The Euler equation is studied numerically by the high order Runge - Kutta method of discontinuous Galerkin method .. In this paper, we study the slope limiter in the high resolution numerical flow format, and use the ghost fluid method to suppress the numerical oscillation. The results show that the slope limiter is stable in calculation, high accuracy, high precision and high resolution can be achieved in the calculation; numerical calculation method by using the discontinuous Runge-KuttaGalerkin method, computing environment interface processing method using ghost fluid method, slope limiter is very efficient and accurate, and has broad prospects in engineering application.