# What are some of the differences between using a Lagrangian and Eulerian framework to quantify passive scalar dynamics?

On one hand, one may seed the domain with particles and track their trajectories in the Lagrangian sense by implementing a Lagrangian particle tracking model. On the other hand, one may use the Eulerian approach and solve a scalar transport advection/diffusion equation for a passive scalar similar to what is done for the density transport equation (except density is not a passive quantity because it is typically coupled to the momentum equation, whereby it influences the velocity field through the buoyancy term).

From my perspective the first option is simpler from a numerical implementation standpoint since all you need is to perform an integration in time, but it is missing diffusion. For most flows, molecular diffusion can be neglected so that may not even be an issue. The second option involves solving the advection/diffusion equation which has stability and other types of concerns. Are there any other considerations that I should think about?

I think this greatly depends on what kind of physics you are trying to model even though for some problems both approaches are viable.

## Lagrangian vs. Eulerian Framework

For certain problems Lagrangian frameworks are better suited than their Eulerian counterparts. For instance, if one is interested in studying the current patterns or sedimentation problems in oceanography it is more natural to do Lagrangian particle tracking than solving for a concentration field. Especially since, at least for some of such problems, diffusion would be meaningless and numerical diffusion inherent in Eulerian methods adversely affect the result.

On the other hand, certain problems are better formulated in Eulerian framework. Probably the most prominent example here is interface tracking. Evolving interfaces using Lagrangian framework, a.k.a front tracking methods, are very hard to implement, especially for 3D problems. This is because motion of individual particles do not satisfy entropy conditions and often lead to ambiguous definition of "interfaces location". To prevent this, connectivity must be defined for neighboring points which is hard to handle when the interface undergoes topological changes. However this class of problems are very naturally expressed in the Eulerian framework, e.g. in the level-set framework.

## Pros and Cons

Apart from which formulation would be more natural for a given problem, there are pros and cons to both methods.

1. Lagrangian formulation is essentially diffusion-free. This could be viewd as a blessing or short-coming depending on what you are doing.
2. Lagrangian formulation is highly parallelizable since essentially all particles are advected independently.
3. Lagrangian formulation is essentially CFL-free as one is integrating along the characteristics. Note that this property may also be exploited in the Eulerian framework through the so-called "Semi-Lagrangian" methods
4. Eulerian framework is more suited for problems that involve interfaces. They can very naturally express complicated topological changes, e.g. as in the level-set method. Moreover, for this class of problems, Lagrangian methods cannot be used if the interface develops shocks and/or rarefaction fans.
5. It is easier to develop high-order accurate methods in the Lagrangian framework.
6. Reconstruction of "concentration fields" from individual particle locations is expensive and the final result may not be very smooth
• For nr.2 I would just comment that if you are deriving the Lagrangian velocity from a Eulerian velocity field, which is what is most common in NS-solvers. Then the parallelization becomes more difficult as particles cross sub-domains held by individual processors. It may even cause unbalance as particles may tend to cluster together. Nov 13, 2013 at 18:58
• @IsopycnalOscillation That is correct if you are thinking in terms of domain-decomposition (which I admit is probably more relevant). On the other hand, in each sub-domain, threading (OpenMP, GPU, etc.) can give you near ideal speed up for these type of problems. Nov 13, 2013 at 20:13

Heat is a passive scalar (under assumptions of incompressibility). But there is no heat transfer from boundaries without diffusion. So you can't really do any such heat transfer problems in a purely Lagrangian framework (unless you go stochastic).

Same thing holds for permeable boundaries where the scalar diffuses into the domain.