Arbitrary high order discontinuous Galerkin schemes based on the GRP method for compressible Euler equations

Combined with the discontinuous Galerkin (DG) framework, the generalized Riemann problem (GRP) method is applied to design a GRP-DG scheme with high order accuracy for compressible Euler equations. Since numerical fluxes with second order accuracy in time are derived by the GRP method, the reconstruction steps for physical variables in the new scheme are halved compared with the traditional Runge-Kutta discontinuous Galerkin (RK-DG) scheme. The numerical results are also improved due to more introduced physical information. Several numerical examples verify the validity of the proposed schemes.