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Adaptive discontinuous evolution Galerkin method for dry atmospheric flow. (English) Zbl 1349.76292

Summary: 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 a nonlinear part that models advection. Time integration is realized by the IMEX type approximation using the semi-implicit second-order backward differentiation formula (BDF2). Moreover in order to approximate efficiently small scale phenomena, adaptive mesh refinement using the space filling curves via the AMATOS function library is employed. Four standard meteorological test cases are used to validate the new discontinuous evolution Galerkin method for dry atmospheric convection. Comparisons with the Rusanov flux, a standard one-dimensional approximate Riemann solver used for the flux integration, demonstrate better stability and accuracy, as well as the reliability of the new multidimensional DEG method.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76R10 Free convection
86-08 Computational methods for problems pertaining to geophysics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
86A10 Meteorology and atmospheric physics

Software:

HLLE; HE-E1GODF; AMATOS
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Full Text: DOI

References:

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