We present an energy conserving space discretisation based on a Poisson bracket that can be used to derive the dry compressible Euler as well as thermal shallow water equations. It is formulated using the compatible finite element method, and extends the incorporation of upwinding for the shallow water equations as described in Wimmer, Cotter, and Bauer (2020). While the former is restricted to DG upwinding, an energy conserving SUPG method for the (partially) continuous Galerkin thermal field space is newly introduced here. The energy conserving property is validated by coupling the Poisson bracket based spatial discretisation to an energy conserving time discretisation. Further, the discretisation is demonstrated to lead to an improved thermal field development with respect to stability when upwinding is included. An approximately energy conserving scheme that includes upwinding for all prognostic fields with a smaller computational cost is also presented. In a falling bubble test case used for the Euler equations, the latter scheme is shown to resolve small scale features at coarser resolutions than a corresponding scheme derived directly from the equations without the Poisson bracket framework.

^{1}; Colin J. Cotter

^{2}; Werner Bauer

^{3}

@article{SMAI-JCM_2021__7__267_0, author = {Golo A. Wimmer and Colin J. Cotter and Werner Bauer}, title = {Energy conserving {SUPG} methods for compatible finite element schemes in numerical weather prediction}, journal = {The SMAI Journal of computational mathematics}, pages = {267--300}, publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles}, volume = {7}, year = {2021}, doi = {10.5802/smai-jcm.77}, language = {en}, url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.77/} }

TY - JOUR AU - Golo A. Wimmer AU - Colin J. Cotter AU - Werner Bauer TI - Energy conserving SUPG methods for compatible finite element schemes in numerical weather prediction JO - The SMAI Journal of computational mathematics PY - 2021 SP - 267 EP - 300 VL - 7 PB - Société de Mathématiques Appliquées et Industrielles UR - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.77/ DO - 10.5802/smai-jcm.77 LA - en ID - SMAI-JCM_2021__7__267_0 ER -

%0 Journal Article %A Golo A. Wimmer %A Colin J. Cotter %A Werner Bauer %T Energy conserving SUPG methods for compatible finite element schemes in numerical weather prediction %J The SMAI Journal of computational mathematics %D 2021 %P 267-300 %V 7 %I Société de Mathématiques Appliquées et Industrielles %U https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.77/ %R 10.5802/smai-jcm.77 %G en %F SMAI-JCM_2021__7__267_0

Golo A. Wimmer; Colin J. Cotter; Werner Bauer. Energy conserving SUPG methods for compatible finite element schemes in numerical weather prediction. The SMAI Journal of computational mathematics, Volume 7 (2021), pp. 267-300. doi : 10.5802/smai-jcm.77. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.77/

[1] Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., Volume 39 (2002) no. 5, pp. 1749-1779 | DOI | MR | Zbl

[2] et al. PETSc users manual, Argonne National Laboratory, 2019

[3] Efficient management of parallelism in object-oriented numerical software libraries, Modern software tools for scientific computing, Springer, 1997, pp. 163-202 | DOI | Zbl

[4] Energy–enstrophy conserving compatible finite element schemes for the rotating shallow water equations with slip boundary conditions, J. Comput. Phys., Volume 373 (2018), pp. 171-187 | DOI | MR | Zbl

[5] Towards a geometric variational discretization of compressible fluids: the rotating shallow water equations, J. Comput. Dyn., Volume 6 (2019) no. 2158-2491, p. 1 | MR | Zbl

[6] A compatible finite element Discretisation for the Moist Compressible Euler Equations (2019) (https://arxiv.org/abs/1910.01857)

[7] A structure-exploiting numbering algorithm for finite elements on extruded meshes, and its performance evaluation in Firedrake, Geoscientific Model Development, Volume 9 (2016) no. 10, pp. 3803-3815 | DOI

[8] Stability of the SUPG finite element method for transient advection–diffusion problems, Comput. Methods Appl. Mech. Eng., Volume 193 (2004) no. 23-26, pp. 2301-2323 | DOI | MR | Zbl

[9] et al. Mixed finite element methods and applications, 44, Springer, 2013 | DOI

[10] The mathematical theory of finite element methods, 15, Springer, 2007

[11] Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., Volume 32 (1982) no. 1-3, pp. 199-259 | DOI | MR | Zbl

[12] Application of the piecewise parabolic method (PPM) to meteorological modeling, Monthly Weather Review, Volume 118 (1990) no. 3, pp. 586-612 | DOI

[13] Linear energy-preserving integrators for Poisson systems, BIT Numer. Math., Volume 51 (2011) no. 1, pp. 91-101 | DOI | MR | Zbl

[14] Mixed finite elements for numerical weather prediction, J. Comput. Phys., Volume 231 (2012) no. 21, pp. 7076-7091 | DOI | MR | Zbl

[15] A quasi-Hamiltonian discretization of the thermal shallow water equations, J. Comput. Phys., Volume 379 (2019), pp. 1-31 | DOI | MR

[16] , Proceedings of the Exascale Applications and Software Conference (2013)

[17] A global hexagonal C-grid non-hydrostatic dynamical core (ICON-IAP) designed for energetic consistency, Quarterly Journal of the Royal Meteorological Society, Volume 139 (2013) no. 670, pp. 152-175 | DOI

[18] Slate: extending Firedrake’s domain-specific abstraction to hybridized solvers for geoscience and beyond., Geoscientific Model Development, Volume 13 (2020) no. 2, pp. 735-761 | DOI

[19] Solver composition across the PDE/linear algebra barrier, SIAM J. Sci. Comput., Volume 40 (2018) no. 1, p. C76-C98 | DOI | MR | Zbl

[20] A guide to numerical methods for transport equations, Ph. D. Thesis, University Erlangen-Nuremberg (2010)

[21] Petrov–Galerkin flux upwinding for mixed mimetic spectral elements, and its application to geophysical flow problems, Comput. Math. Appl., Volume 89 (2021), pp. 68-77 | DOI | MR | Zbl

[22] A mixed mimetic spectral element model of the 3D compressible Euler equations on the cubed sphere, J. Comput. Phys., Volume 401 (2020), 108993 | MR | Zbl

[23] Energetics of climate models: net energy balance and meridional enthalpy transport, Reviews of Geophysics, Volume 49 (2011) no. 1 | DOI

[24] Stabilized high-order Galerkin methods based on a parameter-free dynamic SGS model for LES, J. Comput. Phys., Volume 301 (2015), pp. 77-101 | DOI | MR | Zbl

[25] Automated generation and symbolic manipulation of tensor product finite elements, SIAM J. Sci. Comput., Volume 38 (2016) no. 5, p. S25-S47 | DOI | MR | Zbl

[26] Choice of function spaces for thermodynamic variables in mixed finite-element methods, Quarterly Journal of the Royal Meteorological Society, Volume 144 (2018) no. 712, pp. 900-916 | DOI

[27] An inherently mass-conserving iterative semi-implicit semi-Lagrangian discretization of the non-hydrostatic vertical-slice equations, Quarterly Journal of the Royal Meteorological Society, Volume 136 (2010) no. 648, pp. 799-814

[28] Hamiltonian description of the ideal fluid, Rev. Mod. Phys., Volume 70 (1998) no. 2, p. 467 | DOI | MR | Zbl

[29] Noncanonical Hamiltonian density formulation of hydrodynamics and ideal magnetohydrodynamics, Phys. Rev. Lett., Volume 45 (1980) no. 10, p. 790 | DOI | MR

[30] Scale-selective dissipation in energy-conserving finite element schemes for two-dimensional turbulence, Quarterly Journal of the Royal Meteorological Society, Volume 143 (2017) no. 705, pp. 1734-1745 | DOI

[31] A variational H(div) finite element discretisation for perfect incompressible fluids, IMA J. Numer. Anal., Volume 38 (2017) no. 3, pp. 1388-1419 | DOI

[32] Compatible finite element spaces for geophysical fluid dynamics, Dynamics and Statistics of the Climate System, Volume 1 (2016) no. 1 | DOI

[33] Firedrake: automating the finite element method by composing abstractions, ACM Trans. Math. Softw., Volume 43 (2016) no. 3, p. 24 | MR | Zbl

[34] Conservation laws for primitive equations models with inhomogeneous layers, Geophys. Astrophys. Fluid Dyn., Volume 70 (1993) no. 1-4, pp. 85-111 | DOI | MR

[35] Symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics, Adv. Geophys., Volume 32 (1990) no. 287–338, p. 2

[36] Higher-order compatible finite element schemes for the nonlinear rotating shallow water equations on the sphere, J. Comput. Phys., Volume 375 (2018), pp. 1121-1137 | DOI | MR | Zbl

[37] Numerical solutions of a non-linear density current: a benchmark solution and comparisons, Int. J. Numer. Methods Fluids, Volume 17 (1993) no. 1, pp. 1-22 | DOI

[38] An energy consistent discretization of the nonhydrostatic equations in primitive variables, Journal of Advances in Modeling Earth Systems, Volume 12 (2020) no. 1 | DOI

[39] A standard test set for numerical approximations to the shallow water equations in spherical geometry, J. Comput. Phys., Volume 102 (1992) no. 1, pp. 211-224 | DOI | MR | Zbl

[40] Energy conserving upwinded compatible finite element schemes for the rotating shallow water equations, J. Comput. Phys., Volume 401 (2020), p. 109016 | DOI | MR | Zbl

[41] Geophysical fluid dynamics: understanding (almost) everything with rotating shallow water models, Oxford University Press, 2018 | DOI

*Cited by Sources: *