The HLL solver assumes a two-wave configuration: Two signals, separating three states. That works for hyperbolic equations, but as soon as you have a larger system, such as presented by the Euler equations, it is no longer correct and your end result may well be completely wrong. In a general case, I would advice implementing HLLC, which restores the third wave. It makes the code much more flexible.

For some more information, I would recommend the link below. In any case, the pros and cons of such solvers are explained in any good book on numerical hydrodynamics

As already answered by Allard, the HLLC is better as it resolves the system better than the HLL.

Allard, you didn't include the link you mentioned in your answer.

I know the difference between the two solvers.

The question was  HLLC 1st order accurate vs. HLL 2nd order accurate (MUSCL or WAF).

Although, I don't think there is a clear answer to this.

Sorry about that. This is the link

http://link.springer.com/chapter/10.1007%2Fb79761_10

Alternatively, you can also look at:

http://www.reading.ac.uk/web/FILES/maths/CKong-riemann.pdf

HLLC solver i is better than HLL solver because it can solve exactly stationary contact discontinuity.. It s an essential feature for problem envolving large density ratio and phase change. For example It avoid fluid mixing between 2 fluids of different density which are at rest and have the same presuure ...

for more information the following reference can be usefull :

• Riemann Solvers and Numerical Methods for Fluid Dynamics. A practicalIntroduction. E.F. TORO, Spinger Verlag, 1997.

For ALE formulation :

• On the computation of multi-material flows using ALE formulation. H. LUO, J.D. BAUM and R. LÖHNER,Journal of Computational Physics 194 : 304-328, 2004.
• Numerical experiments using a HLLC-type scheme with ALE formulation for compressible two-phase flows five-equation models with phase transition Frederic Daube, Pascal Galon, Zhenlan Gao, Etienne Blaud  Computers & Fluids 05/2014; 48:112–138.