On the performance of numerical solvers for a chemistry submodel in three-dimensional air quality models

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 6, NO. D7, PAGES 2,7-2,88, SEPTEMBER 6, 2 On the performance of numerical solvers for a chemistry submodel in three-dimensional air quality models. 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.. Introduction the 3-D AQMs with operator splitting, chemical solvers do not act as a stand-alone submodel. Several other operators must be A three-dimensional (3-D) air quality model (AQM) applied between two calls to the chemical solvers. Therefore it contains a set of mass conservation equations of chemical is not clear that in a given 3-D AQM whether a particular species. These equations must be solved numerically to study chemical solver based on the stand-alone box model results is the evolution of chemical pollutants in the atmosphere. Besides actually optimal. In this study we have explored this issue and transport and mixing, many physical and chemical processes analyzed how the presence of other operators may affect the that affect the atmospheri concentrations of chemical species operational performances of chemical solvers and their are also included in the model formulation [Reynolds et al., accuracy. 973; McRae et al., 982; Carmichael et al., 986; Chang et Many numerical techniques have been developed to solve al., 987; Hovet al., 989; Chang et al., 99]. Because of the the stiff differential equations describing atmospheric gas overall mathematical complexity and the differences among phase chemical reactions. Among these techniques, the mathematical characteristics of individual physical and implicit methods are commonly used due to their intrinsic chemical processes, operator splitting methods are commonly stability [Gear, 97; Chang et al., 97; Dekker and Verwer, used in these models, which approximates the entire system by 98; Hairer and Warmer, 99; Sandu et al., 997]. However, a sequence of operators, each representing only one or a few such methods often require the inversion of large matrices or closely related physical or chemical processes [Chang et al., iterative solution of nonlinear equations that can be 99]. Almost all AQMs spend a considerable amount of their computationally expensive and difficult to implement in 3-D computational resources in solving the chemical kinetics models. As an alternative, methods using a quasi-steady-state equations describing the gas phase chemistry components approximation (QSSA) which utilize heuristic approximations [Dunker, 986; Odman et al., 992]. Because the gas phase based on physical reasoning to reduce the stiffness of the chemistry submodel usually constitutes a separate operator in mathematical system have been proposed [Hesstvedt et al., the 3-D AQMs, it is feasible to use less expensive box models 978; Vetwet and Loon, 99; Sandu et al., 997]. While this other than the 3-D ones to test the performances of the selected type of methods is widely used, it is less reliable as a general numerical technique, i.e., the chemical solver, for solving the numerical solver both in numerical accuracy and in stability. A stiff ordinary equations that describe the gas phase chemistry. natural extension to this is the hybrid scheme where the stiff It is noted that all previous studies except one [Chock et al., part of the system is solved with a more expensive scheme, 99] on the performances of various numerical techniques while the rest of the system is solved using a more economical using box models have treated the solving procedure of gas method [Young and Boris, 977; Odman et al., 992; Saylor phase chemistry as a stand-alone model. In real applications of and Ford, 99; Gong and Cho, 993; Sun et al., 99; Chock et al., 99]. In this type of schemes it is the iteration algorithm Now at Illinois State Water Survey, Champaign, Illinois, USA. between these two sets which carries the heuristic burden. Since these solvers utilize different numerical techniques in Copyright 2 by the American Geophysical Union dealing with stiffness in the equation set, the changes in their Paper number 2JD2. performances caused by various model constraints, for 8-227//2JD2 $9. example, the length of the integration step, are also expected to 2,7

2 2,76 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 operators, the matrix operator M is also a sum of distinct matrix operators in the form of M = radvectio n q-kdiffusio n 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 section 3. The observational data and 3-D AQM data used in various numerical procedures and the search for the values of II i)c. l n+ cn+ln.l -C.I = (T + K + G + D)nC. I + E n () Ot At ' optimal control parameters for chemical solvers are described in section. Section describes the comparison of chemical where Co = Co l (nat) with a selected time step At. Although solvers using a hierarchy of box models to simulate and this is only one of the simplest approximation possible, it illustrate the role of other operators (submodels) of a 3-D suffices for the present purpose of introducing the concept of model with operator splitting. Section 6 introduces a new operator splitting technique and how it relates to the chemical modeling procedure for the box models which includes a more solvers for a box model. Therefore the full system becomes realistic emulation of the effect of operator splitting on chemical solvers. Finally, we summarize our findings. Col n+l = [I + At(T + K + G + D)]Col n + AtE, () where I is the identity matrix. For a suitably small At this equation can be approximated by 2. Air Quality Model and the Operator Splitting Method The atmosphericoncentration 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 () 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 jkl 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 () is a sum of distinct ' Col n+l = (I + atrx + atx)( + ato)( + AtO)Col n + AtE. (6) An expansion of the multiplicative terms will yield equation () 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 C ol n+a = (I + AtT)Col n+c C n+l =C n+d ol ol + AtE., (7) 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 () 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

3 HUANG AND CHANG: CHEMICAL SOLVERS EVALUATION USING BOX.MODEL 2,77 each stage a different temporary At* can be used and then This scheme has been widely used to obtain the "reference repeated so long as at the end of the stage the total time passed solution" in solving stiff ordinary differential equations. is equal to the overall At chosen. For example, with this VODE-FLC uses an implicit fixed-leading-coefficient BDF consideration, the actual operator splitting technique used in method that is also a multistep, variable step, and variable the SJVAQS/AUSPEX Regional Modeling Adaptation Project order method [Byrne and Hindmarsh, 97' Hindmarsh and (SARMAP) Air Quality Model (SAQM) has the form Byrne, 976, 977' Brown et al., 989]. It is theoretically more suitable than LSODE for problems with frequent changes in cn+l ol = I+ T At ml I+ At K m2 I+ At Jacobian array coefficients and time step size [Brown et al., G ml m2 m3 989]. RADAU uses an implicit Runge-Kutta approach, ß (8) which is a one step method [Hairer et al., 993]. The solvers of both the Hesstvedt et al. [ 978] and the State University of New York air quality model (SUNY/AQM) [Huang, 999] 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 ( At n I + D C.i + AtE m SJVAQS denotes the San Joaquin Valley Air Quality Study and AUSPEX denotes the Atmospheric Utilities Signatures, Predictions and Experiments; m i for i = to 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 ). Solvers using implicit methods are LSODE, VODE-FLC, and RADAU. LSODE uses the implicit fixed-coefficient backward differentiation formula (BDF), which is a multistep, variable step, and variable order method [Hindmarsh, 983]. different in many respectsuch 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, 993; Sun et al., 99; Chock et al., 99] or a highly accurate explicit method [Young and Boris, 977' Odman et al., 992; Saylot and Ford, 99]. 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. [993] 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, RADAU, SUNY/AQM solver, and the IEH solver of Sun et al. [ 99] and Chock et al. [ 99], 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, 999]. The modified solver uses Newton-Raphson iteration routine to solve the solutions of two strongly coupled groups { 3, NO, NO 2, O lb, and O 3P} and {HO, HO 2, HONO, and HNO}. All of the selected chemical solvers have been well documented in their Table. General Features of Selected Chemical Solvers Chemical Solver Implicit Versus Explicit One Step Versus Multisteps Order of Accuracy Fixed Time Step Versus Variable Time Step LSODE implicit multisteps SVODE implicit multisteps RADAU 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 variable order variable order fifth order first order second/first order second/first order first order variable and second order first order variable variable variable fixed variable variable semifixed variable fixed

4 2,78 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..2 SULF..7 NO NO 3.x x HNO H ALD.2. x -6 HCHO.. papers. The reader is referred to the referenced papers for further detail.. 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 (CBM) [Gery et al., 989] and the Statewide Air Pollution Research Center chemical mechanism (SAPRC) [Carter, 99], were adopted in box model simulations. They are different both in number of species integrated (3 versus ) and in number of reactions involved (83 versus 37), and the approach of species lumping. The rates of emissions and photolysis for a 3 day box model simulation at a min time interval have been taken from the SAQM simulation of the SARMAP high-ozone episode August 3-6, 99. The initial concentrations of species have been taken from observations at PST, August, 99, 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 3K and one atmosphere. Only day and night values of dry deposition velocity (Table 2) have been used in the box model simulations in section. 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 min intervals. The reference solution for the comparison of each case was obtained by using LSODE with a relative error tolerance value (RTOL) of. x -7 and an absoluterror tolerance value (ATOL) of. x -. The Jacobian matrix has been updated at every step (msbp = ). 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 to 7 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 CBM and the SAPRC chemical mechanisms are listed in Tables 3 and, respectively.. 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 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 CBM Box Model Simulations Chemical Solvers Parameters Setting LSODE SVODE RADAU Hesstvedt et al. SUNY/AQM Odman et al. Gong and Cho IEH Hertel et al. RTOL = lx -, ATOL = lx -, MSBP = 2 RTOL = lx -, ATOL = lx -6, MSBP = 2, MSBJ = 2 RTOL = lx -, ATOL = lx -6 At = s, number of iterations = 3 rmi n = 3 s, EPS = 2% error tolerance value = lx -3, r6min = s, EPS - 3% TOLX = lx -, TOLF = lx', Armi n -- s, Arma x = 8 s RTOL = lx '2, ATOL = lx -6 TOLX = lx -, TOLF = lx '6, At = s artol is the relativ 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.

5 HUANG AND CHANG: CHE CAL SOLVERS EVALUATION USING BOX MODEL 2,79 Table. Similar to Table 3 Except for SAPRC Box Model Simulations Chemical Solvers LSODE SVODE RADAU Hesstvedt et al. SUNY/AQM Odman et al. Gong and Cho IEH Hertel et al. Parameters Setting RTOL = lx -2, ATOL = lx -6, MSBP = 2 RTOL = lx '2, ATOL = lx '6, MSBP = 2, MSBJ = 2 RTOL = lxl '2, ATOL = lxl -6 At = 2 s, number of iterations = 3 Train = 3 s, EPS = % error tolerance value = lx '3, T_min = s, EPS = 2% TOLX = lx -3, TOLF = lx -6, Armi n = 7 s, Arma x '- 9 s RTOL = lx '2, ATOL = lx -6 TOLX = x -6, TOLF = x -7, At = 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 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 often determined by meteorological conditions, e.g., the advection time step, and is normally around a few minutes. Sun chemical solver. et al. [99] and Chock et al. [99] have both suggested that.. Methodology the comparison of performances of a chemical solver should The difference in box model comparisons using various box always focus on the first few minutes from initiation, i.e., the model evaluation procedures was first illustrated by solving the transition stage. Therefore comparison between the following ordinary differential equation; performances of any two chemical solvers clearly should always be made within the transition stage. The inclusion of dc = (p_ LC)che m + Semiss _ VdC, the result from the quasi-equilibrium stage will most likely dt (9) smear the true performances of a chemical solver within a 3-D AQM. For a box model simulation associated with external where c is concentration of chemical species, P is the source and/or sink, e.g., emissions and dry deposition, previous production term of gas phase photochemical reaction, L is the studies often did not stress the description on how the sources and sinks were implemented. Since a 3-D AQM is often loss terms of gas phase photochemical reaction, Semis s is the source term due to emission, and V d is the dry deposition implemented with an operator splitting method, the modeled velocity. concentrations of chemical species after current gas phase Three box model evaluation procedures for solving equation chemistry integration are modified by other physical processes. (9) have been compared. () CON case: this procedure solves This indicates that gas phase chemistry integration for the next equation (9) in a continuous manner, which treats all terms on always starts with a set of initial conditions that is perturbed the right-hand side of the equation as one source function. and away from the equilibrium stage, which most of the solvers, 2.E+2.6E+2 -.2E+2 8.E+.E+ IIIIIIlllllllilllllllllllllllllllll' Ozone CB.E+ III I' I"l l'l' I"' I'''"' I =''"' I''l t TIME (HRs) Figure. Ozone simulation using a box model with initial conditions only at FSD (thick line) and HST (thin line). ' dc= (p_ LC)chem + $emiss - Vd c dt = fchem + femiss + fdrydeposition -- ftotal, () cf + = 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/, c +b - c +a + (femission + fchemreaction )Ate2, () C +c ----C +b + (femission + fchemreaction)at 2,,,+ = c? + C + fdrydeposition /Xt 2. ci (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:

6 - 2,8 HUANG AND CHANG: CHE C SOLVERS EVALUATION USING BOX MODEL ' t (c) CBM(HST) o -. 2 '(b) rolo a 8 6 =o < 2 2 cpu (sec) 2 SAPRC(FSD) '... I... I... I cpu (sec) ''... I... I ' ' ''2'' "'2 ' o (d) SAPRC(HST)... I... I... I cpu (sec) [] LSODE O VODE O RADAU ß x Hesstvedt ffi SUNY AQM ß Odman ß Gong&Cho? IEH Hertel Figure 2. Performance charts of daytime ozone simulations using gas phase chemistry only: (a,b) results at FSD using CBM and SAPRC chemical mechanism, respectively; (c,d) results at HST using CBM 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/6, with a single processor in seconds..2. Result and discussion 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 + = c +e q- fdrydeposition Ate2' (2).2.. CBM 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 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 importanto 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., 997], 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

7 HUANG AND CHANG: CHE C SOLVERS EVALUATION USING BOX MODEL 2,8 - (a) CON(FSD) - (d) CON(HST) o 2 < (b) OPX(FSD) (e) OPX(HST) V cpu (sec) - (c) OP(FSD) - (f) OP(HST) o 2 < av 9. o.o,, 2 2 o 2 < A []... I... I... I... I cpu (sec) [] LSODE O VODE RADAU /x Hesstvedt ß SUNY AQM ß Odman ß Gong&Cho V IEH Ill Hertel Figure 3. Performance charts of daytime ozone simulations using CBM 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. Th evaluation procedure used is labeled at the top right comer of each panel. procedures (case CON), they become a substitute choice for 3-D AQM applications where the operator splitting impact is performances of solver RADAU is similar to the Gear solvers imbedded (cases OPX and OP). except for that the difference in performances between cases.2.2. SAPRC simulations. Figures 2b and 2d show the OPX and OP (-8 times) has been larger than that of the Gear performance plots of chemical solver solving only the gas solver (.-2. times). This implies that different solvers due to phase chemistry at FSD and HST. Figure shows the results of their unique numerical techniques have various degrees of cases CON, OPX, and OP at FSD and HST. With more species response to the magnitude of the operator splitting impact. and reactions than the CBM chemical mechanism, the Therefore a question arose. What is the proper magnitude of increase of computational time for chemical solvers is different the operator splitting impacto be added into a box model because of different numerical techniques used. In general, the simulation? This issue will be explored further in section 6. comparison results for chemical solvers in SAPRC simulations The results of cases CON, OPX, and OP also suggesthat it is are in a good agreement with that of CBM simulations. The better to solve chemistry and emission together to reduce the distinction of performances among selected chemical solvers is magnitude of the operator splitting impact, since emission clearer than in the CBM runs. often has a huge operator splitting impact on the solution at One factor in evaluating the performances of a chemical each time step. The results for both station show that two solver is the change of performance caused by the variation in hybrid solvers, Hertel et al. and IEH, have consistently better the complexity of the chemical mechanism. The computational performances throughout. Although the QSSA solvers had time generally increases with a more complex chemical worse performances than three fully implicit solvers and a mechanism. The magnitude of increase depends on the hybrid solver of Odman et al. in the traditional evaluation numerical technique of a chemical solver, e.g., functional results of cases OPX and OP. In addition, Gear solvers also require a large memory allocation for storage. The

8 ,, 2,82 HUANG AND CHANG: CI-[E CAL SOLVERS EVALUATION USING BOX MODEL - - (a) CON(FSD) (d) CON(HST) O= '' ' ' I' '"i"' I... I OPX(FSD) - (e) 2 2 OPX(HST) v... I... I... I... I cpu (sec) - (c) OP(FSD) - (f) OP(HST) A... I... I... I... I I... I''[" ' LSODE VODE o RADAU A Hesstvedt IB SUNY AQM ß Odman ß Gong&Cho V IEH Ill Hertel Figure. 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 chemical reactions, LSODE has the greatest increase in CPU time by a factor of 2. to 3., while VODE has an increase of twofold. The SUNY/AQM solver shows varied degrees of proportional to the power of the total number of predicted CPU time increase, between a factor of. to 3, dependent species. For example, most computation is spent on solving the upon the cases solved. The solver of Odman et al. shows about grouped species for the chemical solver of Hertel et al. Since a factor of 2 increase in CPU time. The RADAU solver the number of grouped species remains unchanged among experiences the least impact, except for case OPX at HST. The various chemical mechanisms, the variation of computational rest of the solvers have a % increase in CPU time with the time is mainly due to the variation of functional evaluation. increase of complexity in the chemical mechanism. Therefore the impact of the changes of the complexity in chemical mechanism on the chemical solver of Hertel et al. 6. Performances of Numerical Techniques may be minimized (-% increase). On the other hand, for a Under Operator Splitting 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. increase). Figure shows the factor of increase in CPU time from simulations using the CBM chemical mechanism to The results of section 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. [997] simulations using the SAPRC mechanism at FSD for cases used emission as the only operator splitting impact for a hour OPX and OP, while Figure 6 shows the results at HST. With a 6% increase in integrated species and a 6% increase in simulation, which may be too large, based on the results in section. Sun et al. [99] and Chock et al. [99] both

9 HUANG AND CHANG: CHE C SOLVERS EVALUATION USING BOX MODEL 2,83 I o I LSODE VODE RADAU - Hesstvedt SUNY AQM Odman m 2 Gong & Cho IEH Hertel [-'] case OPX I case OP Figure. 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 CBM 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.. Methodology As described in section 2, in a 3-D AQM simulation, the concentration of any simulated species is modified by other processesuch 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. [99] and Chock et al. [99] 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 result 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?) i,out I model C n c.n.+ll, (3) _'""in Imodel (C?) -cin, out Imodel

10 2,8 HUANG AND CHANG: CHEMICAL SOLVERS EVALUATION USING BOX MODEL m 2 o LSODE VODE RADAU m 2 Hesstvedt SUNY AQM - Gong & Cho IEH Hertel case OPX I case OP Figure 6. Similar to Figure except at HST..n.+l] to different chemical solver. It is noted that when different perturbation factor = - t,,n Imodel models implement different numerical schemes in solving C n ' other physical processes, e.g., advection, the magnitude of the /,out Imodel () time series of the perturbation factor may be different. F,n+l Nevertheless, the inclusion of the perturbation factor in a box a where ' i, out] model is the 3-D model concentration Onf+l species model evaluation procedure provides a much more realistic i at the end of chemical integration time n(at), Ci, in Imodel is scenario for examining the performances of a chemical solver. the 3-D model concentration of species i at the beginning of From the results of section it is better to solve for emission chemical integration time n+l (At), C? is the concentration of species i in the box model computation at time n(at), and along with gas phase chemistry to improve the performances of C +l is the concentration of species i in the box model a chemical solver. Therefore 3 day box model simulations with computation at time n+l(at). Often, the unrealistically large a time step of min which solve gas phase chemistry and number for the perturbation factor could be generate due to emission together were performed. To compare the new box model evaluation procedure with the method used in Chock et near-zero values of concentration from 3-D AQM runs, e.g., al. [99], two box model evaluation procedures have been the depletion of nitric oxide because of ozone at night. A cap performed to introduce operator splitting impacthus to reveal may be applied to limit the values of the computed perturbation factor. The cap used in this study is two: the importance of solution accuracy propagation. One simulation has been performed by substituting the initial cn+l conditions directly with the concentration of chemical species i,in Imodel obtained from the 3-D AQM run at each chemical integration, perturbation factor = minimu 2., C /,out n Imodel. referred to as case NPR. The other simulation has been () conducted by modifying the solution of concentrations of The time series of the computed perturbation factors can be chemical species from the previous time step with computed obtained using 3-D model concentrations of chemical species. perturbation factors as the initial conditions of the current time This time series represents the approximation of the changes at step, referred to as case WPR. a station with a similar nature, e.g., urban or rural conditions, 6.2. Result and Discussion and can be generally applied to any box model evaluation with different solvers. These relative changes of the concentration Figures 7a and 7c show the results of case NPR using the of chemical species actually come from model processes other CBM mechanism at FSD and HST, respectively. Figures 7b than gas phase chemistry; therefore they do not vary according and 7d show the results of case WPR using the CBM

11 HUANG AND CHANG: CI-{E C SOLVERS EVALUATION USING BOX MODEL 2,8 - (a) NPR(FSD) - (c) NPR(HST) < (b) WPR(FSD) - (d) WPR(HST) v [] < []. ' '' I... I... I [] LSODE O VODE o RADAU A Hesstvedt t! SUNY AQM ß Odman ß Gong&Cho V IEH Hertel Figure 7. Performance charts of daytime ozone simulations using CBM 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. - (a) NPR(FSD) - (c) NPR(HST) < Y-/...,,.,;7- v cpu (sec) 8 (b) WPR(FSD) - (d) WPR(HST) a 6 E - - < lily A [] ß.. q...,.',..., !... I... I [] LSODE O VODE o RADAU A Hesstvedt t! SUNY AQM ß Odman ß Gong&Cho V IEH Hertel Figure 8. Similar to Figures 7a-7d except for using the SAPRC chemical mechanism.

12 2,86 HUANG AND CHANG: CHE C SOLVERS EVALUATION USING BOX MODEL I o LSODE VODE o RADAU Hesstvedt SUNY AQM Odman - Gong & Cho $ 3 $ 3 I l IEH. 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 CBM 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 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 RADAU. 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 CBM 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, the results show that the chemical solvers 2EH, of Hesstvedt et at., and RADAU 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 at FSD and HST, respectively. Similar to the result in section, LSODE has greatest impact with a factor of 3 increase in CPU time, while VODE has about a factor of 2 increase. The RADAU 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. 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. 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

13 HUANG AND CHANG: CHE C SOLVERS EVALUATION USING BOX MODEL 2,87 LSODE VODE I RADAU Hesstvedt o SUNY AQM Odman ;u 2 Gong & Cho IEH o Hertel I-'] casenpr [Y] casewpr Figure. 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 RADAU, IEH, of Hesstvedt et al., Gong and Cho, and Hertel et al. have less impact than Gear solvers (% 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 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., 997]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.

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