A 1D exact treatment of shock waves within spectral methods in plane geometry

We present a very exact numerical technique for solving 1D Euler equations coupled with the transport equations for the entropy and the chemical abundances with or without shock formation. Two moving grids are used before and after the shock formation. Quantities are expanded on both sides of the matching point in Chebychev polynomials series. After the shock is formed, Rankine-Hugoniot conditions are used to determine the velocity of the shock and the matching conditions across the shock. Typical results are presented.