Adaptive discontinuous evolution Galerkin method for dry atmospheric flow

We present a new adaptive genuinely multidimensional method within the framework of the discontinuous Galerkin method. The discontinuous evolution Galerkin (DEG) method couples a discontinuous Galerkin formulation with approximate evolution operators. The latter are constructed using the bicharacteristics of multidimensional hyperbolic systems, such that all of the infinitely many directions of wave propagation are considered explicitly. In order to take into account multiscale phenomena that typically appear in atmospheric flows nonlinear fluxes are split into a linear part governing the acoustic and gravitational waves and to the rest nonlinear part that models advection. Time integration is realized by the IMEX type approximation using the semi-implicit second-order backward differentiation formulas (BDF2) scheme. Moreover in order to approximate efficiently small scale phenomena adaptive mesh refinement using the space filling curves via AMATOS function library is applied. Three standard meteorological test cases are used to validate the new discontinuous evolution Galerkin method for dry atmospheric convection. Comparisons with the standard one-dimensional approximate Riemann solver used for the flux integration demonstrate better stability, accuracy as well as reliability of the new multidimensional DEG method.