In many research areas, it is common to model advection-diffusion problems

with Lagrangian particle methods. This is the same as solving a stochastic

differential equation, with drift and diffusion coefficients derived from the

advection-diffusion equation. But there is also a necessary condition for the

particle method to be equivalent to the Eulerian advection-diffusion equation,

is that it satisfies the well-mixed condition (Thomson, 1987), which says that

if particles are well mixed, they have to stay well mixed later on. This is just a

statement with respect to second law of thermodynamics, which is entropy. A

commonly used implementation of reflecting boundary conditions for particle

methods is analysed. We find that in some cases, this reflecting scheme will

give rise to oscillations in concentration close to the boundary, which we call

the boundary artifact.

We analyse the reflection scheme in the Lagrangian model, and compare

it to Neumann boundary conditions in the Eulerian model. We find that if

the diffusivity has a non-zero derivative at the boundary, this violates one of

the conditions for equivalence with the advection-diffusion equation, which

is that the drift coefficient in the SDE must be Lipschitz continuous. This

seems to be the origin of the boundary artifact. We analyse the artifact

further, and describe two different types of boundary artifact.

We suggest different approaches to dealing with the problem, and find

that the problem can in practice be handled by adjusting the diffusivity

close to the boundary. Support and motivation for such a change is found in

the concept of the ”unresolved basal layer” (Wilson & Flesch, 1993), which is

a pragmatic idea stating that closer than some distance from the boundary,

we simply cannot know the details of the turbulent motion.