A study of spectral element and discontinuous Galerkin methods for the Navier-Stokes equations in nonhydrostatic mesoscale atmospheric modeling: Equation sets and test cases
Abstract
We present spectral element (SE) and discontinuous Galerkin (DG) solutions of the Euler and compressible Navier–
Stokes (NS) equations for stratified fluid flow which are of importance in nonhydrostatic mesoscale atmospheric modeling.
We study three different forms of the governing equations using seven test cases. Three test cases involve flow over mountains
which require the implementation of non-reflecting boundary conditions, while one test requires viscous terms (density
current). Including viscous stresses into finite difference, finite element, or spectral element models poses no additional
challenges; however, including these terms to either finite volume or discontinuous Galerkin models requires the introduction
of additional machinery because these methods were originally designed for first-order operators. We use the local
discontinuous Galerkin method to overcome this obstacle. The seven test cases show that all of our models yield good
results. The main conclusion is that equation set 1 (non-conservation form) does not perform as well as sets 2 and 3 (conservation
forms). For the density current (viscous), the SE and DG models using set 3 (mass and total energy) give less
dissipative results than the other equation sets; based on these results we recommend set 3 for the development of future
multiscale research codes. In addition, the fact that set 3 conserves both mass and energy up to machine precision motives
us to pursue this equation set for the development of future mesoscale models. For the bubble and mountain tests, the DG
models performed better. Based on these results and due to its conservation properties we recommend the DG method. In
the worst case scenario, the DG models are 50% slower than the non-conservative SE models. In the best case scenario, the
DG models are just as efficient as the conservative SE models.
Description
The article of record as published may be found at http://dx.doi.org/10.1016/j.jcp.2007.12.009
Rights
This publication is a work of the U.S. Government as defined in Title 17, United States Code, Section 101. Copyright protection is not available for this work in the United States.Collections
Related items
Showing items related by title, author, creator and subject.
-
A study of spectral element and discontinuous Galerkin methods for the Navier-Stokes equations in nonhydrostatic mesoscale atmospheric modeling: Equation sets and test cases
Giraldo, F.X.; Restelli, M. (2008);We present spectral element (SE) and discontinuous Galerkin (DG) solutions of the Euler and compressible Navier-Stokes (NS) equations for stratified fluid flow which are of importance in nonhydrostatic mesoscale atmospheric ... -
A study of spectral element and discontinuous Galerkin methods for the Navier–Stokes equations in nonhydrostatic mesoscale atmospheric modeling: Equation sets and test cases
Giraldo, Francis X.; Restelli, M. (2008);We present spectral element (SE) and discontinuous Galerkin (DG) solutions of the Euler and compressible Navier– Stokes (NS) equations for stratified fluid flow which are of importance in nonhydrostatic mesoscale atmospheric ... -
Development and evaluation of a hydrostatic dynamical core using the spectral element/discontinuous Galerkin methods
Giraldo, F.X.; Choi, S.-J. (Copernicus Publications, 2014-06-26);In this paper, we present a dynamical core for the atmospheric primitive hydrostatic equations using a unified formulation of spectral element (SE) and discontinuous Galerkin (DG) methods in the horizontal direction with ...