that largely depends on the problem you are trying to solve. You need to specify in more detail whether you talk about single-phase or multiphase flow. Another question is what you aim for, accuracy of the solution, or computational efficiency. Also accessibility might be a critieria. For instance, FVM is available through the Open Source code OpenFOAM, there are also many LBM codes available and well-documented. Spectral methods are sometimes used for studying hydrodynamic instabilities like miscible viscous fingering.

that largely depends on the problem you are trying to solve. You need to specify in more detail whether you talk about single-phase or multiphase flow. Another question is what you aim for, accuracy of the solution, or computational efficiency. Also accessibility might be a critieria. For instance, FVM is available through the Open Source code OpenFOAM, there are also many LBM codes available and well-documented. Spectral methods are sometimes used for studying hydrodynamic instabilities like miscible viscous fingering.

Hello,

I agree with Dr. Berg that this is highly dependent on the problem; mainly, if your equations are conservation laws or not. If they are conservation laws, then shocks would occur, in this case a FEM or FVM would be preferred. In practice, really a FVM would be preferred due to numerical stability and the fact FVM are made for these sort of problems. A method for the conservation laws with shocks would be Godunov's scheme. If you were really interested in powerful method which uses FEM and FVM together see discontinuous galerkin method.

If the equations in question are not conservative in nature, then spectral methods would be preferred. Spectral methods usually assume smoothness in solution. If boundary conditions are periodic then the classical pseduo-spectral method would be fine, if there are not periodic boundary conditions I would suggest chebyshev spectral methods.

If you are interested in books for these subjects:

FEM- Gilbert Strang

FVM- Randall J. LeVeque

Spectral Methods -Trefethen 

I know nothing about LBM so I cannot comment on them. I assumed from this explanation that  your equation(s) are hyperbolic in nature on a regular geometry. 

If they are different than hyperbolic, I.E parabolic or elliptic then chebyshev spectral method should fit the bill.

Lastly, if the computational domain is irregular, then spectral element method could be applied. 

As the others have said, this depends on the what you are trying to simulate. The LBM can be advantages for multi-phase problems, complex and irregular boundaries, or non-Newtonian problems.

FVM

For strongly convective flows FVM, for diffusive flows (bottom of the ocean) FEM!!

In my experience with simulation of turbulent flows using Direct Numerical Simulations (DNS) I can tell you that if your geometry is simple (i.e. a parallelepiped) and your flow is homogeneous in one or more directions you can take advantage of spectral methods, with give you the best resolution compared to other methods such as FVM, FEM, LBM, etc.. This translate into lower errors for the same number of grid points, which is desirable when computing high Reynolds numbers. With that you can save a lot of computational time to other tasks or to evolve further in time the flow of interest. If one of your directions is not periodic, you can use Chebyshev or Compact Finite Differences (to distribute your grid points with more flexibility).

If you geometry is complex, then you should move into FVM, FEM, LBM,  or Spectral Elements; which give you more flexibility in the geometry election, but meshing the domain can be quite challenging and achieving high-order quite hard (i.e. a lot of computations for a given error threshold).  There are a number of commercial and open-source codes that use those approach. As already discussed in this thread, FVM: OpenFoam and the SU2 from Stanford (soon to be available with a FEM solver as well). SU2 is growing and being adopted really fast, perhaps you want to take a look. A quite famous code is NEK5000, that uses Spectral Elements (turbulent channels, pipes, and wings have been already simulated).

Regards

Thank you very much all respondents. Your answers are very helpful and helped me to synthesize a valuable summary on these numerical methods.

For the continuum assumption (Kn>1) and FEM is good when the flow is elliptic or highly diffusive. Spectral methods, although highly accurate, have limitations when geometry become complex.

Even for Kn > 0.1, FVM is preferred within the Moment Methods approach (FVM is the only method used to date that I know of) because the fields contain shocks and are highly hyperbolic.

 thank you for those answers