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.
- Lagrangian formulation is essentially diffusion-free. This could be viewd as a blessing or short-coming depending on what you are doing.
- Lagrangian formulation is highly parallelizable since essentially all particles are advected independently.
- 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
- 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.
- It is easier to develop high-order accurate methods in the Lagrangian framework.
- Reconstruction of "concentration fields" from individual particle locations is expensive and the final result may not be very smooth