[2] Sergei K. Godunov is credited with introducing the first exact Riemann solver for the Euler equations,[3] by extending the previous CIR (Courant-Isaacson-Rees) method to non-linear systems of hyperbolic conservation laws.
More recent research shows that an exact series solution to the Riemann problem exists, which may converge fast enough in some cases to avoid the iterative methods required in Godunov's scheme.
[5] The HLLE solver (developed by Ami Harten, Peter Lax, Bram van Leer and Einfeldt) is an approximate solution to the Riemann problem, which is only based on the integral form of the conservation laws and the largest and smallest signal velocities at the interface.
[6][7] The stability and robustness of the HLLE solver is closely related to the signal velocities and a single central average state, as proposed by Einfeldt in the original paper The HLLC (Harten-Lax-van Leer-Contact) solver was introduced by Toro.
More advanced techniques exist, like using the Roe average velocity for the middle wave speed.