JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 106, NO. D17, PAGES 20,175-20,188, SEPTEMBER 16, 2001

On the performance of numerical solvers for a chemistry submodel in three-dimensional air quality models 1. Box model simulations

Ho-Chun Huangl and Julius S. Chang Atmospheric Sciences Research Center, University at Albany, State University of New York, Albany

Abstract. The performance of numerical techniques in solving differential equations of the gas phase chemistry submodel (i.e., the chemical solver) is one of the most important factors in determining the overall computational cost for a three-dimensional (3-D) Air Quality Model (AQM). The estimated performance of a chemical solver in an AQM is often obtained by using simple box model analysis. In the present work some essential characteristics of the computational environment of any AQM, the operator splitting technique, have been identified and shown that different evaluation procedures will result in different conclusions for the relative performances of chemical solvers. A new box model evaluation procedure incorporating the impact of operator splitting has been designed to better mimic the true performances of various chemical solvers. Among the chemical solvers tested, the Hertel solver has the best overall performance and is the most robust in dealing with diverse computational environments.

1. Introduction

A three-dimensional (3-D) air quality model (AQM) contains a set of mass conservation equations of chemical species. These equations must be solved numerically to study the evolution of chemical pollutants in the atmosphere. Besides transport and mixing, many physical and chemical processes that affect the atmospheric concentrations of chemical species are also included in the model formulation [Reynolds et al., 1973; McRae et al., 1982; Carmichael et al., 1986; Chang et al., 1987; Hovet al., 1989; Chang et al., 1991]. Because of the overall mathematical complexity and the differences among mathematical characteristics of individual physical and chemical processes, operator splitting methods are commonly used in these models, which approximates the entire system by a sequence of operators, each representing only one or a few closely related physical or chemical processes [Chang et al., 1991]. Almost all AQMs spend a considerable amount of their computational resources in solving the chemical kinetics equations describing the gas phase chemistry components [Dunker, 1986; Odman et al., 1992]. Because the gas phase chemistry submodel usually constitutes a separate operator in the 3-D AQMs, it is feasible to use less expensive box models other than the 3-D ones to test the performances of the selected numerical technique, i.e., the chemical solver, for solving the stiff ordinary equations that describe the gas phase chemistry. It is noted that all previous studies except one [Chock et al., 1994] on the performances of various numerical techniques using box models have treated the solving procedure of gas phase chemistry as a stand-alone model. In real applications of

1Now at Illinois State Water Survey, Champaign, Illinois, USA.

Copyright 2001 by the American Geophysical Union

Paper number 2000JD000121. 0148-0227/01/2000JD000121 $09.00

the 3-D AQMs with operator splitting, chemical solvers do not act as a stand-alone submodel. Several other operators must be applied between two calls to the chemical solvers. Therefore it is not clear that in a given 3-D AQM whether a particular chemical solver based on the stand-alone box model results is

actually optimal. In this study we have explored this issue and analyzed how the presence of other operators may affect the operational performances of chemical solvers and their accuracy.

Many numerical techniques have been developed to solve the stiff differential equations describing atmospheric gas phase chemical reactions. Among these techniques, the implicit methods are commonly used due to their intrinsic stability [Gear, 1971; Chang et al., 1974; Dekker and Verwer, 1984; Hairer and Warmer, 1991; Sandu et al., 1997]. However, such methods often require the inversion of large matrices or iterative solution of nonlinear equations that can be computationally expensive and difficult to implement in 3-D models. As an alternative, methods using a quasi-steady-state approximation (QSSA) which utilize heuristic approximations based on physical reasoning to reduce the stiffness of the mathematical system have been proposed [Hesstvedt et al., 1978; Vetwet and Loon, 1994; Sandu et al., 1997]. While this type of methods is widely used, it is less reliable as a general numerical solver both in numerical accuracy and in stability. A natural extension to this is the hybrid scheme where the stiff part of the system is solved with a more expensive scheme, while the rest of the system is solved using a more economical method [Young and Boris, 1977; Odman et al., 1992; Saylor and Ford, 1995; Gong and Cho, 1993; Sun et al., 1994; Chock et al., 1994]. In this type of schemes it is the iteration algorithm between these two sets which carries the heuristic burden.

Since these solvers utilize different numerical techniques in dealing with stiffness in the equation set, the changes in their performances caused by various model constraints, for example, the length of the integration step, are also expected to

20,175

20,176 HUANG AND CHANG: CHE•CAL SOLVERS EVALUATION USING BOX MODEL

be different. Most previous studies have not included the influences of these constraints in the operation of the full AQM. As a result, it is difficult, through literature review, to assess whether a particular chemical solver is cost-effective for a given application. Further, new studies are still being pursued where some critical constraints remain ignored.

Section 2 first describes a conceptual description for the operator splitting methods used in 3-D AQMs. Then, the nine chemical solvers selected for this study are summarized in section 3. The observational data and 3-D AQM data used in various numerical procedures and the search for the values of optimal control parameters for chemical solvers are described in section 4. Section 5 describes the comparison of chemical solvers using a hierarchy of box models to simulate and illustrate the role of other operators (submodels) of a 3-D model with operator splitting. Section 6 introduces a new modeling procedure for the box models which includes a more realistic emulation of the effect of operator splitting on chemical solvers. Finally, we summarize our findings.

2. Air Quality Model and the Operator Splitting Method

The atmospheric concentration of a given chemical species considered in a 3-D AQM is governed by atmospheric transport, source emissions, deposition and removal, and chemical transformation. The corresponding constituent mass conservation equation is

•)C = _V(•C)+ X7(KeX7C)+ Pchem _ Lche m + E Ot

OC

where C is the concentration of a chemical species; V is the three-dimensional velocity vector at each grid point in the model domain; K e is the eddy diffusivity used to parameterize the subgrid scale fluxes of trace species' Pchern and Lcher n are the production and loss rates due to chemical reactions, respectively; E is the source emission rate; (OC/•t)cloud is the change rate of the concentration of chemical species due to

cloud effects' and (•)C/•}t)dry is the change rate due to dry deposition.

For a typical Eulerian 3-D AQM, the model domain is divided into a number of grid volumes centered at (x i, yj, z/•) with (i, j, k) denoting the center of the grid volume and Cijkl representing the volume-averaged concentration of chemical species l for this grid. Each of the terms, or mathematical operators, on the fight-hand side of equation (1) is discretized over this grid system by appropriate discrete approximation. The resulting mathematical system is a matrix differential equation of the form

&C.l = m(C.l )C.l + E (2) Ot '

where Co t is the vector [C.. ] organized in some manner with 1jkl regard to the three coordinate indices i, j, and k. M(Co I ) is a matrix operator with matrix elements mijkl, each of which can be a nonlinear function of Co I . E is the emissions vector with mostly zeros as its elements except for those locations where the chemical species are emitted. It is important to note that because the fight-hand side of equation (1) is a sum of distinct

operators, the matrix operator M is also a sum of distinct matrix operators in the form of

M = radvectio n q-Kdiffusio n q- Ggaschem q- Ddryde p . (3)

The meanings of these operators are clear from the subscripts and henceforth used without the subscripts. If left-hand side of equation (2) is further approximated by time differencing, then we have

1

II cn+ln i)C. l n+• __ .l -C.I = (T + K + G + D)nC. I + E n (4) Ot At '

where Co• = Co l (nAt) with a selected time step At. Although this is only one of the simplest approximation possible, it suffices for the present purpose of introducing the concept of operator splitting technique and how it relates to the chemical solvers for a box model. Therefore the full system becomes

Col n+l = [I + At(T + K + G + D)]Col n + AtE, (5) where I is the identity matrix. For a suitably small At this equation can be approximated by

Col n+l = (I + atrX• + atx)(• + ato)(• + AtO)Col n + AtE. (6)

An expansion of the multiplicative terms will yield equation (5) plus a number of other terms each multiplied by at least (At) 2. The sum of these additional terms is the error of the approximation used and can be studied by using ever smaller At. The terms in equation (6) actually can be represented by a set of sequential operations in the manner

c.," : + ato)c., ,

Col n+b = (I + AtG)Col n+a,

n+c = (I + AtK)Col n+b , (7)

C n+a = (I + AtT)Col n+c ol

C n+l =C n+d ol ol + AtE.

One enormous gain in considering this set of operator equations is that each of them involves only one physical or chemical process, such as advection by winds or gas phase chemical reactions. While the vector Co I is of dimension (max i) x (max j)x (max k)x (max l), each stage of the above sequence involves only a limited set of indices. For example, because the coefficients of the advection and diffusion

operators are not functions of species index l, all steps involving operator T are identical. Therefore a do-loop on index I will take care of all chemical species. This follows for all the other sequential stages with only a change of appropriate indices.

Another major consequence of using the operator splitting method (equation (7)) is that since each stage in this procedure involves only one operator, one type of physical or chemical process, we can use a specific numerical technique tailored to the mathematical characteristics of the corresponding physical or chemical process. This approximation then defines the original discretization of equation (1) which was only conceptually defined. In short, each stage of this sequence can be solved with a different approximation with "best" representation of the process under consideration. In fact, at

HUANG AND CHANG: CHEMICAL SOLVERS EVALUATION USING BOX.MODEL 20,177

each stage a different temporary At* can be used and then repeated so long as at the end of the stage the total time passed is equal to the overall At chosen. For example, with this consideration, the actual operator splitting technique used in the SJVAQS/AUSPEX Regional Modeling Adaptation Project (SARMAP) Air Quality Model (SAQM) has the form

cn+l At ml At m2 At ol = I+•T I+ K I+ G ml m2 m3

( At n I + D C.i + AtE

m4

ß (8)

SJVAQS denotes the San Joaquin Valley Air Quality Study and AUSPEX denotes the Atmospheric Utilities Signatures, Predictions and Experiments; m i for i = 1 to 4 is the number of integrations repeated by an operator, and At/m i is the optimal time step for the numerical scheme implemented in that operator.

Because the advection and diffusion operators are multidimensional they can be further split into a number of one-dimensional operators. This just adds more terms to the above equation. It is clear that in an actual operation of a 3-D AQM the chemical solver representing the operator G will only be applied for a limited number of steps, then the results will be modified by a sequence of other operators. Only after that will the chemical solver be applied again. Therefore an integration of gas phase chemical reactions is always interrupted by other operations and simply cannot be done continuously. In a typical box model study, a chemical solver is never interrupted so that it can maintain solution accuracy and a good estimate of the changes for the next time step. This is the central critical difference between traditional box model analysis and actual 3-D model applications.

3. Selected Chemical Solvers for Evaluation

Nine solvers were selected because of their unique numerical algorithms. Among these nine solvers, three of them use the implicit method; two are derived on the basis of quasi-steady-state approximation (QSSA); and the rest use the hybrid method (Table 1).

Solvers using implicit methods are LSODE, VODE-FLC, and RADAU5. LSODE uses the implicit fixed-coefficient backward differentiation formula (BDF), which is a multistep, variable step, and variable order method [Hindmarsh, 1983].

This scheme has been widely used to obtain the "reference solution" in solving stiff ordinary differential equations. VODE-FLC uses an implicit fixed-leading-coefficient BDF method that is also a multistep, variable step, and variable order method [Byrne and Hindmarsh, 1975' Hindmarsh and Byrne, 1976, 1977' Brown et al., 1989]. It is theoretically more suitable than LSODE for problems with frequent changes in Jacobian array coefficients and time step size [Brown et al., 1989]. RADAU5 uses an implicit Runge-Kutta approach, which is a one step method [Hairer et al., 1993].

The solvers of both the Hesstvedt et al. [ 1978] and the State University of New York air quality model (SUNY/AQM) [Huang, 1999] which evolved from the SARMAP air quality model are based on QSSA, which is actually an extreme case of the hybrid method. However, the two QSSA solvers are different in many respects such as different treatments for HO x computation, a different combination of solution formulas (lumped species), and a different control of the time step.

A solver using the hybrid method is designed to solve different groups of species with different numerical algorithms. The fast-reacting species that cause the stiffness of the system are solved using either an implicit method [Gong and Cho, 1993; Sun et al., 1994; Chock et al., 1994] or a highly accurate explicit method [Young and Boris, 1977' Odman et al., 1992; Saylot and Ford, 1995]. The slowly varying species are often solved with a simple explicit method. The separation of species depends on the comparison between the length of integration and the lifetime of the species. As a different approach, the solver of Hertel et al. [1993] first identifies the strongly coupled species within a chemical system and solves them as a coupled system. The solutions of these coupled species are then used to solve the remaining species using the Euler backward method.

Computer codes of several solvers selected in this study, including LSODE, VODE-FLC, RADAU5, SUNY/AQM solver, and the IEH solver of Sun et al. [ 1994] and Chock et al. [ 1994], are directly obtained from their authors, while codes of the rest solvers are programed by us on the basis of relevant papers. For a general application to various chemical mechanisms, we have applied a modified procedure to the chemical solver of Hertel et al. [Huang, 1999]. The modified solver uses Newton-Raphson iteration routine to solve the solutions of two strongly coupled groups { 03, NO, NO 2, O lb, and O 3P} and {HO, HO 2, HONO, and HNO4}. All of the selected chemical solvers have been well documented in their

Table 1. General Features of Selected Chemical Solvers

Chemical Solver Implicit One Step Versus Versus

Explicit Multisteps

LSODE implicit multisteps SVODE implicit multisteps RADAU5 implicit one step Hesstvedt ed al. explicit one step SUNY/AQM mixed one step Odman et al. mixed one step Gong and Cho mixed one step Sun and Chock (IEH) mixed multisteps and

two steps Hertel et al. implicit one step

Order of Accuracy

variable order variable order

fifth order first order

second/first order second/first order

first order

variable and second order

first order

Fixed Time Step Versus

Variable Time Step

variable variable variable

fixed variable

variable semifixed variable

fixed

20,178 HUANG AND CHANG: CHEMIC• SOLVERS EVALUATION USING BOX MODEL

Table 2. Dry Deposition Velocities of Day and Night

Species Daytime (cm/s) Nighttime (cm/s)

SO 2 0.5 0.2 SULF 0.4 0.07

NO 2 0.35 0.06 NO 3.0x10 -9 6.6x10 -7

03 0.6 0.2 HNO 3 3.0 1.0 H202 0.7 0.3 ALD 0.2 1.1 x 10 -6 HCHO 0.4 0.04

papers. The reader is referred to the referenced papers for further detail.

4. Numerical Experiment Design

Two stations, Fresno downtown (FSD) and Hollister (HST) from the SARMAP simulation domain, were selected. FSD

represents the urban case where NOx emission strength was stronger, while HST represents the rural case where NOx emission was weaker, but isoprene emission was stronger. Two different chemical mechanisms, the Carbon Bond Mechanism IV (CBM4) [Gery et al., 1989] and the Statewide Air Pollution Research Center chemical mechanism (SAPRC) [Carter, 1990], were adopted in box model simulations. They are different both in number of species integrated (35 versus 51) and in number of reactions involved (83 versus 137), and the approach of species lumping.

The rates of emissions and photolysis for a 3 day box model simulation at a 5 min time interval have been taken from the

SAQM simulation of the SARMAP high-ozone episode August 3-6, 1990. The initial concentrations of species have been taken from observations at 0400 PST, August 4, 1990, at FSD and HST. To assess the impact results only from the operator splitting, the temperature and pressure for the box model simulation have been held fixed at 300K and one

atmosphere. Only day and night values of dry deposition velocity (Table 2) have been used in the box model simulations

in section 5. The time series of the data needed in section 6, the concentrations of chemical species at the beginning and the end of each gas phase chemical integration, have also been obtained from the SAQM simulation at 5 min intervals.

The reference solution for the comparison of each case was obtained by using LSODE with a relative error tolerance value (RTOL) of 1.0 x 10 -7 and an absolute error tolerance value (ATOL) of 1.0 x 10 -11. The Jacobian matrix has been updated at every step (msbp = 1). The solution accuracies in the following figures have been defined in the root-mean-square (RMS) of relative error between the reference solution and the modeled solutions during the day from 1100 to 1700 PST.

The computational performance of a chemical solver is often dependent on the selected values of the control parameters. To search for the best performance of each chemical solver for the comparison study, a sensitivity test prior to the box model simulation was performed to select optimal values of these control parameters. The optimal values of the control parameters were then carefully chosen for the best performance, i.e., higher solution accuracy with reasonably shorter computational time. With this additional selection process of optimal values for control parameters, intentional bias toward any of the selected solvers hopefully can be removed from the results presented in the following sections. Using the box model simulation with the procedure of solving gas phase chemistry, emissions, and dry deposition continuously and conditions at site FSD. The control parameter values of selected chemical solvers with the CBM4 and the

SAPRC chemical mechanisms are listed in Tables 3 and 4, respectively.

5. Comparison of Box Model Performance

The traditional box model evaluations of the performances of chemical solvers often use a set of initial conditions and run

for a period of hours or days. Figure 1 shows the results of a 3 day box model simulation using two sets of initial data from an urban station (FSD) and a rural station (HST). For simulation without source and sink terms other than gas phase reactions, the potential oxidation capacity is fixed, and the chemical system will take a few hours to reach a quasi-equilibrium stage. The period from the start of the integration to the state of quasi-equilibrium, which will be referred to as the transition

Table 3. Values of Control Parameters for Various Chemical Solvers in CBM4 Box Model Simulations

Chemical Solvers Parameters Setting

LSODE SVODE

RADAU5 Hesstvedt et al.

SUNY/AQM Odman et al.

Gong and Cho IEH

Hertel et al.

RTOL = lx10 -4, ATOL = lx10 -5, MSBP = 20 RTOL = lx10 -4, ATOL = lx10 -6, MSBP = 20, MSBJ = 20 RTOL = lx10 -5, ATOL = lx10 -6 At = 15 s, number of iterations = 3 rmi n = 3 s, EPS = 2%

error tolerance value = lx10 -3, r6min = 1 s, EPS - 3% TOLX = lx10 -4, TOLF = lx10' , Armi n -- 5 s, Arma x = 8 s RTOL = lx10 '2, ATOL = lx10 -6 TOLX = lx10 -4, TOLF = lx10 '6, At = 150 s

aRTOL is the relative error tolerance value, ATOL is the absolute error tolerance value, MSBP is the number of passes to update the Jacobian Matrix, MSBJ is the number of passes needed to update the J Matrix, rmi n is the value of the minimum time step allowed, EPS is the limit for changes of any concentration of chemical species in one time step, TOLX and TOLF are the convergence criteria for the Newton-Raphson iteration, and Armi n and ATma x are the smallest and largest time steps applied in two time zones during the simulation.

HUANG AND CHANG: CHE•CAL SOLVERS EVALUATION USING BOX MODEL 20,179

Table 4. Similar to Table 3 Except for SAPRC Box Model Simulations

Chemical Solvers Parameters Setting

LSODE SVODE RADAU5 Hesstvedt et al.

SUNY/AQM Odman et al.

Gong and Cho IEH Hertel et al.

RTOL = lx10 -2, ATOL = lx10 -6, MSBP = 20 RTOL = lx10 '2, ATOL = lx10 '6, MSBP = 20, MSBJ = 20 RTOL = lxl 0 '2, ATOL = lxl 0 -6 At = 20 s, number of iterations = 3 Train = 3 s, EPS = 1% error tolerance value = lx10 '3, T_min = 1 s, EPS = 2% TOLX = lx10 -3, TOLF = lx10 -6, Armi n = 7 s, Arma x '- 9 s RTOL = lx10 '2, ATOL = lx10 -6 TOLX = 1 x 10 -6, TOLF = 1 x 10 -7, At = 150 s

stage, requires much more computational time than any other later stage. On the other hand, the amount of time for each gas phase chemistry integration in an operational 3-D AQM is often determined by meteorological conditions, e.g., the advection time step, and is normally around a few minutes. Sun et al. [1994] and Chock et al. [1994] have both suggested that the comparison of performances of a chemical solver should always focus on the first few minutes from initiation, i.e., the transition stage. Therefore comparison between the performances of any two chemical solvers clearly should always be made within the transition stage. The inclusion of the result from the quasi-equilibrium stage will most likely smear the true performances of a chemical solver within a 3-D AQM. For a box model simulation associated with external source and/or sink, e.g., emissions and dry deposition, previous studies often did not stress the description on how the sources and sinks were implemented. Since a 3-D AQM is often implemented with an operator splitting method, the modeled concentrations of chemical species after current gas phase chemistry integration are modified by other physical processes. This indicates that gas phase chemistry integration for the next always starts with a set of initial conditions that is perturbed and away from the equilibrium stage, which most of the solvers

Ozone

2.0E+02

1.6E+02 -

•,• 1.2E+02

• 8.0E+01

4.0E+01

IIIIIIlllllllilllllllllllllllllllll'

CB4 '

0.0E+00 III I' I"l•l'l' I"'• I'''"' I =''"' I''l•t 0. 12. 24. 36. 48. 60. 72.

TIME (HRs)

Figure 1. Ozone simulation using a box model with initial conditions only at FSD (thick line) and HST (thin line).

can take advantage of. Therefore the implementation of the source and sink terms in the box models that differ from those

in the 3-D AQM will not reveal the true performances of a chemical solver.

5.1. Methodology

The difference in box model comparisons using various box model evaluation procedures was first illustrated by solving the following ordinary differential equation;

dc = (p _ LC)che m + Semiss _ VdC, (9) dt

where c is concentration of chemical species, P is the production term of gas phase photochemical reaction, L is the loss terms of gas phase photochemical reaction, Semis s is the source term due to emission, and V d is the dry deposition velocity.

Three box model evaluation procedures for solving equation (9) have been compared. (1) CON case: this procedure solves equation (9) in a continuous manner, which treats all terms on the right-hand side of the equation as one source function.

dc= (p _ LC)chem + $emiss - Vd c dt

= fchem + femiss + fdrydeposition -- ftotal, (10)

cf +1 = c? + ftotalAt ,

where At is the model time step, C? and C? +l are the concentrations of species i at time n(At) and n+l(At), respectively. (2) OPX case: the procedure treats dry deposition as the only operator splitting impact and solves chemistry and emission continuously. The integrated process has been implemented in a symmetric sequence:

c? +a = c? + fdrydeposition At/• 2,

c• +b - c• +a + (femission + fchemreaction )Ate2, (11)

C• +c ----C• +b + (femission + fchemreaction)At•2,

,,+• = c ? + C /Xt•2. ci + fdrydeposition

(3) OP case: similar to case OPX except that all chemical and physical processes in equation (9) have been treated in an operator splitting manner:

20,180 HUANG AND CHANG: CHE•C• SOLVERS EVALUATION USING BOX MODEL

14

•'12

•10 • 8 • 6 • 4

o

0 5 10 15 20

cPu (sec)

14-

25

•. 12 rolo a• 8 • 6

=o 4 < 2

'(b) SAPRC(FSD) -

'• .... I .... I .... I .... 0 5 10 15 20 25

cPu (sec)

14-

12 t (c)• •10

CBM4(HST)

0 ''1 .... I .... I 0 5 10 15' ' ' '2'0' "'25

CPU (sec) 14

•' 12 1 (d) SAPRC(HST) •1o 8

6

4

2

0 .... I .... I .... I .... 0 5 10 15 20 25

cPu (sec)

[] LSODE O VODE O RADAU5

ß x Hesstvedt ffi SUNY AQM ß Odman

ß Gong&Cho ? IEH 111 Hertel

Figure 2. Performance charts of daytime ozone simulations using gas phase chemistry only: (a,b) results at FSD using CBM4 and SAPRC chemical mechanism, respectively; (c,d) results at HST using CBM4 and SAPRC chemical mechanism, respectively. The vertical axis shows solution accuracy in root-mean-square (RMS) of relative error in percent. The horizontal axis is the computational time, on a SUN ES/6000, with a single processor in seconds.

C• +a = C? + fdrydeposition Ate2,

c• +b = c• +a + femission Ate2,

c• +c = c• +b + fchemreaction Ate2,

c• +d = c• +c + fchemreaction Ate2,

c• +e = c• +d q- femission Ate2,

c• +1 = c• +e q- fdrydeposition Ate2'

(12)

5.2. Result and discussion

5.2.1. CBM4 simulations. Figures 2a and 2c show the performances of chemical solvers in box model runs, including gas phase chemistry integration only at FSD and HST. Figure 3 shows the result of cases CON, OPX, and OP at FSD and HST. The results from Figures 2a and 2c show that the fixed time step structure may not be able to take advantage of the situation where a larger time step is allowed during the equilibrium stage for achieving the same accuracy. This result implies that in a very clean environment, such as the upper atmosphere where nearly no emission and transport of ozone precursors are detected, a solver using a variable time step may have a considerable gain in performances. The results of Figure 3 clearly show that different evaluation procedures implemented into a box model can produce different conclusions about the

relative performances of chemical solvers. The performances of chemical solvers decrease when an operator splitting impact is presented. Figure 1 shows that the integration including only gas phase chemistry leads to an equilibrium state. However, the modification of solutions exited from gas phase chemistry integration due to dry deposition and/or emission actually keeps the simulated chemical species within the transition stage. Therefore instead of speeding up the integration with a set of solutions closer to the equilibrium stage, it often requires the chemical solver to spend more computational time, either with smaller time steps or more iteration steps, to obtain the solution for each new gas phase chemical integration. It is important to recognize that the operator splitting plays a role in shifting the state of chemical species away from the equilibrium stage and hence produces a different scenario from a traditional box model evaluation. Although some studies emphasize the restarting cost of a chemical solver as an influences on the performances of a chemical solver [Sandu et al., 1997], it is necessary for the chemical solver to restart the procedure since the solution has jumped to a different stage due to the impact of operator splitting and cannot be solved continuously.

The Gear solvers, LSODE and VODE, with lesser constraints actually have performed very well in case CON, but their performances quickly degrade with increase of operator splitting impact in case OPX and OP. One may choose a Gear solver solely based on the result of a box model simulation using an evaluation procedure of case CON, when in fact, it is not suitable for implementation in a 3-D AQM based on the

HUANG AND CHANG: CHE•C• SOLVERS EVALUATION USING BOX MODEL 20,181

10-

o 2 <

10-

10-

o 2 <

(a) CON(FSD)

0 5 10 15 20 25

CPU (sec)

(b) OPX(FSD)

0 5 10 15 20 25

CPU (sec)

(c) OP(FSD)

av 9. o.o, , 0 5 10 15 20 25

CPU (sec)

10-

10-

10-

o 2 <

(d) CON(HST)

0 5 10 15 20 25

CPU (sec)

(e) OPX(HST)

V

0 5 10 15 20 25

cPu (sec)

(f) OP(HST)

A • []

.... I .... I .... I .... I .... 1

0 5 10 15 20 25

cPu (sec)

[] LSODE O VODE 0 RADAU5

/x Hesstvedt ß SUNY AQM ß Odman

ß Gong&Cho V IEH Ill Hertel

Figure 3. Performance charts of daytime ozone simulations using CBM4 chemical mechanism: (left (a-c)) results using three box model evaluation procedures at FSD; (right (d-f)) results using three box model evaluation procedures at HST. The evaluation procedure used is labeled at the top right comer of each panel.

results of cases OPX and OP. In addition, Gear solvers also

require a large memory allocation for storage. The performances of solver RADAU5 is similar to the Gear solvers except for that the difference in performances between cases OPX and OP (4-8 times) has been larger than that of the Gear solver (1.5-2.5 times). This implies that different solvers due to their unique numerical techniques have various degrees of response to the magnitude of the operator splitting impact. Therefore a question arose. What is the proper magnitude of the operator splitting impact to be added into a box model simulation? This issue will be explored further in section 6. The results of cases CON, OPX, and OP also suggest that it is better to solve chemistry and emission together to reduce the magnitude of the operator splitting impact, since emission often has a huge operator splitting impact on the solution at each time step. The results for both stations show that two hybrid solvers, Hertel et al. and IEH, have consistently better performances throughout. Although the QSSA solvers had worse performances than three fully implicit solvers and a hybrid solver of Odman et al. in the traditional evaluation

procedures (case CON), they become a substitute choice for 3-D AQM applications where the operator splitting impact is imbedded (cases OPX and OP).

5.2.2. SAPRC simulations. Figures 2b and 2d show the performance plots of chemical solvers solving only the gas phase chemistry at FSD and HST. Figure 4 shows the results of cases CON, OPX, and OP at FSD and HST. With more species and reactions than the CBM4 chemical mechanism, the

increase of computational time for chemical solvers is different because of different numerical techniques used. In general, the comparison results for chemical solvers in SAPRC simulations are in a good agreement with that of CBM4 simulations. The distinction of performances among selected chemical solvers is clearer than in the CBM4 runs.

One factor in evaluating the performances of a chemical solver is the change of performance caused by the variation in the complexity of the chemical mechanism. The computational time generally increases with a more complex chemical mechanism. The magnitude of increase depends on the numerical technique of a chemical solver, e.g., functional

20,182 HUANG AND CHANG: CI-[E•CAL SOLVERS EVALUATION USING BOX MODEL

10- 10-

10-

(a) CON(FSD)

O= '' ' ' I' '"i"' I .... I .... 0 5 10 15 20 25 0

CPU (sec)

10- OPX(FSD)

v•

.... I .... I .... I .... I ....

0 5 10 15 20 25

cPu (sec) ,

(c) OP(FSD)

A

.... I .... I .... I .... I ....

0 5 10 15 20 25

(d) CON(HST)

5 10 15 20 25

CPU (sec)

10-

0

0

(e) OPX(HST)

0 5 10 15 20 25

CPU (sec) ,

(f) OP(HST)

.... I .... I''["•'1 ...... 5 10 15 20 25

CPU (sec) CPU (sec)

13 LSODE 0 VODE o RADAU5

A Hesstvedt IB SUNY AQM ß Odman

ß Gong&Cho V IEH Ill Hertel

Figure 4. Similar to Figures 3a-3f except for using the SAPRC chemical mechanism.

evaluation versus matrix computation. The degree of increase in computational time ranges from an increase proportional to the difference in the number of species to an increase proportional to the power of the total number of predicted species. For example, most computation is spent on solving the grouped species for the chemical solver of Hertel et al. Since the number of grouped species remains unchanged among various chemical mechanisms, the variation of computational time is mainly due to the variation of functional evaluation. Therefore the impact of the changes of the complexity in chemical mechanism on the chemical solver of Hertel et al.

may be minimized (-50% increase). On the other hand, for a chemical solver that contains matrix computation such as LSODE, the degree of CPU time variation is much lager due to the changes in the complexity of the chemical mechanism (- factor of 2.5 increase).

Figure 5 shows the factor of increase in CPU time from simulations using the CBM4 chemical mechanism to simulations using the SAPRC mechanism at FSD for cases OPX and OP, while Figure 6 shows the results at HST. With a 46% increase in integrated species and a 65% increase in

chemical reactions, LSODE has the greatest increase in CPU time by a factor of 2.5 to 3.0, while VODE has an increase of twofold. The SUNY/AQM solver shows varied degrees of CPU time increase, between a factor of 1.5 to 3, dependent upon the cases solved. The solver of Odman et al. shows about a factor of 2 increase in CPU time. The RADAU5 solver

experiences the least impact, except for case OPX at HST. The rest of the solvers have a 50% increase in CPU time with the

increase of complexity in the chemical mechanism.

6. Performances of Numerical Techniques Under Operator Splitting

The results of section 5 show that the magnitude of operator splitting impact results in different degrees of change in the performances of different solvers. Therefore the introduction of the strength of operator splitting impact into a box model evaluation becomes an important issue. Sandu et al. [1997] used emission as the only operator splitting impact for a 1 hour simulation, which may be too large, based on the results in section 5. Sun et al. [1994] and Chock et al. [1994] both

HUANG AND CHANG: CHE•C• SOLVERS EVALUATION USING BOX MODEL 20,183

5 I

• 2

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o

LSODE

5

4

• 2

1

0

5

4

• 2

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VODE

I RADAU5

5

4-

SUNY AQM Hesstvedt Odman

5

4

Gong & Cho IEH

5

4

m 2

1

0

Hertel

[-'] case OPX I• case OP Figure 5. Variation of CPU time due to the changes of the complexity of chemical mechanism at FSD. The Y axis is the ratio of CPU time using SAPRC chemical mechanism over that using CBM4 chemical mechanism. Each panel shows the data of CPU time increase of cases OPX and OP using a chemical solver labeled at the bottom of the panel.

considered the different system states encountered by the chemical solvers as the inclusion of the contributions from

other processes in full model simulations. However, it is difficult to determine if the initial conditions used in their

simulation actually reflect the realistic conditions in 3-D AQMs. Therefore a new evaluation procedure using the box models has been developed in this study to properly introduce the operator splitting impact based on 3-D AQM simulations.

6.1. Methodology

As described in section 2, in a 3-D AQM simulation, the concentration of any simulated species is modified by other processes such as transport between two gas phase chemistry integrations. Therefore if the chemical solver implemented produces any error, it should propagate through other physical and chemical processes and return to the next gas phase chemistry module. With the nonlinear nature of each process the solution passed through, it is difficult to assess the absolute magnitude of changes due to the error generated. Sun et al. [1994] and Chock et al. [1994] has introduced the operator splitting impact by updating the initial conditions with the combination of the maxima and minima of concentrations of

chemical species at each time step. However, the results simply cannot reflect the proper magnitude of solution modification. As an alternative, the substitution of initial conditions at the

beginning of every chemical integration with the concentrations of species from 3-D AQM runs may be more reasonable to introduce the absolute magnitude of changes due to operator splitting. However, if the chemical solver implemented in the box model is different from the chemical solver used in the 3-D AQM, the use of the concentrations of chemical species from 3-D AQM as the initial conditions also may not correctly introduce operator splitting impact. To avoid dependency on the 3-D AQM using different chemical solvers in selecting operator splitting impact, a perturbation factor is used, which is related to the relative changes of concentration of chemical species due to modification by processes other than gas phase chemistry.

At the beginning of each time step, the concentration of chemical species is modified by the amount of perturbation computed on the basis of the scale of relative change and the concentration of chemical species from the previous time step;

C[t+l=c[t+[i'inlmodel i'øutlmode ?) C n i,out I model

c.n.+ll , (13) _ '""in Imodel (C?) -cin, out Imodel

20,184 HUANG AND CHANG: CHEMICAL SOLVERS EVALUATION USING BOX MODEL

m 2

1

o

LSODE

0

VODE

1

0

RADAU5

5

4

Hesstvedt SUNY AQM

5

4

m 2

1

0

5

4-

Gong & Cho IEH Hertel

case OPX I• case OP

Figure 6. Similar to Figure 5 except at HST.

•.n.+l] perturbation factor = -•t,,n Imodel

C n ' /,out Imodel (14) F,n+l

a where '•i, out] model is the 3-D model concentration Onf+l species i at the end of chemical integration at time n(At), Ci, in Imodel is the 3-D model concentration of species i at the beginning of chemical integration at time n+l (At), C? is the concentration of species i in the box model computation at time n(At), and C• +l is the concentration of species i in the box model computation at time n+l(At). Often, the unrealistically large number for the perturbation factor could be generated due to near-zero values of concentration from 3-D AQM runs, e.g., the depletion of nitric oxide because of ozone at night. A cap may be applied to limit the values of the computed perturbation factor. The cap used in this study is two:

cn+l i,in Imodel perturbation factor = minimu 2., C n . /,out Imodel (15)

The time series of the computed perturbation factors can be obtained using 3-D model concentrations of chemical species. This time series represents the approximation of the changes at a station with a similar nature, e.g., urban or rural conditions, and can be generally applied to any box model evaluation with different solvers. These relative changes of the concentration of chemical species actually come from model processes other than gas phase chemistry; therefore they do not vary according

to different chemical solver. It is noted that when different

models implement different numerical schemes in solving other physical processes, e.g., advection, the magnitude of the time series of the perturbation factor may be different. Nevertheless, the inclusion of the perturbation factor in a box model evaluation procedure provides a much more realistic scenario for examining the performances of a chemical solver.

From the results of section 5 it is better to solve for emission

along with gas phase chemistry to improve the performances of a chemical solver. Therefore 3 day box model simulations with a time step of 5 min which solve gas phase chemistry and emission together were performed. To compare the new box model evaluation procedure with the method used in Chock et al. [1994], two box model evaluation procedures have been performed to introduce operator splitting impact thus to reveal the importance of solution accuracy propagation. One simulation has been performed by substituting the initial conditions directly with the concentration of chemical species obtained from the 3-D AQM run at each chemical integration, referred to as case NPR. The other simulation has been

conducted by modifying the solution of concentrations of chemical species from the previous time step with computed perturbation factors as the initial conditions of the current time step, referred to as case WPR.

6.2. Result and Discussion

Figures 7a and 7c show the results of case NPR using the CBM4 mechanism at FSD and HST, respectively. Figures 7b and 7d show the results of case WPR using the CBM4

HUANG AND CHANG: CI-{E•C• SOLVERS EVALUATION USING BOX MODEL 20,185

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< 2 0

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0 5 10 15 20 25

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.

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[] v

'•'1' I .... I .... I .... 5 10 15 20 25

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10-

0 5 10 15 20 25

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• 2 <

(d) WPR(HST)

[]

0 5 10 15 20 25

CPU (sec) CPU (sec)

[] LSODE O VODE o RADAU5

A Hesstvedt t•! SUNY AQM ß Odman

ß Gong&Cho V IEH 111 Hertel

Figure 7. Performance charts of daytime ozone simulations using CBM4 chemical mechanism: (left (a,b)) results of using two box model evaluation procedures at FSD; (right (c,d)) results of using two box model evaluation procedures at HST. The evaluation procedure used is labeled at the top right comer of each panel.

10-

<

(a) NPR(FSD)

•Y-/•...,,. ,;1171-•0v .... 0 5 10 15 20 25

• 8 (b) a• 6

E 4-

• 2- <

0

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WPR(FSD)

lily

A []

ß .. q .... ,•.'• , .... , .... 5 10 15 20 25

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10-

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0 5 10 15 20 25

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(d) WPR(HST)

..... ! .... I .... I ....

0 5 10 15 20 25

CPU (sec) CPU (sec)

[] LSODE O VODE o RADAU5

A Hesstvedt t•! SUNY AQM ß Odman

ß Gong&Cho V IEH •1 Hertel

Figure 8. Similar to Figures 7a-7d except for using the SAPRC chemical mechanism.

20,186 HUANG AND CHANG: CHE•C• SOLVERS EVALUATION USING BOX MODEL

5 I

• 2

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o

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Gong & Cho

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SUNY AQM

5 5

4 4

$ 3 $ 3

• 2 • 2

1 I l 0 0

IEH

5

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• 2

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o

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Odman

. Hertel

[-I caseNPR I• caseWPR

Figure 9. Variation of CPU time due to the changes of the complexity of chemical mechanism at FSD. The Y axis is the ratio of CPU time using SAPRC chemical mechanism over that using CBM4 chemical mechanism. Each panel shows the data of CPU time increase of cases NPR and WPR using a chemical solver labeled at the bottom of the panel.

mechanism at FSD and HST, respectively. Figures 8a and 8c show the results of case NPR using the SAPRC mechanism at FSD and HST, respectively. Figures 8b and 8d show the results of case WPR using the SAPRC mechanism at FSD and HST, respectively. In case NPR, CPU time required for the restarting procedure becomes the only factor in determining the performances of chemical solvers, since all solvers have produced accurate solutions over a 5 min integration interval with both mechanisms at both sites. By substituting a new set of initial conditions, the errors produced by different solvers actually have been ignored at every time step. One actually compensates for these errors artificially to produce the same set of concentrations of chemical species for all solvers encountered at the next integration step. However, the results for case WPR have shown that chemical solvers produced larger solution error when the propagation of error of the chemical solver is included in the evaluation procedure with the exception of solver RADAU5. This is because in a 3-D AQM run the modification of concentrations of chemical species due to other processes are either directly related to the absolute magnitude of the concentration of chemical species at the current grid cell, e.g., dry deposition, or depend on the relative magnitude of the concentration of chemical species among the current grid cell and neighboring grid cells, e.g., advection. Therefore the changes in performances depend on how well the given chemical solver handles the impact of operator splitting added to the system. Although case NPR is a good measure to test the restarting cost of chemical solvers, case WPR apparently produces a more realistic situation than

case NPR. The chemical solver of Hertel et at. apparently is the best solver in case NPR because it has the fastest time at both

stations with both chemical mechanisms. It also has a good performance in the case WPR with the exception of CBM4 simulations at FSD. Further simulations show that by reducing the time step for iterations the performances of the Hertel solver can be better than the other solvers. As in section 5, the results show that the chemical solvers 12EH, of Hesstvedt et at., and RADAU5 become more competitive in case WPR.

The increase in CPU time with the more complex chemical mechanism, S APRC, for cases NPR and WPR are also shown in Figures 9 and 10 at FSD and HST, respectively. Similar to the result in section 5, LSODE has greatest impact with a factor of 3 increase in CPU time, while VODE has about a factor of 2 increase. The RADAU5 solver also has a factor of 2 increase in

CPU time, except for case NPR at FSD. The SUNY/AQM solver has varied degrees of increase in CPU time ranging from a factor of 3 to 4. The solver of Odman et at. has an increase in

CPU time twofold to fourfold. The solver of Hertel et al. shows an increase in CPU time from a factor of 1.5 to 2. The rest of

the solvers have behaved the same as in the prior analysis.

7. Summary

The computational efficiency of a chemical submodel is crucial for the performances of a 3-D AQM. The computational speed, solution accuracy, and memory allocation are three major factors in evaluating the performances of a chemical solver. IN evaluating the

HUANG AND CHANG: CHE•C• SOLVERS EVALUATION USING BOX MODEL 20,187

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;u 2

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I-'] caseNPR [Y] caseWPR

Figure 10. Similar to Figure 9 except at HST.

performances of various selected chemical solvers using a box model, this study has demonstrated that the relative performances of chemical solvers can be changed by different box model evaluation procedures. To correctly evaluate the performances of a chemical solver, it is important to include the impact of operator splitting in the box model simulations. Our results also indicate that it may be more efficient to solve gas phase chemistry together with emission in box model simulations.

Since the inclusion of operator splitting impact creates a nonequilibrium state for modeled chemical species, the performances of a chemical solver in the transition period is more critical than that at the quasi-equilibrium stage. The comparison among the tested cases show that different chemical solvers have different sensitivities to the impact of operator splitting because of the different numerical techniques implemented. A new box model evaluation procedure using the perturbation factor has been developed to mimic the proper magnitude of operator splitting impact received. The perturbation factor reflects the relative change in concentrations of chemical species because of the modification of model physical and chemical processes other than gas phase chemistry. The use of the perturbation factor also removes the dependency on data obtained from any AQM. Because the new procedure also includes the error propagation of a chemical solver occurring in 3-D AQM simulations, our results have shown that it is more appropriate than the direct substitution with the initial conditions from a 3-D model simulation.

An optimal chemical solver should have the least impact from changes in complexity of chemical mechanisms. Our

results have shown that the computational time of model simulations increases with the increase of complexity of the model chemical mechanism. The magnitude of the increase depends on the numerical technique used with a chemical solver. Among the solvers tested, the solvers RADAU5, IEH, of Hesstvedt et al., Gong and Cho, and Hertel et al. have less impact than Gear solvers (50% versus a factor of 3 increase in computation time).

The chemical solver of Hertel et al. has the best overall

performances among those chemical solvers tested. Although the chemical solver IEH has also performed very well in most of the tests, the chemical solver of Hertel et al. may be more favorable because of its lower system memory requirement than the chemical solver IEH. As will be reported in a separate paper, we have applied the Hertel solver to a 3-D AQM, our results shown that the new box model evaluation procedure can indeed reflect the performance changes of different chemical solvers in 3-D models (a factor of 5 speedup in our evaluation procedure versus a factor of 2 speedup in real 3-D AQM applications). To search for the best performance of a chemical solver, it is necessary to apply the new box model evaluation procedure to reflect the realistic constraints in 3-D AQM simulations.

During the course of this study, newer versions of the fully implicit solvers have been devised with the added sparse matrix procedure to speed up the modified Newton iteration [Sandu et al., 1997]o It is an ongoing study to test the new versions of solvers against the current best one (the solver of Hertel et al. in this study) under the proposed new box model procedure in the future.

20,188 HUANG AND CHANG: CI-••C• SOLVERS EVALUATION USING BOX MODEL

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J. S. Chang, Atmospheric Sciences Research Center, University at Albany, State University of New York, 251 Fuller Road, Albany, NY 12203, USA.

H.-C. Huang, Illinois State Water Survey, 2204 Griffith Drive, Champaign, IL 61820-7495, USA. (huang2 @uiuc.edu)

(Received November 3,2000; revised April 10, 2001; accepted May 1, 2001.)