Why the Euler scheme in particle tracking is not enough: the shallow-sea pycnocline test case

Abstract

During the last decades, the Euler scheme was the common "workhorse" in particle tracking, although it is the lowest-order approximation of the underlying stochastic differential equation. To convince the modelling community of the need for better methods, we have constructed a new test case that will show the shortcomings of the Euler scheme. We use an idealised shallow-water diffusivity profile that mimics the presence of a sharp pycnocline and thus a quasi-impermeable barrier to vertical diffusion. In this context, we study the transport of passive particles with or without negative buoyancy. A semi-analytic solutions is used to assess the performance of various numerical particle-tracking schemes (first-and secondorder accuracy), to treat the variations in the diffusivity profile properly. We show that the commonly used Euler scheme exhibits a poor performance and that widely used particle-tracking codes shall be updated to either the Milstein scheme or second-order schemes. It is further seen that the order of convergence is not the only relevant factor, the absolute value of the error also is.

Vertical distribution of buoyant Microcystis blooms in a Lagrangian particle tracking model for short‐term forecasts in Lake Erie

“…After evaluating the performance of 10 random walk schemes [ Gräwe , ], Gräwe et al . [] recommended the use of either the Milstein scheme or higher order schemes such as S1.5 or PC2. The order of accuracy (rate of convergence) of numerical approximations to stochastic differential equations is separated into weak and strong cases, where the weak case relates to the accuracy of the ensemble particle distribution, while the strong case relates to the accuracy of particle trajectories [ Gräwe , ].…”

“…These artifacts can be easy to misinterpret as features of interest. Several studies have applied Lagrangian particle models to the 1‐D column case with steady, idealized diffusivity profiles [ Gräwe et al ., ; Ross and Sharples , ; Visser , ]. In 3‐D ocean and lake models, use of random‐walk vertical mixing schemes can be challenging because the required time step varies spatially and temporally over the model domain.…”

Simulating diffusion processes in discontinuous media: Benchmark tests

“…This scheme no longer works when is discontinuous (see e.g. [19,27] for numerical tests). The latter case is still a challenging problem.…”

“…Many interpretations and simulation techniques have been proposed during the last twenty years. Some numerical methods consider only the mathematical aspect of the simulation [12-14, 30, 31, 36-38] while others are driven by applications in a specific field: in geophysics [1-4, 9, 11, 22, 24-27, 33, 35, 43-46, 48, 55], fluid/gas dynamics [21], ecology [8,42,47], brain imaging [16], astrophysics [34,56], meteorology [54], oceanography [19,20,53], molecular dynamics [7,39], among others.…”

“…In [26], the different schemes are also qualitatively evaluated for symmetry. In [52] and [19], which are related to oceanography, density and residence times are compared with respect to known values. In [4,47], the first and second spatial moments in a two-layer aquifer system are estimated in a long time regime.…”