Running an Atmospheric Chemistry Scheme from a Large Air Pollution Model by Using Advanced Versions of the Richardson Extrapolation

Abstract

AbstractAtmospheric chemistry schemes, which are described mathematically by non-linear systems of ordinary differential equations (ODEs), are used in many large-scale air pollution models. These systems of ODEs are badly-scaled, extremely stiff and some components of their solution vectors vary quickly forming very sharp gradients. Therefore, it is necessary to handle the atmospheric chemical schemes by applying accurate numerical methods combined with reliable error estimators. Three well-known numerical methods that are suitable for the treatment of stiff systems of ODEs were selected and used: (a) EULERB (the classical Backward Differentiation Formula), (b) DIRK23 (a two-stage third order Diagonally Implicit Runge-Kutta Method) and (c) FIRK35 (a three-stage fifth order Fully Implicit Runge-Kutta Method). Each of these three numerical methods was applied in a combination with nine advanced versions of the Richardson Extrapolation in order to get more accurate results when that is necessary and to evaluate in a reliable way the error made at the end of each step of the computations. The code is trying at every step (A) to determine a good stepsize and (B) to apply it with a suitable version of the Richardson Extrapolation so that the error made at the end of the step will be less than an error-tolerance TOL, which is prescribed by the user in advance. The numerical experiments indicate that both the numerical stability can be preserved and sufficiently accurate results can be obtained when each of the three underlying numerical methods is correctly combined with the advanced versions of the Richardson Extrapolation.KeywordsAtmospheric chemical schemesStiff systems of ODEsImplicit Runge-Kutta MethodsAdvanced versions of the Richardson Extrapolation