For non linear system of hyperbolic PDE, The finite volume methods work well (because of inherent conservation). Godunov scheme is a very elegant solution philosophy. For linear system, it is nothing but upwinding, but for nonlinear system, there are entropy issues. Also the Riemann solution at the local Riemann problem can not be determined if the global solution is the centered expansion fan. For this issue, we apply entropy fix. (or local linearisation e.g. Roe's) There are lot of solutions provided for this. Which one gives most consistent and accurate results?
Godunov's method has an exact Riemann solver so no entropy fix is needed. A Roe solver (of which there are a few variants) uses a local linearization which has no diffusion to "fill in" the rarefaction fan, so it needs an entropy fix. Other approximate Riemann solvers, including Lax-Friedrichs, Rusanov, and the HLL family are inherently diffusive and do not need an entropy fix. Not surprisingly, an exact Riemann solver will give the best results for a given spatial discretization, but may be much more computationally expensive. Approximate Riemann solvers are more attractive with high order methods like WENO. Two books that you may find useful are