  We propose an approximate solver for multi-medium Riemann problems with
materials described by a family of general Mie-Gr\"uneisen equations of state,
which are widely used in practical applications. The solver provides the
interface pressure and normal velocity by an iterative method. The
well-posedness and convergence of the solver is verified with mild assumptions
on the equations of state. To validate the solver, it is employed in computing
the numerical flux on phase interfaces of a numerical scheme on Eulerian grids
that was developed recently for compressible multi-medium flows. Numerical
examples are presented for Riemann problems, air blast and underwater explosion
applications.
