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Synopsis
This work examines three chemical computing facets that appear to be relevant to current advancements. The first step is to introduce extrapolation techniques for differential-algebraic equation numerical treatment. At a certain point in its development, the related extrapolation code LIMEX has become sophisticated enough to seriously challenge Petzold’s more widely used multi-step algorithm DASSL. The second approach is the adaptive method of lines for partial differential equations, like the ones that come up in combustion problems. There includes a thorough discussion of both static and dynamic regridding approaches. Lastly, some novel approaches to solving the kinetic equations resulting from polymer reactions are discussed. One novel aspect of the proposed method is the use of sets of orthogonal polynomials over a Galerkin procedure.

A trapezoidal rule was used to discretize the model with respect to time. The resulting non-linear algebraic equalities (AE) were solved using two different approaches: (1) formulating the dynamic model equations with the modelling language AMPL [51] and solving them with the large-scale solver LANCELOT [52], and (2) formulating the model in the modelling language DIVA [49] as a code generator, temporal discretization, and solution with the non-linear equation solver NLEQIS [50]. Due to LANCELOT’s specific architecture for large-scale sparse non-linear equations, the second method was quicker.

For numerical treatment of ODE systems arising in air pollution models, it is also necessary to compare the partitioning algorithms with other well-liked and effective algorithms; see, for instance, Chock et al. [46], Jay et al. [145], Odman et al. [187], Sandu et al. [211], [213], [212], Shieh et al. [221], Verwer and van Loon [259], Verwer and Simpson [260], Verwer et al. [258]. Comparisons with the extrapolation time-integration methods of Deuflhard and his colleagues ([63], [64], [65], [66], [67]) and the Runge-Kutta type methods examined in Butcher [37] are necessary because they are promising options to consider when working with large air pollution models. Note 5.13.