I have graduated from Moscow Physical-Technical Institute in 1981, received doctor's degree in 1990, now I work in Mexico. I received reliable results. Then I have extended my modification for nonlinear case. In this case it is necessary to take into account some nonlinear properties of Riemann's problem solution and the quadratic spline for density. But I have some problem, I think you know it.
Somebody calls it as nonlinear nonstability.
I do not know how to avoid this nonphysical oscillations. They are generated by rigged body boundary conditions. So I would like to have contact with specialists in this area to solve this and some other problems, conserning modifications of Godunov's method for some other kinds of equations, lenearized Navier-Stokes equations for example, different state equations and etc. The Piecewise Parabolic method for Gas-Dynamical Simulations.