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A Variational Multiscale Stabilized Finite Element Method to solve the Euler Equations for Nonhydrostatic Stratified Benchmarks

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Multiscale Numerics for the Atmosphere and Ocean

In this talk we present a Variational Multiscale Stabilization (VMS) for Compressible Euler Equations applied to the Finite Element (FE) solution of nonhydrostatic stratified flows. The VMS method was firstly presented by Hughes and co-workers [1] in the context of incompressible flows. In the present work, recentely presented in [3], we extend these concepts to Compressible Flows. In the framework of nonhydrostatic atmospheric dynamics, we test the algorithm for problems at low Mach numbers. A general version of the current compressible VMS technique was originally devised for Computational Fluid Dynamics (CFD) of compressible flows without stratification [2]. The present work is justified by the previously observed good performance of VMS and by the advantages that an element-based Galerkin formulation offers on massively parallel architectures, a challange for both CFD and Numerical Weather Prediction (NWP). Unphysical vertical oscillations that may appear for not well-balanced approximations are a relevant problem in NWP , especially in the proximity of steep topography. In that respect, to properly discretize the dominant hydrostatics, a particular interpolation technique is proposed. To evaluate the performance of the method in this context, some standard test cases of stratified environments are presented.

References [1] T. Hughes, Multiscale Phenomena: Greens functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, CMAME 127 (1995) 387401.

[2] M. Moragues, M. Vazquez, G. Houzeaux, R. Aubry, Variational Multiscale Stabilization of Compressible flows in Parallel Architectures, Parallel CFD 2010 , Taiwan, May 2010.

[3] S. Marras, M. Moragues, M. Vazquez, O. Jorba, G. Houzeaux, A Variational Multiscale Stabilized Finite Element Method for the Solution of the Euler Equations of Nonhydrostatic Stratified Flows, J. Comput. Phys. (submitted 2012)

This talk is part of the Isaac Newton Institute Seminar Series series.

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