Using the same stencils, the compact schemes can get higher accuracy and resolution compared with the traditional ones. But it will bring about spurious oscillations if the compact schemes are used directly. There are several methods to settle this problem. The TVD algorithm was selected in our study. Firstly, two five-order upwind compact schemes were introduced and Fourier analysis was used to compare their dissipation and dispersion characteristics. Secondly, two different TVD methods were applied to Euler equations. The performance of the numerical algorithm was assessed by performing preliminary simulations on some problems, such as the oblique shock reflection problem. The algorithm applied here is proven to have good resolution properties and robust of capturing shock waves and contacts, but it still has the problems of accuracy degree and lack of dissipation.