  In physically inviscid fluid dynamics, "shock capturing" methods adopt either
an artificial viscosity contribution or an appropriate Riemann solver
algorithm. These techniques are necessary to solve the strictly hyperbolic
Euler equations if flow discontinuities (the Riemann problem) are to be solved.
A necessary dissipation is normally used in such cases. An explicit artificial
viscosity contribution is normally adopted to smooth out spurious heating and
to treat transport phenomena. Such a treatment of inviscid flows is also widely
adopted in the Smooth Particle Hydrodynamics (SPH) finite volume free
Lagrangian scheme. In other cases, the intrinsic dissipation of Godunov-type
methods is implicitly useful. Instead "shock tracking" methods normally use the
Rankine-Hugoniot jump conditions to solve such problems. A simple, effective
solution of the Riemann problem in inviscid ideal gases is here proposed, based
on an empirical reformulation of the equation of state (EoS) in the Euler
equations in fluid dynamics, whose limit for a motionless gas coincides with
the classical EoS of ideal gases. The application of such an effective solution
to the Riemann problem excludes any dependence, in the transport phenomena, on
particle smoothing resolution length $h$ in non viscous SPH flows. Results on
1D shock tube tests, as well as examples of application for 2D turbulence and
2D shear flows are here shown. As an astrophysical application, a much better
identification of spiral structures in accretion discs in a close binary (CB),
as a result of this reformulation is also shown here.
