AbstractFor Computational Aeroacoustics (CAA), good wave propagation properties are a crucial point for numerical schemes which solve the linearized Euler equations (LEE) in the time domain. A recently developed high order scheme with excellent wave propagation properties is the Discontinuous Galerkin (DG) method. The DG schemes are based on the Finite Element Method (FEM), using discontinuous piecewise polynomial basis functions to represent the solution. The efficiency can be highly improved for CAA applications if a quadrature‐free formulation of the DG method is used, but nevertheless the DG schemes consume much computer storage due to the Runge‐Kutta time integration procedure and the high number of degrees of freedom to be stored for each element. We managed to develop a high order single‐step quadrature‐free DG method which does not use a Runge‐Kutta time integration procedure but which applies the ADER approach of Toro, Schwartzkopff et al. and where the accuracy of the scheme can further be improved using additionally a reconstruction technique working on the degrees of freedom of the element and its neighbors.