  Shock waves, vorticity waves, and entropy waves are fundamental discontinuity
waves in nature and arise in supersonic or transonic gas flow, or from a very
sudden release (explosion) of chemical, nuclear, electrical, radiation, or
mechanical energy in a limited space. Tracking these discontinuities and their
interactions, especially when and where new waves arise and interact in the
motion of gases, is one of the main motivations for numerical computation for
the gas dynamics equations. In this paper, we discuss some historic and recent
developments, as well as mathematical challenges, in designing and formulating
efficient numerical methods and algorithms to compute weak entropy solutions
for the Euler equations for gas dynamics.
