  We are interested in the numerical solution of large systems of hyperbolic
conservation laws or systems in which the characteristic decomposition is
expensive to compute. Solving such equations using finite volumes or
Discontinuous Galerkin requires a numerical flux function which solves local
Riemann problems at cell interfaces. There are various methods to express the
numerical flux function. On the one end, there is the robust but very diffusive
Lax-Friedrichs solver; on the other end the upwind Godunov solver which
respects all resulting waves. The drawback of the latter method is the costly
computation of the eigensystem.
  This work presents a family of simple first order Riemann solvers, named
HLLX$\omega$, which avoid solving the eigensystem. The new method reproduces
all waves of the system with less dissipation than other solvers with similar
input and effort, such as HLL and FORCE. The family of Riemann solvers can be
seen as an extension or generalization of the methods introduced by Degond et
al. \cite{DegondPeyrardRussoVilledieu1999}. We only require the same number of
input values as HLL, namely the globally fastest wave speeds in both
directions, or an estimate of the speeds. Thus, the new family of Riemann
solvers is particularly efficient for large systems of conservation laws when
the spectral decomposition is expensive to compute or no explicit expression
for the eigensystem is available.
