Why the <Emphasis Type="Italic">Euler</Emphasis> scheme in particle tracking is not enough: the shallow-sea pycnocline test case |
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Authors: | Ulf Gräwe Eric Deleersnijder Syed Hyder Ali Muttaqi Shah Arnold Willem Heemink |
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Institution: | 1.Leibniz Institute for Baltic Sea Research (IOW),Warnemünde,Germany;2.Institute of Mechanics, Materials and Civil Engineering (IMMC),Université catholique de Louvain,Louvain-la-Neuve,Belgium;3.Earth and Life Institute (ELI), G. Lema?tre Centre for Earth and Climate Research (TECLIM),Université catholique de Louvain,Louvain-la-Neuve,Belgium;4.Delft Institute of Applied Mathematics (DIAM),Delft University of Technology,Delft,the Netherlands |
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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 second-order
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. |
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