Capturing Infinitely Sharp Discrete Shock Profiles with the Godunov Scheme

It is well-known that this problem generally does not admit smooth solutions for large times so that weak solutions in the sense of distributions are considered. Due to the presence of discontinuities, these are generally not uniquely determined by v0 and the validity of an entropy criterion is added for the admissibility of discontinuities. More precisely, (1) is supplemented with the following entropy inequality