Abstract
Abstract. A rigorous methodology for the evaluation of integration schemes for Lagrangian particle dispersion models (LPDMs) is presented. A series of one-dimensional test problems are introduced, for which the Fokker–Planck equation is solved numerically using a finite-difference discretisation in physical space and a Hermite function expansion in velocity space. Numerical convergence errors in the Fokker–Planck equation solutions are shown to be much less than the statistical error associated with a practical-sized ensemble (N = 106) of LPDM solutions; hence, the former can be used to validate the latter. The test problems are then used to evaluate commonly used LPDM integration schemes. The results allow for optimal time-step selection for each scheme, given a required level of accuracy. The following recommendations are made for use in operational models. First, if computational constraints require the use of moderate to long time steps, it is more accurate to solve the random displacement model approximation to the LPDM rather than use existing schemes designed for long time steps. Second, useful gains in numerical accuracy can be obtained, at moderate additional computational cost, by using the relatively simple “small-noise” scheme of Honeycutt.
Highlights
State-of-the-art Lagrangian particle dispersion models (LPDMs hereafter), for example FLEXPART (Stohl et al, 2005) and NAME (Jones et al, 2007), are key scientific tools for the study of the long-range transport and dispersal of the transport of atmospheric trace gases and aerosols
Speaking, LPDMs are formulated as stochastic differential equations (SDEs hereafter). (It is notable that it is possible to include jump processes (Platen and Liberati, 2010) as a representation of non-local convective parameterisations (Forster et al, 2007), but we will not be concerned here with this possibility.) the numerical analysis of solution techniques for SDEs (e.g. Kloeden and Platen, 1992; Milstein and Tretyakov, 2004) is a mature subject in mathematics, LPDMs have not, generally speaking, exploited developments in the subject, and are typically formulated using numerical schemes adapted from those used for ordinary differential equations
The numerical accuracy above is sufficient for benchmarking our LPDM solutions, because the statistical error associated with reasonable-sized ensembles (N = 106) of the LPDM is of order E(t1) ≈ 10−2, as will be discussed below
Summary
State-of-the-art Lagrangian particle dispersion models (LPDMs hereafter), for example FLEXPART (Stohl et al, 2005) and NAME (Jones et al, 2007), are key scientific tools for the study of the long-range transport and dispersal of the transport of atmospheric trace gases and aerosols. The aim of the present work, is to introduce a rigorous framework for the testing and evaluation of numerical schemes for LPDMs. The framework is based on a standard one-dimensional dispersion model problem. Our approach to evaluating a given LPDM numerical scheme is to cross-validate its performance against a numerical solution of the corresponding Fokker–Planck equation The FPE describes the time evolution of the probability density function (pdf) of the stochastic process, and is formulated in position-velocity space, and so in the context of the current problem of dispersion in one spatial dimension is a partial differential equation in 2 + 1 dimensions. 3, the methodology for using the FPE solution to assess specific numerical schemes for the LPDM is presented, and in Sect.
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