What are the limitations of Smoothed-Particle Hydrodynamics? I've been excited by some of the possibilities of Smoothed-Particle Hydrodynamics (SPH). I have seen some very exciting demonstrations of their use in 3D graphics, but I am wondering how well the formulas stack up to actual fluids. What is left out? How would I measure this objectively? 
 A: The Idea of SPH is that once you put that Smoothing Kernel, you are killing small wavelenght components of the function you are approximating, so, you it's not too different from grid methods, but you do it on a 'smoother' way. Other problems relate to the construction of the kernels, like the limitation of 1st order reproducibility, which is necessary to assure positivity of you reference density.
Also, SPH always had problems with dealing with boundaries, there is no natural way (as far as I know) to include boundary conditions directly on SPH formalism. You end up needing either to include 'ghost particles' which constitute and 'softly' enforce the boundary conditions, or to put it by brute force on the external potential.
This problems, in my view, are not entirely gratutious, part of the point of creating SPH was exactly that: to be able to deal with boundary-free and higly deformable boundaries that you find in explosion and astrophysical simulations (2 of the first real-life applications of the SPH method).
As for actual fluids, there is all kinds of simulations done with it: Atmospheric Dynamics, High-Explosives Design, Galactic Evolution, Astrophysics, Relativistic Heavy-Ion Collisions/QGP dynamics, you name it.
There are things that SPH simply sucks, it's true, like simulating Schrödinger Equation (via Madelung's Eqiation), but many other methods also suck, so that's no problem intrinsic to SPH. 
So, as far as what you can do with it, I believe that it's past the point that you can put into question whether is valid or not to use SPH. If it's the right method for your problem, it's another question all together.
Mesh-full methods, specially Finite-Elements Method, have received a lot of attention from the mathematicians, which have proven quality of the algorithms and the conditions for convergence for the most important class of PDEs, and this have almost (if not already) a century. Finally SPH and other mesh-free methods, are being given the same kind of treatment. It may not be as 'relevant' for the practicing engineer, but the correctness, conditions for convergence and quality of the algorithms is of fundamental importance if a Numerical Method is to thrive in the future. 
So, as for your happiness (as for mine too), SPH is not some method to do nice fluid flows videos in games, although some of those 3D graphics with SPH are indeed simply gorgeous.
If you wish, I can link some references, also, I can discuss why I believe SPH have a great future in research in many areas. It's good to see other people getting interested in SPH.
A: For those interested I recommend the book of "Smoothed Particle Hydrodynamics A Meshfree Particle Method by G. R. Liu, M. B. Liu (z-lib.org)". It is a little bit old (2002) but it really gives a extremelly comprehensive explanation about SPH.
In a nutshell, SPH is a mesh free method (contrary to lagrangian or eulerian approaches) that allows to obtain the values of a function in a point in space and time by the sum of the properties of the surrounding particles. However the way of solving the problem is limited to the same steps:

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*governing equations with proper boundary conditions (BC) and/or initial conditions (IC),

*domain discretization technique for creating particles,

*numerical discretization technique (weak form, strong form, particle methods),

*numerical technique to solve the resultant algebraic equations or ordinary differential equations (ODE).

The formulation for the NS equations change when compared to the typical Lagrangian approach. But it is still necessary to comply with a:

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*Continuity equation

*Particle approximation of momentum

*Particle approximation of energy

It is important to know that, for all of the above there are several formulations that are more or less mature, or that have different advantages and drawbacks.
