Abstract. We present a high-order formulation for solving hyperbolic conservation laws using the Discontinuous Galerkin Method (DGM). We introduce an orthogonal basis for the spatial discretization and use explicit Runge-Kutta time discretization. Some results of higher-order adaptive refinement calculations are presented for inviscid Rayleigh Taylor flow instability and shock reflexion problems. The adaptive procedure uses an error indicator that concentrates the computational effort near discontinuities.