSAand MY&B solvers. This approach sacrifices some


Table 1. Initial mixing ratios (ppb) for the single-cell test cases

Species Urban Rural Species Urban Rural

NO 7.5 0.02 HC5 16.6 1.58NO

22.5 0.08 HC8 16.3 0.95

HONO 0.01 0.001 OL2 7.0 0.6HNO

31.0 0.02 OLT 8.0 0.3

O3

50 20 OLI 5.0 0.0SO

220 1 ISO 0.0 0.6

CO 1000 120 TOL 12.0 1.7PAN 0.4 0 XYL 8.0 0.3NMHC (total) 100 10 HCHO 2.5 0.1ETH 5.5 1.58 ALD 2.5 0.1HC3 16.6 2.21

Water vapor mixing ratios are set to a constant value of 1.04]104 ppm.

degree of computational efficiency, but providesa seemingly more general solution procedure rela-tively free of mechanism specifics. The relative meritsof such considerations need to be evaluated for modelapplications where an optimal balance between accu-racy, efficiency, and generality is required.

3. COMPARISON METHODOLOGY AND RESULTS

In most previous comparative studies, methods ofinterest have been tested against the Gear’s (1971)method, which employs a variable-order backwarddifferentiation, constrained by strict tolerance limitsto provide extremely accurate solutions for a particu-lar test problem. Further, most of these studies havefocused on examining the relative accuracy and effi-ciency of selected solution techniques for limited box-model calculations. While instructive in regard to theperformance of the solution methodology, such calcu-lations are rarely indicative of the actual performanceof the numerical scheme under the highly varyingconditions experienced in typical 3-D calculations,where the ambient concentrations could be dramati-cally influenced by other interacting processes such asspatially inhomogeneous emissions distributions andvarying meteorological conditions (e.g. boundary-layer growth and effects of winds and clouds).

We have used the second generation regional aciddeposition model chemical mechanism (RADM2) ofStockwell et al. (1990) as the reference mechanism inour comparisons. The mechanism includes signifi-cantly expanded organic chemistry compared to thecarbon bond mechanism IV (CBM-IV) (Gery et al.,1989) used in several previous comparative studies.We also chose the RADM2 mechanism because it islikely to represent the future direction of using ex-panded chemical mechanisms; as more information isgained about reaction intermediates, previously lum-ped single reactions are being expanded into severalseparate elementary reactions. It is therefore impera-tive to examine and reevaluate the performance ofchemical solution techniques for these expanded

mechanisms, as they are likely to impose additionalcomputational burden.

Further, in each of the integration methods de-scribed in Section 2, one or two adjustable parameters(e.g. time-step size or error tolerance) typically dictatethe accuracy and computational speed of the tech-nique. The relative accuracy of either of these methodscan be increased at the expense of more computa-tional effort. Determining the optimal balance be-tween these two desired properties in integration ofatmospheric kinetic equations is a challenging prob-lem. The performance of integration methodologiesmust be evaluated in the context of the relativetradeoffs between desired accuracy and computa-tional efficiency. We therefore designed two sets ofexperiments to examine the relative performance ofthe numerical schemes described in Section 2. Thefirst set of experiments involves the traditional ap-proach, in which single-cell calculations are used toinvestigate specific optimal parameter values such astime-step range and error tolerances. The second setof experiments involves incorporating the solutiontechniques within a comprehensive 3-D modelingframework and examining their relative performancein the presence of other interacting physical processes.

3.1. Single-cell test cases

Table 1 shows the initial conditions for the two testproblems considered. The initial conditions for theinorganic components and total NMHC are adoptedfrom Gong and Cho (1993), who used the RADM1chemical mechanism of Stockwell (1986), a prede-cessor to the RADM2 mechanism used here. Theconditions for the first test represent an urban mixwhile those for the second test represent a rural mix(hereafter, referred to as the urban and rural cases,respectively). In both cases, we then partitioned thetotal NMHC concentration into individual organiccomponents based on the fractions suggested byStockwell et al. (1990) for typical urban and ruralNMHC composition. Simulations were conductedfor a period of 48 h using temporally varying photo-lysis rates (calculated for 36°N latitute and 78°W


Table 2. RMS for selected species for the urban case

RADM RADM RADM MQSSA MQSSA MQSSA MY&B MY&B MY&BSpecies 1 s 10 s 30 s 1 s 10 s 30 s 1 s 10 s 30 s

ACO3

0.0284 0.0322 1.0000 0.0433 0.0524 0.1003 0.0344 0.0398 0.0533ALD 0.0485 0.0477 0.5219 0.0487 0.0452 0.0284 0.0500 0.0494 0.0466CO 0.0038 0.0038 0.0286 0.0037 0.0037 0.0038 0.0038 0.0038 0.0038ETH 0.0012 0.0011 0.0532 0.0012 0.0012 0.0004 0.0012 0.0012 0.0012H

2O

20.0837 0.0801 28.770 0.0839 0.0843 0.0858 0.0852 0.0872 0.1023

HCHO 0.0746 0.0767 0.7913 0.0496 0.0526 0.0755 0.0477 0.0483 0.0494HNO

30.0263 0.0291 141.50 0.0205 0.0293 0.0869 0.0158 0.0173 0.0251

HNO4

0.0634 0.0655 0.9804 0.0613 0.0631 0.1350 0.0607 0.0593 0.0717HO 0.0731 0.0697 27.040 0.0770 0.0716 0.0566 0.0804 0.0793 0.0805HO

20.0952 0.0939 4.6520 0.1253 0.1223 0.1077 0.1334 0.1328 0.1337

HONO 0.0631 0.0963 85050.0 0.0694 0.1014 0.2047 0.0730 0.1061 0.1479ISO 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000N

2O

50.3748 0.4199 496.90 0.3358 0.4391 1.2920 0.2270 0.2352 0.2920

NO 0.0441 0.0476 24650.0 0.0501 0.0619 0.1462 0.0512 0.0583 0.1018NO

20.0635 0.0722 4294.00 0.0555 0.0674 0.1512 0.0380 0.0422 0.0571

NO3

0.3360 0.3878 2.1180 0.3404 0.4305 1.1830 0.1994 0.2075 0.2511O

30.0198 0.0226 1.0000 0.0273 0.0334 0.0635 0.0261 0.0264 0.0288

OLN 0.2532 0.3222 0.7875 0.2208 0.2274 0.2563 0.2198 0.2210 0.2374ONIT 0.0314 0.0467 0.7590 0.0511 0.0597 0.0752 0.0515 0.0639 0.0857PAN 0.0641 0.0762 4.7120 0.0702 0.0857 0.2072 0.0375 0.0401 0.0484SO

20.0045 0.0044 0.1619 0.0047 0.0046 0.0013 0.0046 0.0046 0.0046

SULF 0.0424 0.0402 5.3340 0.0578 0.0548 0.0293 0.0574 0.0589 0.0567TOL 0.0175 0.0162 0.7008 0.0243 0.0217 0.0042 0.0242 0.0236 0.0214TPAN 0.0626 0.0815 5.7690 0.0830 0.0969 0.1794 0.0597 0.0657 0.0716XYL 0.0794 0.0731 0.9733 0.1126 0.0997 0.0156 0.1125 0.1094 0.0983

longitude) with ambient temperature and pressure setto standard conditions. The simulations were initiatedat 1000 GMT on 2 August, 1983. In these box-modelcalculations, the total 48 h simulation is split up into6 min subintervals, representing transport time stepsin an operator-splitting environment, where photoly-sis rates are updated and the solvers restarted. Notethat the use of temporally varying photolysis ratesallows us to examine how well the various solutionmethodologies track the relatively large concentra-tion variations that occur during the rapid changes ofphotolysis rates at sunrise and sunset. In addition, itmay be noted that no emissions are considered inthese box-model calculations.

A comparison of the solution methodologies for thetwo representative test cases is presented in Tables2 and 3, which tabulate the root-mean-square error(RMS) for each species k, defined as

RMSk"S

1

M

M+

m/1AcL mk!cm

kcL mkB2

(25)

where cL mk

and cmk

represent the exact and computedsolutions for species k at the end of step m. Con-sequently, the RMS is based on sampling speciesconcentrations at 480 steps. These tables show therelative performance of these solvers as theminimum-time-step parameter is varied from 1 to30 s, while the maximum allowable time step is set to5 min in all cases. The LSODE simulation which istaken as the ‘‘exact’’ solution is based on setting the

tolerance parameters ATOL to 10~7 ppm and RTOLto 10~7.

The specification of the minimum and maximumallowable time steps for the solvers has importantimplications for both the accuracy and computationalperformance of the solution methodologies. Atmo-spheric chemistry problems typically contain reac-tions with time constants as small as 10~9 s; therefore,in the absence of such time-step bounds, extremelysmall time steps can be chosen based on the time-stepselection criteria described in Section 2, resulting insignificant computational effort which may be unac-ceptable for practical applications. On the other hand,the specification of an upper bound of 5 min serves toprotect the solution methodology from taking too-large step sizes that may induce instabilities in theexplicit schemes. Verwer et al. (1996) point out thatstep sizes below 1 s are redundant because specieswith a time constant smaller than 1 s almost instan-taneously achieve their steady state when perturbed.As the RMSs in Tables 2 and 3 show, the relativeaccuracy of each individual solver deteriorates as theminimum time step is relaxed from 1 to 30 s. Reason-able accuracies are obtained for the MQSSA, MY&B,and RADM solvers with the minimum time step seteither at 1 or 10 s for both the urban and rural testcases; for the RADM solver in the urban test case,however, substantial deterioration in solutionaccuracy is observed when the minimum time step isincreased to 30 s (Table 2). A more global examinationof the accuracy of the solvers can be obtained by


Table 3. RMS for selected species for the rural case

RADM RADM RADM MQSSA MQSSA MQSSA MY&B MY&B MY&BSpecies 1 s 10 s 30 s 1 s 10 s 30 s 1 s 10 s 30 s

ACO3

0.0414 0.0419 0.0429 0.0394 0.0394 0.0403 0.0501 0.0501 0.0510ALD 0.0194 0.0199 0.0214 0.0172 0.0172 0.0176 0.0169 0.0169 0.0168CO 0.0046 0.0046 0.0046 0.0037 0.0037 0.0038 0.0039 0.0039 0.0040ETH 0.0007 0.0007 0.0008 0.0008 0.0008 0.0008 0.0008 0.0008 0.0008H

2O

20.0865 0.0857 0.0842 0.0798 0.0799 0.0788 0.0791 0.0791 0.0795

HCHO 0.0487 0.0502 0.0536 0.0297 0.0297 0.0315 0.0292 0.0292 0.0292HNO

30.0216 0.0227 0.0255 0.0423 0.0423 0.0451 0.0240 0.0240 0.0238

HNO4

0.0751 0.0764 0.0862 0.0659 0.0659 0.0724 0.0545 0.0545 0.0543HO 0.0575 0.0570 0.0556 0.0692 0.0692 0.0687 0.0714 0.0714 0.0725HO

20.0759 0.0762 0.0810 0.0569 0.0569 0.0555 0.0635 0.0635 0.0629

HONO 0.0325 0.0312 0.0291 0.0247 0.0247 0.0240 0.0217 0.0217 0.0216ISO 0.1031 0.1026 0.1002 0.1251 0.1252 0.1253 0.1225 0.1226 0.1280N

2O

50.0946 0.0986 0.1718 0.2393 0.2393 0.2578 0.1010 0.1010 0.1026

NO 0.0610 0.0603 0.0597 0.0438 0.0438 0.0494 0.0411 0.0411 0.0407NO

20.0568 0.0608 0.0708 0.1088 0.1089 0.1190 0.0537 0.0537 0.0554

NO3

0.0742 0.0785 0.0892 0.1568 0.1568 0.1664 0.0796 0.0796 0.0801O

30.0045 0.0050 0.0062 0.0143 0.0143 0.0124 0.0144 0.0144 0.0121

OLN 1.5250 1.6190 1.7520 0.4605 0.4607 0.4708 0.3597 0.3598 0.3659ONIT 0.0793 0.0765 0.0691 0.0484 0.0485 0.0498 0.0445 0.0445 0.0449PAN 0.0616 0.0650 0.0745 0.0847 0.0848 0.0957 0.0556 0.0556 0.0556SO

20.0020 0.0020 0.0020 0.0021 0.0021 0.0021 0.0021 0.0021 0.0021

SULF 0.0586 0.0588 0.0587 0.0690 0.0690 0.0698 0.0678 0.0679 0.0694TOL 0.0123 0.0123 0.0122 0.0159 0.0159 0.0160 0.0152 0.0154 0.0160TPAN 0.1709 0.1561 0.1250 0.1169 0.1169 0.1294 0.0619 0.0620 0.0631XYL 0.0529 0.0526 0.0515 0.0644 0.0644 0.0646 0.0631 0.0632 0.0650

Table 4. SDA values for the urban and rural cases

Solver Urban case Rural case

RADM—1s 2.22 1.76RADM—10s 2.13 1.73RADM—30s! — 1.69MQSSA—1s 2.27 2.57MQSSA—10s 2.20 2.57MQSSA—30s 1.83 2.54MY&B—1s 2.35 2.71MY&B—10s 2.30 2.71MY&B—30s 2.21 2.70

! RADM—30s solution was unstable for urban case.

comparing the number of significant digits for theaverage (SDA) of RMS

kas suggested by Verwer et al.

(1996) and Sandu et al. (1996), where the SDA isdefined as

SDA"!log10 A

1

n

n+k/1

RMSkB (26)

where n is the number of species (58 in the RADM2mechanism used here). Table 4 summarizes the globalaccuracies of the MQSSA, MY&B, and RADM sol-vers for the three minimum-time-step specificationsfor both the urban and rural test cases. Again it can beobserved that the general accuracy of the solversdeteriorates as the minimum-time-step specification isrelaxed from 1 to 30 s. In general, for the given set ofsimulation conditions and solution method para-meters, the MQSSA and MY&B solvers give similaraccuracies for the various cases, and provide a slightlymore accurate solution for the rural case than theRADM solver. In terms of the CPU time needed, thecomputational requirements for all solution method-ologies were close and in general ranged from 1 to3.5 s for the urban case and 1 to 2.5 s for the rural caseon a CRAY C90; in general, CPU time increases asthe minimum-time-step specification is made stricter.

Saylor and Ford (1995) and Sandu et al. (1996) havepresented a methodology to evaluate the performanceof various integration techniques by constructingplots of the RMS or SDA as a function of the CPUtime required for similar box-model calculations.While their results are indicative of the tradeoffs be-tween accuracy and computational effort, both stud-

ies also caution that a more robust comparison is onewhere the integration techniques are tested withina real 3-D model calculation. Since the primary objec-tive of the box-model calculations presented here is toexamine the suitable range of adjustable parameterssuch as time-step bounds, we have refrained frompresenting a detailed analysis similar to theirs. Rather,we have attempted to use these box-model calcu-lations as a means of determining optimal values forthe adjustable parameters in the solvers, and thenused them in the 3-D simulations discussed next.

3.2. ¹hree-dimensional chemistry calculations

As indicated in the previous discussion, single-cellcalculations at best represent the approximate en-vironment that an ODE solver would experience ina 3-D atmospheric chemistry model. However, theactual environment that the solver experiences in such


a model could deviate significantly from theseidealized conditions. Consequently, performancemeasures based on such calculations cannot be re-garded as exhaustive. Since the ODE solvers are in-tended to operate in a comprehensive model, a moremeaningful comparison of different ODE solvers isone in which the methods are tested in situ withina 3-D simulation.

To test the performance of the solvers under morerealistic and disparate chemical regimes, we designeda comprehensive 3-D test. The 3-D modeling systemchosen for this work is the Multiscale Air QualitySImulation Platform (MAQSIP), a modular air qual-ity modeling system that is a prototype for the U.S.EPA’s Models-3 concept (Dennis et al., 1996). Themodular and flexible structure of MAQSIP facilitatesthe incorporation of alternative physical representa-tions and numerical algorithms. We have exploitedthis flexibility to use the modeling system as a test-bedenvironment to test the alternative chemical integra-tion schemes described in Section 2. The modelingsystem was configured to include detailed treatmentof horizontal and vertical advection, turbulent diffu-sion based on K-theory, anthropogenic and naturalemission sources for the eastern United States basedon the NAPAP inventory, dry deposition, and mixingand attenuation of photolysis rates due to the pres-ence of clouds.

We incorporated the MQSSA and MY&B solversinto the 3-D modeling system so that they were asstructurally similar as possible, given the differences inthe specific integration formulae and species classi-fication schemes discussed in Section 2. Further, someeffort was spent to exploit vectorization to aid incomputational efficiency. In particular, we used vec-torization over horizontal grid-cell dimensioning andstructured the codes so that each individual cell hadits own internal ‘‘chemical integration clock’’. Similarpractices have been employed in the implementationof the chemistry codes in other models (e.g. Chang etal., 1987; Young et al., 1993; Jacobson and Turco,1994). We should point out that although our imple-mentation may not exploit vectorization tothe fullest in its current form, the approach has beenadequate both in terms of efficiency and memoryrequirements for all of our testing and applications todate. Further exploitation of vectorization principlesis underway and is likely to make the solvers evenmore efficient.

We also incorporated the RADM solver as an alter-native solution methodology in the 3-D framework toprovide a base solution methodology that is in use incurrent modeling systems that include the RADM2mechanism. While we have taken extreme care torepresent the solution methodology correctly, itshould be noted that because our motivation to in-clude the RADM solver was based more on accuracyconcerns, and because the solution methodology isintricately intertwined with the mechanism features,we devoted more effort toward implementing the

methodology’s algorithmic features than toward com-putational efficiency. Consequently, the CPU timesdiscussed later for the RADM solver, while represen-tative, may not match those observed by the originalauthors. Lastly, we included the SMVGEAR (Jacob-son and Turco, 1994) which uses the same basic nu-merical method as LSODE, to provide a higher-order- accurate solution in the 3-D calculations; this istaken as the ‘‘exact’’ solution in the comparisons pre-sented below.

We included each alternative chemical solver in themodeling system and performed a series of 3-D simu-lations for a 24 h period over a domain encompassingthe eastern United States. The horizontal domain wasdiscretized using an 80 km resolution grid with 32computational nodes in the east—west direction and35 nodes in the north—south dimension. The verticaldomain, ranging from the surface to 100 mb, wasdiscretized using six levels. We obtained initial condi-tions, emissions, and all pertinent meteorological vari-ables from previously constructed data sets used inRADM simulations. The simulations were initiated at0000 GMT on 2 August, 1983. It may further be notedthat in these calculations, the transport of radicalspecies is not considered.

3.2.1. Initial applications. We computed percent-age differences for each species prediction at eachcomputational node for each solver with respect tothe SMVGEAR ‘‘exact’’ solution and examined thedistribution of these differences over the simulationperiod as a function of time of day. The SMVGEAR‘‘exact’’ solution is based on setting the absolute toler-ance to 10~9 ppm and the relative tolerance to 10~3.For the MQSSA and MY&B solvers, the variousadjustable parameters such as minimum and max-imum time-step bounds and tolerance parameterswere made identical to those in the most accuratebox-model configurations (i.e. the 1 s minimum-time-step case), whereas for the RADM solver they weremade identical to those in the operational version ofRADM. This exercise is essential to determinewhether optimal values of such parameters diagnosedfrom box-model calculations result in similar accu-racy levels when they are used for the more varyingconditions the solvers experience in actual 3-Dmodeling environments.

Figures 1—3 present comparisons of such percent-age difference distributions for selected species for thesurface-level predictions (representing a total of 1120sampling points at each sampling instant) for theMQSSA, MY&B, and RADM solver cases. In theseplots, the bottom and top of each box represent 5 and95% of the data. Within each box, the middle linerepresents the median value of the data (50%), whilethe lower and upper dotted lines represent 25 and75% of the data, respectively. A qualitative compari-son of these distributions for the various solvers indi-cates that, for most of the representative speciesshown, the 75th percentile generally follows the sametemporal trend for all three solvers. For NO

2, NO,


Fig. 1. Percentage difference percentile plots for 3-D test case with MQSSA (factor"0.01 and e"0.005).


Fig. 2. Percentage difference percentile plots for 3-D test case with MY&B (factor"0.01 and e"0.005).


Fig. 3. Percentage difference percentile plots for 3-D test case with the RADM solver.


PAN, and O3, the largest differences tend to occur

around hour 11 of the simulation (0600 local time) atthe onset of photolysis. Further, at times of day whenthe percentage distributions show maximum variation,the distributions are skewed on the higher end. Thisskewness is generally associated with sampling pointswhere NO

xconcentrations are relatively small (not

shown). For all solvers, the 75th percentile for O3

iswithin 7% of the exact solution, for PAN it is within35% of the exact solution, for OH it is within 20%, andfor H

2O

2it is within 5%. Much larger differences

are observed for NO and NO2. It may also be noted

that in these plots, large percentage differences aregenerally associated with sampling points where con-centrations are relatively low. An immediate con-clusion from this cursory analysis is that one cannotnecessarily expect the same level of accuracy whentolerance parameters diagnosed from a small set ofbox-model calculations are used in a more realistic3-D model.

3.2.2. ¹ime-step selection. The choice of time-stepselection scheme greatly affects the accuracy of anychemical integration methodology. Although fixedtime steps have been used with some of these schemes(or variants thereof ), variable time-step control ishighly desirable from a computational perspectivegiven spatial and temporal variation in the stiffness ofatmospheric chemistry. To further explore this issue,we examined the sensitivity of the solution proceduresto time-step control. For brevity, we present resultsfrom only the MQSSA experiments, since this solver’saccuracy for the given set of parameters was thelowest; the general conclusions also apply to theMY&B solver. Here, time-step control is maintainedthrough equation (13). Further, only those speciesthat have concentrations greater than factor]CK ,where CK "C

NO#C

NOÈ#C

OÊ, are considered in the

time-step determination. The procedure can be re-garded as a formal error-control methodology. In factone could view the parameter factor as an absolutetolerance, and e clearly is a relative tolerance. Weperformed additional simulations wherein the factorwas reduced to 0.001 from the value of 0.01 used in theprevious simulations. Figure 4 summarizes the per-centage difference distributions for selected species forthis case. Comparison of Figs 1 and 4 clearly indicatesthe impact of the size of factor on the accuracy of theresults. In general, the 75th and 95th percentiles of thepercentage difference distributions are reduced as theerror tolerance is tightened, due to more accuratepredictions at lower concentration ranges. Addition-ally, reducing the absolute tolerance in the time-stepdetermination procedure results in consideration ofcharacteristic time scales of faster-reacting species,resulting in a substantially more accurate solution.The impact of tightening the tolerance on computa-tional demand is discussed later.

3.2.3. Species lumping. A variety of conceptualmethods that center around the concept of ‘‘lumped’’species have also been suggested as means of improv-

ing accuracy of various chemical integration schemes(e.g. Hesstvedt et al., 1978; Sillman, 1991; Hertel et al.,1993). Lumped entities essentially represent linearcombinations of individual chemical species thateither exhibit strong chemical coupling (e.g. odd hy-drogen, odd nitrogen, odd oxygen) or as a group areconservative (e.g. total nitrogen). Lumped-species ap-proaches contribute to improved solution accuracy intwo ways: (1) by combining species with strong chem-ical coupling involving rapid interconversions, theyhelp reduce the stiffness of the overall chemical sys-tem; and (2) they help reduce any chemical massimbalance in the solution by ensuring conservation ofa group of species. While such detailed conceptuallumping methods can be built to exploit the featuresof individual chemical mechanisms (e.g. Chang et al.,1987; Sillman, 1991; Hertel et al., 1993), certain gen-eral lumped entities based on the common set ofinorganic chemical reaction can still be introduced ina generalized framework applicable to different chem-ical mechanisms. In this study we have focused on thelatter.

We considered the following lumped entities:

[NOx]"[NO]#[NO

2] (27a)

[HOx]"[OH]#[HO

2] (27b)

[PAO3]"[PAN]#[ACO

3] (27c)

[N2N

5]"[N

2O

5]#[NO

3] (27d)

[NOy]"[HNO

3]#[HNO

4]#[HNO

2]

#2[N2O

5]#[NO

3]#[NO]#[NO

2]

#[OLN]#[ONIT]#[PAN]#[TPAN].

(27e)

In the above expressions, ACO3

represents acetyl-peroxy radical, OLN represents a peroxy radical in-volving nitrogen, and TPAN and ONIT are organicnitrogen species (Stockwell et al., 1990). Following thesolution of each of the lumped entities, the concentra-tions of HO

2and PAN were rediagnosed directly

from the lumped group using equations (27b) and(27c), respectively. The concentrations of NO andNO

2, however, were readjusted using their molar

ratios (based on the unlumped solution) and the com-puted NO

xconcentration, and a similar procedure

was adopted for N2O

5and NO

3; this procedure was

found suitable for avoiding unrealistic negative con-centrations. Finally, any changes in total moles ofnitrogen over the time step were accounted forthrough the NO

ylump. Figure 5 presents percentage

difference distributions for the MQSSA solver withthe additional lumping procedure described above;the adjustable parameters for this case were set to beidentical to the initial simulation (as in Fig. 1). Com-parison of Figs 1 and 5 clearly indicates the effect ofincorporating species lumping. Significant reductions


Fig. 4. Percentage difference percentile plots for 3-D test case with MQSSA (factor"0.001 and e"0.005).


Fig. 5. Percentage difference percentile plots for 3-D test case with MQSSA and species lumping (fac-tor"0.01 and e"0.005).


Fig. 6. Time-series plots for selected species at spatial locations where maximum percentage differencesoccurred for the case with MQSSA (factor"0.01 and e"0.005).

in percentage differences of predicted concentration(compared to the exact solution) are apparent. Theaddition of the lumping procedure improves massconservation of the numerical integration scheme and

reduces the error of predictions when relatively re-laxed tolerance parameters are used. The impact ofintroducing species lumping on solution accuracy isfurther illustrated by Fig. 6, which presents time-series


Table 5. Percentage of sampling points over the 24 h simulation period with percentage difference less than 5%

MQSSA MY&BMQSSA MQSSA MQSSA MQSSA Lump MY&B Lump

Species( factor"

0.001,( factor"

0.001,( factor"

0.01,( factor"

0.01,( factor"

0.01,( factor"

0.01,( factor"

0.01, 1RADMname e"0.005) e"0.001) e"0.001) e"0.005) e"0.005) e"0.005) e"0.005) Solver

ACO3

97.270 99.900 86.630 69.710 64.710 80.230 65.540 89.270ALD 98.370 99.810 85.580 78.240 87.380 82.340 85.320 82.030CO 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00ETH 100.00 100.00 99.230 99.590 100.00 100.00 100.00 100.00H

2O

299.880 100.00 93.390 90.280 97.770 95.500 98.740 99.200

HCHO 99.440 99.970 88.040 64.180 76.300 75.730 81.110 69.030HNO

396.930 99.930 72.890 63.990 51.950 72.490 53.010 92.790

HNO4

95.710 99.570 80.610 65.070 68.040 70.930 69.120 81.910HO 97.690 99.870 83.110 63.630 71.640 78.130 74.800 96.240HO

278.300 93.620 63.420 50.410 64.790 59.560 65.850 63.780

HONO 93.360 99.990 82.090 66.550 70.360 67.470 69.160 86.930ISO 96.860 99.660 85.550 66.440 71.640 74.530 72.650 92.460N

2O

585.070 95.550 82.320 76.100 76.980 76.010 76.830 83.720

NO 84.230 96.720 70.550 47.020 51.630 52.480 50.650 82.910NO

274.420 93.750 52.710 20.120 30.070 32.850 28.500 72.900

NO3

80.380 94.480 76.320 68.080 71.390 68.600 71.560 86.070O

399.850 99.990 89.170 89.500 97.410 98.360 98.450 98.130

OLN 90.060 96.420 82.770 72.070 71.970 75.030 72.170 72.790ONIT 91.860 99.860 73.790 36.540 39.800 54.800 41.070 40.940PAN 87.280 97.210 64.130 45.680 56.440 56.340 55.750 84.700SO

2100.00 100.00 96.080 97.140 100.00 100.00 100.00 100.00

SULF 99.940 100.00 98.400 95.150 99.050 99.150 99.670 66.150TOL 99.850 100.00 89.860 81.800 88.980 93.530 93.900 99.420TPAN 98.980 100.00 80.590 72.370 67.900 79.220 68.470 96.670XYL 99.500 100.00 91.180 76.980 83.870 86.030 86.960 98.530

comparisons for selected species at grid points wherethe maximum percentage error was observed for theselected species in the MQSSA solution. These figuresshow that these maximum differences generally occurat points where the individual species concentrationsare relatively small compared to their typical ambientlevels, as mentioned earlier. Further, the lumping pro-cedure enables more accurate resolution of the tem-poral gradients, thereby reducing the differencescompared to the exact solution.

3.2.4. Summary. Table 5 presents another measureof the accuracy of the various solvers for varyingadjustable tolerance parameters. In this table, wetabulate the percentage of sampling points, includingall spatial nodes over the 24 h simulation, that hada percentage difference less than 5% compared to theexact solution. Again it can be observed that theglobal accuracy of the MQSSA solver increases as thetolerance parameters are tightened. The usefulness ofincluding lumping techniques is also apparent bycomparing the tabulated results for the MQSSA sol-ver with and without lumping for various species.Finally, we extend the concept of the SDA valuesdiscussed for the box-model calculations to providean additional assessment of global accuracy for thevarious cases as follows:

RMSmk"S

1

J

J+j/1

AcL mjk!cm

jkcL mjk

B2

(28)

and

SDA"!log10 A

1

n

1

M

n+k/1

M+

m/1

RMSmkB (29)

where cL mjk

and cmjk

represent the exact and computedsolutions for species k at spatial node j at the end ofhour m of the simulation. Table 6 quantifies the globalaccuracy for the various cases in these 3-D calcu-lations, and presents the computational effort in termsof the total CPU time for each case. As expected, moreaccurate solutions can be obtained at the expense ofcomputational effort. More importantly, a robusttime-step determination scheme can result in moreaccurate and efficient solutions (contrast the resultsfor the various MQSSA cases). Comparisons of theaverage SDA values and CPU times for both theMQSSA and MY&B solvers, with and without spe-cies lumping, indicate that species lumping can pro-vide substantial solution accuracies at the cost ofrelatively modest additional computational effort.Finally, comparisons of SDA values and CPU timesfor SMVGEAR with varying tolerances, indicate thatsignificant gains in computational effort can beachieved through relaxation of the tolerance levels.For instance, setting ATOL"10~3 and RTOL"

10~3 resulted in substantial computational improve-ments and makes SMVGEAR’s overall performancemore comparable with that of explicit schemes.This suggests that with some experimentation and


Table 6. Average SDA values and CPU times for 3-Dcalculations

Solver Average CPU! timeSDA (s)

MQSSA( factor"0.001, e"0.001) 4.92 1900MQSSA( factor"0.001, e"0.005) 3.46 700MQSSA( factor"0.01, e"0.005) 0.86 370MQSSA Lump( factor"0.01, e"0.005) 1.95 400MY&B( factor"0.01, e"0.005) 1.75 358MY&B Lump( factor"0.01, e"0.005) 1.94 388RADM 2.02 700SMVGEAR(ATOL"10~3, RTOL"10~3) 3.06 528SMVGEAR(ATOL"10~9, RTOL"10~2) 6.10 1000SMVGEAR"

(ATOL"10~9, RTOL"10~3) — 1180

! CPU times reported for a CRAY C90."Base solution considered as ‘‘exact’’ solution.

judicious specification of tolerance levels, especiallythe absolute tolerance, a Gear-type solver can alsoprovide a computationally efficient solution with rela-tively good accuracy. This result is consistent with thefindings of Saylor and Ford (1995).

4. SUMMARY AND CONCLUSIONS

Even with increasing computational power, there isa need for accurate and efficient chemical integrators,since the comprehensive modeling systems need toaddress increasingly complex chemistry issues emerg-ing from more detailed and complex model applica-tions. We have investigated four solution methods inthis study for a variety of test problems including botha box-model environment and in situ within a com-prehensive 3-D modeling system. While the box-model approach has been commonly used in the pastin various comparative studies, the conclusions fromsuch investigations were limited in that they wereeither based on idealized conditions or for a limitedset of chemical conditions. Designing such experi-ments to encompass the wide range of chemical condi-tions that occur in the real atmosphere is a highlychallenging if not an impossible task.

This study extends previous investigations by ex-amining in detail the performance of various chemicalintegration schemes within the actual environment ofa 3-D modeling system. To achieve this, we havedesigned a modular 3-D test bed modeling systeminto which various solution methodologies can beincorporated with relative ease, and have analyzed the

performance of these solution methods by examiningvarious local and global accuracy measures. For thevarious integration methods considered in this study,accuracy levels obtained for both box-model and de-tailed 3-D calculations are examined and compared.These comparisons indicate that an optimal set ofsolution method tolerance-related parameters diag-nosed based on limited set of conditions may notnecessarily result in comparable accuracy levels whenused to simulate the more varying conditions of a 3-Dmodeling environment. Further, our results indicatethat significant improvements in the accuracy of theexplicit schemes considered in this study can beachieved by the design of a more robust time-stepdetermination procedure. Additionally, the solutionaccuracy can be improved by adding a limited set oflumped species. This procedure can be implementedin a manner that maintains the generalized nature ofthe solution method and incurs only a modest in-crease in computational effort. Our analysis also indi-cate that with some experimentation and judiciouserror tolerance level specification, implicit Gear-typesolution methods can also yield accurate and com-putationally efficient solutions.

Finally, as also indicated by several comparativestudies in the past, a tradeoff exists between the accu-racy and computational efficiency of ODE solvers foratmospheric chemistry problems, and this must beconsidered in comparing the integration schemes. Theexamination of these tradeoffs through detailed 3-Dcalculations such as those presented in this study isa step toward addressing the more challenging issue ofwhat level of accuracy is adequate in model calcu-lations. While a comprehensive assessment of thisissue is beyond the scope of this study, certain con-clusions can be arrived at based on our results. Sinceany solution method can be made more accurate atthe expense of additional computational effort, froma pragmatic perspective it appears that the desiredaccuracy is dependent on the intended use of themodel predictions. For instance, for all the cases con-sidered in this study we noticed that the domain-widepeak ozone concentration predicted by the varioussolvers with varying parameter values was within 5%of the exact prediction and occurred at the samespatial point of the modeling domain. While this maybe adequate for using the model in a screening mode,we caution that this measure indicates neither theglobal accuracy nor the global sensitivity of the finalsolution. On the other hand, our 3-D calculations fora 24 h period, for certain species, show growing errors.The impact of such errors on longer-term chemistrycalculations in regional/global simulations also needsto be considered in practical model applications.A final pragmatic consideration in this regard per-tains to the inherent uncertainties in chemical mecha-nism formulations and in the representations ofother physical processes in a 3-D modeling frame-work. Given the trade-off between solution effici-ency and accuracy, such inherent uncertainties must


be considered in setting accuracy requirements forchemistry solvers for use in practical applications.

Acknowledgments—The study described in this documenthas been funded in part by the U.S. Environmental Protec-tion Agency under cooperative agreement CR822066 toMCNC. R. Mathur also acknowledges partial supportthrough NASA grant NAGW-4701 during the conduct ofthis research. The document has been subject to the U.S.EPA review and approved for publication. Mention of tradenames or commercial products does not constitute endorse-ment or recommendation for use. The authors also gratefullyacknowledge several helpful discussions with Drs JonathanPleim and Francis Binkowski on the RADM solver, the helpof Dr Prasad Kasibhatla in developing the version of theRADM solver used in this study, and Ms. Jeanne Eichingerfor her editorial assistance.

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APPENDIX: HOx LUMPING IN RADM SOLVER

Consider the equation describing the rate of change of OHconcentrations due to chemical reactions:

d[OH]

dt"P

OH!¸

OH[OH] (A1)

Interconversions between OH and HO2

occur through thefollowing gas-phase reactions:

(R1) O3#HO

2P OH#2O

2(R2) HO

2#NO P NO

2#OH

(R3) OH#HO2P H

2O#O

2


In equation (A1),

¸OH

[OH]"¸@OH

#k3

[OH][HO2] (A2)

which may be rewritten as,

¸OH

[OH]"¸@OH

#k3

[OH] ([HOx]![OH]) (A3)

where ¸@OH

represents the sum of reaction rates for all reac-tions except (R3) that result in chemical loss of OH. The rateof production of OH can similarly be expressed as

POH

"P@OH

#Mk1

[O3]#k

2[NO]N([HO

x]![OH]) (A4)

where P@OH

represents all OH production terms except thoseinvolving HO

2as a reactant. Substituting equations (A3) and

(A4) into equation (A1) yields equation (21), where

A"k3

B"¸@OH

#k1[O

3]#k

2[NO]!k

3[HO

x] (A5)

C"P@OH

#k1[OH] [HO

x]#k

2[NO][HO

x].


Documents

A comparison of numerical techniques for solution of atmospheric kinetic equations