The Runge-Kutta Discontinuous Galerkin(RKDG) method with TVB limiter is a high-order accurate and highly parallelizable method for solving the Euler equations.Simultaneously,the immersed boundary method is a novel and general technique for handling a flow with complex geometry without any special needs of the computational mesh.We try to use the two methods for simulating the Euler equations with the above-mentioned problems on Cartesian mesh.Finally,the results of some classical test cases are provided to verify the efficiency of such methods.