Abstract: A numerical study is undertaken to compare the fifth-order finite difference weighted essentially non-oscillatory scheme (FD-WENO5) to the second-order Godunov scheme  MUSCL (monotonic upstream-centered scheme for conservation laws).For quantitative comparison purpose,these methods are tested on a series of problems whose true solutions are known or can be computed with high accuracy,such as the two-dimensional Burgers initial problems,the two-dimensional periodic vortex problem,Richtmyer-Meshkov instability problems and Rayleigh-Taylor instability problems.The results can be referred to as to how to choose the numerical method when solving the problems of gas dynamics and radiation hydrodynamics.

Key words: weighted essentially non-oscillatory scheme, Godunov scheme, Euler equation