In Eqs. (38a-d), Pi represents the production of species i,

Limi represents the chemical loss of species i, υ

i,l is the

stoichiometric coefficient for species i in reaction l, and rl is

the rate of chemical reaction l. The sum l = 1...Ii , in Eq.

(38c) is over all reactions in which species i appears as a

product, and in Eq. (38d) l = 1...Ji is over all reactions in

which species i appears as a reactant.

The rate of chemical reaction l can be expressed as the

product of a rate constant kl and a term that is a function of

the concentration of the reactions. For elementary reactions,

the concentration dependent term is the product of the

reactant concentrations. In terms of mixing ratios, the rate

of reaction l (rl ) takes one of the following forms:

Equation (38a) forms a system of ordinary differential

equations (ODEs) that must be solved numerically. Two

sources of difficulty arise in obtaining solutions to it when

applied to atmospheric chemistry problems. First, the

system is nonlinear due to the contributions of the secondand

third-order reactions. Second, the system of equations

is "stiff" because of the widely varying time scales of the

chemical reactions and the complex interactions among

species. Mathematically, a stiff ODE system is described as

one in which all the eigenvalues of the system (in this case,

Eq. 38a) are negative, and the ratio of the absolute value of

the largest-to-smallest real parts of the eigenvalues is much

greater than one. For atmospheric chemistry problems, the

ratio is often greater than 1010, making the system very stiff

(Gong and Cho [83]). Because the stiff ODEs must be

solved over tens of thousands of cells repeatedly for air

quality simulations, efficient numerical methods must be

employed. The use of a standard explicit method is often

precluded because relatively small time steps are required to

maintain numerical stability and obtain accurate solutions.

On the other hand, classical implicit methods that are both

accurate and stable are not often applied because of the high

computational costs. As described above, the CMAQ

contains four different solvers – two generalized gas-phase

solvers (SMVGEAR and QSSA) and two mechanism

specific solvers (EBI and MEBI). Each solver offers

varying levels of accuracy, computational efficiency, and

flexibility. A brief description of each is included below,

followed by a description of model simulation results that

illustrate the relative differences in computational efficiency

n1051 - n1052 - n1053 - n1054 - n1055 - n1056 - n1057 - n1058 - n1059 - n1060 - n1061 - n1062 - n1063 - n1064 - n1065 - n1066 - n1067 - n1068 - n1069 - n1070 - n1071 - n1072 - n1073 - n1074 - n1075 - n1076 - n1077 - n1078 - n1079 - n1080 - n1081 - n1082 - n1083 - n1084 - n1085 - n1086 - n1087 - n1088 - n1089 - n1090 - n1091 - n1092 - n1093 - n1094 - n1095 - n1096 - n1097 - n1098 - n1099 - n1100

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