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**A study of spectral element and discontinuous Galerkin methods for the Navier-Stokes equations in nonhydrostatic mesoscale atmospheric modeling: equation sets and test cases.**
*(English)*
Zbl 1194.76189

The authors study three forms of stratified fluid flow equations for possible use in next-generation nonhydrostatic mesoscale atmospheric models. Set 1 is not in conservation form; set 2 can be written in conservation form only for inviscid flows, while set 3 can be written in conservation form for either inviscid or viscous flows, and is in fact the compressible Navier-Stokes (NS) equations. The authors compare the results of the three equation sets using two numerical methods: the spectral element (SE) and discontinuous Galerkin (DG) methods, both are local high-order methods, but the DG method is a fully conservative method (both global and local), while the SE method is only globally conservative. Then the authors test five models using seven test cases, three of which involve mountain waves. The five models give similar results and compare well with models found in the literature. Equation set 1, although not in conservation form, performs better than expected, although it tends to yield results slightly different than set 2 and 3. The differences between the five models are quite small, but the main result is that the differences in the simulation are most often attributed to the equation set used; the numerical method also plays a role in the performance of the models. Set 3 (fully compressible NS equations) yields solutions that are less dissipative (for the density current). The comparison of SE and DG methods shows that both methods yield similar solutions for the problems studied; however, for the bubble and mountain tests, the DG method performs better. It is expected that for more stringent tests (i.e. those having very steep gradients such as those produced by the sub-grid scale parametrization) the DG method will prevail.

Reviewer: Titus Petrila (Cluj-Napoca)

### MSC:

76M22 | Spectral methods applied to problems in fluid mechanics |

76M10 | Finite element methods applied to problems in fluid mechanics |

76N15 | Gas dynamics (general theory) |

86A10 | Meteorology and atmospheric physics |

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\textit{F. X. Giraldo} and \textit{M. Restelli}, J. Comput. Phys. 227, No. 8, 3849--3877 (2008; Zbl 1194.76189)

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