Atmospheric Environment 37 (2003) 3741–3748 Development and analysis of a non-splitting solution for three-dimensional air quality models Khoi Nguyen, Donald Dabdub* Department of Mechanical and Aerospace Engineering, University of California at Irvine, Irvine, CA 92697-3975, USA Received 18 June 2002; accepted 30 April 2003 Abstract This paper examines the errors introduced by operator splitting techniques in air quality models. Results are presented for different time steps used in the splitting schemes as well as for different ordering in which the operators are computed. Furthermore, a non-splitting technique is developed to analyze the performance of operator splitting techniques in air quality models. Convergence rates of operator splitting schemes are determined. Research indicates splitting techniques provides at most linear convergence. For fast-reacting species like N 2 O 5 ; the convergence is not achieved when using splitting methods and time steps as small as 10 s: Symmetric and non-symmetric operator splitting does not provide significant difference in accuracy. Furthermore, operator splitting ordering with stiff operators computed last does not produce better results than with non-stiff operators computed last. The non-splitting method developed achieves convergence by reducing time steps, adapting time steps to insure convergence, and eliminating operator splitting. r 2003 Elsevier Ltd. All rights reserved. Keywords: Operator splitting; Convergence; Operator ordering; Stiff integrator; Air quality models; Numerical techniques 1. Introduction Operator splitting methods (Yanenko, 1971) are mathematical techniques used to solve partial differen- tial equations. Splitting methods are commonly em- ployed in three-dimensional air quality models to reduce the computational effort required to solve the govern- ing equations. One of the first air quality models to use operator splitting is described by McRae et al. (1982). Operator splitting methods reduce chemically reactive three-dimensional transport equations into a series of one-dimensional transport equations and decouple chemical kinetics from transport dynamics. Some current models that split transport and chemistry are GATOR (Jacobson, 1997), Models3/CMAQ (Binkowshki and Shankar, 1995), and UAM-AERO (Lurmann et al., 1997). Three-dimensional air quality models solve the atmo- spheric dispersion equations: @c i @t þrðuc i Þ¼rðKrc i Þþ R i ðc 1 ; c 2 ; yc n Þ ð1Þ subjected to initial and boundary conditions c i ðx @O ; y @O ; tÞ¼ c i@O ; ð2Þ @c i ðz ¼ 0; tÞ @z ¼ S þ i þ v g c i ; ð3Þ @c i ðz ¼ H; tÞ @z ¼ 0; ð4Þ c i ðx; y; z; 0Þ¼ c i0 ; ð5Þ where c i ; x; y; z; t; u; K; R i ; S þ i ; v g ; H; and @O are the concentration of species i; space variables, time variable, wind fields, diffusion fields, chemical kinetics, source, settling velocity, inversion height and domain ARTICLE IN PRESS *Corresponding author. Tel.: +1-949-824-6126; fax: +1- 949-824-8585. E-mail addresses: khnguy@eng.uci.edu (K. Nguyen), ddabdub@uci.edu (D. Dabdub). 1352-2310/03/$-see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S1352-2310(03)00348-0