An in-depth comparative performance evaluation of the HLLC (Harten-Lax-van Leer-Contact) and the latest version of the AUSM (Advection Upstream Splitting Method), the AUSM+-up, numerical schemes is carried out with the help of the one-dimensional shock tube problem. The efficiency of schemes is assessed on the basis of the accuracy in capturing of the shock, contact discontinuity, and the expansion fan in the solution. Numerical schemes viz., the upwind difference, the Godunov, the MacCormack, and the basic AUSM scheme are also investigated for their performance while solving the same problem to do a wider comparison. Numerical results from each method are compared against the exact solution to the problem. The HLLC numerical scheme is found to be the most efficient followed by AUSM+-up, which is marginally inferior with respect to the shock capturing accuracy.