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 (

State-of-the-art Lagrangian particle dispersion models (LPDMs hereafter), for
example FLEXPART

Mathematically speaking, LPDMs are formulated as stochastic differential
equations (SDEs hereafter). (It is notable that it is possible to include
jump processes

The aim of the present work, therefore, 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

A solution method based on a Hermite function expansion is introduced in
order to obtain accurate solutions of the FPE with computational efficiency.
Evaluation of the LPDM scheme proceeds by a comparison of pdfs in appropriate
error norm, where the LPDM pdf is generated from an ensemble of solutions,
using the kernel density method

The outline of the work is as follows. In Sect. 2, the SDEs describing the evolution of particle trajectories in the LPDM are introduced, together with the corresponding FPE. A numerical solution scheme for the FPE is described and solutions are obtained and benchmarked for a number of test cases. In Sect. 3, the methodology for using the FPE solution to assess specific numerical schemes for the LPDM is presented, and in Sect. 4 this methodology is then applied to specific schemes discussed above. In Sect. 5 the consequences of our findings are discussed and conclusions are drawn.

Consider a horizontally homogeneous turbulent ABL of uniform density, with a
vertical velocity distribution that is Gaussian with zero mean and standard
deviation

Vertical profiles of vertical velocity fluctuations

For the purposes of numerical solution, it is more convenient

The non-dimensional profiles of

It is simplest to view Eq. (

To specify our test-case problems it is necessary to choose suitable
(non-dimensional) profiles for

In Sect. 4 large ensembles of numerical solutions of Eq. (

Following the standard procedure in stochastic calculus

The FPE Eq. (

Equations (

The non-dimensionalised FPE Eq. (

Before inserting the expansion Eq. (

The initial conditions for Eq. (

It is worth noting that the series (

Based on the analysis above, Eq. (

The set (

To obtain our benchmark solutions of Eq. (

Relative error

Snapshots of particle concentration

Figure 3 shows snapshots of the particle concentration

In this section, a range of textbook, commonly used and new numerical schemes
for LPDMs will first be introduced and then evaluated using the FPE solutions
described above as a benchmark. The task is somewhat simplified because the
equation set (

The LPDM numerical schemes investigated in Sect. 4. Here

EXPLICIT 3.0 scheme tested in Sect. 4, with

It is important to note, however, that it is by no means obvious that a given
scheme will attain its formal weak order when solving Eq. (

Tables 2–3 summarise the SDE numerical schemes to be investigated. The
first, most obvious scheme to test is the Euler–Maruyama (E-M) scheme

A single candidate from a second class of schemes, the so-called
“small-noise” schemes, to be investigated is the HON-SRKII scheme of

A third class of schemes to be investigated is designed to work well with
long time steps. Such schemes are of interest operationally, because the
practical advantages of calculating large ensembles efficiently are thought
to outweigh the disadvantage of loss of accuracy due to time-stepping errors.
The FLEXPART model

The method used to compare the results from a particular scheme, at fixed
time step

The error associated with a given scheme, at time step

L

L

In the results below, in the interests of reproducibility, the error is
presented as a function of the fixed time step

Computational clock times, measured relative to the E-M scheme, for
all of the schemes detailed in Tables 2 and 3. The calculations are for

L

The main results, showing the performance of the six schemes described in
Tables 2–3 over a wide range of time steps

Also plotted in Figs. 4–6, as a dotted black line, is the L

Figure 4 shows results for the constant

Figure 5 shows results for the stable ABL test case at intermediate time

Figure 6 shows the results for the neutral ABL case at intermediate time

Snapshots of reconstructed particle density

To give an impression of where the particle concentration errors are
accumulating, Fig. 7 shows snapshots of particle density

The main contribution of this paper is to introduce a protocol for the
quantitative assessment of SDE numerical schemes, applied to the problem of
dispersion in an idealised atmospheric boundary layer, as modelled by LPDMs.
Accurate solutions of the Fokker–Planck equation (FPE, Eq.

The convergence results obtained in our model test problems are valuable because, due
to the importance of reflection of particles from the surface and top
of the boundary layer, it is not possible to rely on the formal convergence rates of
different SDE schemes

Our results allow the following recommendations to be made, for consideration
by operational modellers.

For our test problems, the best results with respect to accuracy as a
function of

The “long-step” scheme due to

Global calculations often require the use of long time steps for reasons of
computational efficiency. For such calculations, we recommend switching to
the random displacement model (Eq.

The MATLAB source code of the FPE solver can be found online via GitHub and
by searching for the repository “MRE FPE solver”
(

In this appendix we detail some useful properties of the probabilists'
Hermite polynomials

First, the Hermite polynomials are solutions of Hermite's equation

The FPE Eq. (

In the numerical implementation of LPDM Eq. (

At the end of every time step of the
numerical scheme

Higher-order schemes involve intermediate
time steps. Our treatment of intermediate time steps is as follows. First,
the

Here we derive a new long time-stepping scheme – LONGSTEP. The scheme is
designed to give acceptable results when integrating Eq. (

The point where our analysis departs from that of

To evaluate the

Physical concentrations of the particles

Once a sensible kernel function is chosen, and here we use the Gaussian
kernel

J. Gavin Esler acknowledges support from UK Natural Environment Research Council grant NE/G016003/1. H. M. Ramli acknowledges support from a UBD Chancellor's Scholarship and UCL Studentship. Edited by: S. Unterstrasser Reviewed by: two anonymous referees