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.

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.

Published online:
DOI: https://doi.org/10.5802/smai-jcm.77
Keywords: Compatible finite element methods; Hamiltonian mechanics; Poisson bracket; SUPG method
Golo A. Wimmer 1; Colin J. Cotter 2; Werner Bauer 3

1. Imperial College London; current affiliation: Los Alamos National Laboratory
2. Imperial College London
3. Imperial College London, INRIA Rennes
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     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},
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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] D. N. Arnold; F. Brezzi; B. Cockburn; L. D. Marini Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., Volume 39 (2002) no. 5, pp. 1749-1779 | Article | MR 1885715 | Zbl 1008.65080

[2] S. Balay; S. Abhyankar; M. Adams; J. Brown; P. Brune; K. Buschelman; L. Dalcin; A. Dener; V. Eijkhout; W. D. Gropp et al. PETSc users manual, Argonne National Laboratory, 2019

[3] S. Balay; W. D. Gropp; L. C. McInnes; B. F. Smith Efficient management of parallelism in object-oriented numerical software libraries, Modern software tools for scientific computing, Springer, 1997, pp. 163-202 | Article | Zbl 0882.65154

[4] W. Bauer; C. J. Cotter 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 | Article | MR 3854141 | Zbl 1416.65335

[5] W. Bauer; F. Gay-Balmaz 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 4026149 | Zbl 1421.76189

[6] T. M. Bendall; T. H. Gibson; J. Shipton; C. J. Cotter; B. Shipway A compatible finite element Discretisation for the Moist Compressible Euler Equations (2019) (https://arxiv.org/abs/1910.01857)

[7] G.-T. Bercea; A. T. T. McRae; D. A. Ham; L. Mitchell; F. Rathgeber; L. Nardi; F. Luporini; P. H. J. Kelly 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 | Article

[8] P. B. Bochev; M. D. Gunzburger; J. N. Shadid 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 | Article | MR 2055248 | Zbl 1067.76563

[9] D. Boffi; F. Brezzi; M. Fortin et al. Mixed finite element methods and applications, 44, Springer, 2013 | Article

[10] S. Brenner; R. Scott The mathematical theory of finite element methods, 15, Springer, 2007

[11] A. N. Brooks; T. J. R. Hughes 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 | Article | MR 679322 | Zbl 0497.76041

[12] R. L. Carpenter Jr; K. K. Droegemeier; P. R. Woodward; C. E. Hane Application of the piecewise parabolic method (PPM) to meteorological modeling, Monthly Weather Review, Volume 118 (1990) no. 3, pp. 586-612 | Article

[13] D. Cohen; E. Hairer Linear energy-preserving integrators for Poisson systems, BIT Numer. Math., Volume 51 (2011) no. 1, pp. 91-101 | Article | MR 2784654 | Zbl 1216.65175

[14] C. J. Cotter; J. Shipton Mixed finite elements for numerical weather prediction, J. Comput. Phys., Volume 231 (2012) no. 21, pp. 7076-7091 | Article | MR 2969702 | Zbl 1284.86005

[15] C. Eldred; T. Dubos; E. Kritsikis A quasi-Hamiltonian discretization of the thermal shallow water equations, J. Comput. Phys., Volume 379 (2019), pp. 1-31 | Article | MR 3881148

[16] R. Ford; M. J. Glover; D. A. Ham; C. M. Maynard; S. M. Pickles; G. Riley; N. Wood Gung Ho: A code design for weather and climate prediction on exascale machines, Proceedings of the Exascale Applications and Software Conference (2013)

[17] A. Gassmann 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 | Article

[18] T. H. Gibson; L. Mitchell; D. A. Ham; C. J. Cotter 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 | Article

[19] R. C. Kirby; L. Mitchell Solver composition across the PDE/linear algebra barrier, SIAM J. Sci. Comput., Volume 40 (2018) no. 1, p. C76-C98 | Article | MR 3763931 | Zbl 1383.65021

[20] D. Kuzmin A guide to numerical methods for transport equations (2010) (Ph. D. Thesis)

[21] D. Lee 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 | Article | MR 4233329 | Zbl 07336182

[22] D. Lee; A. Palha A mixed mimetic spectral element model of the 3D compressible Euler equations on the cubed sphere, J. Comput. Phys., Volume 401 (2020), 108993 | MR 4020473 | Zbl 1453.86025

[23] V. Lucarini; F. Ragone Energetics of climate models: net energy balance and meridional enthalpy transport, Reviews of Geophysics, Volume 49 (2011) no. 1 | Article

[24] S. Marras; M. Nazarov; F. X. Giraldo Stabilized high-order Galerkin methods based on a parameter-free dynamic SGS model for LES, J. Comput. Phys., Volume 301 (2015), pp. 77-101 | Article | MR 3402719 | Zbl 1349.76127

[25] A. T. T. McRae; G.-T. Bercea; L. Mitchell; D. A. Ham; C. J. Cotter Automated generation and symbolic manipulation of tensor product finite elements, SIAM J. Sci. Comput., Volume 38 (2016) no. 5, p. S25-S47 | Article | MR 3565562 | Zbl 1352.65615

[26] T. Melvin; T. Benacchio; J. Thuburn; C. J. Cotter 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 | Article

[27] T. Melvin; M. Dubal; N. Wood; A. Staniforth; M. Zerroukat 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] P. J. Morrison Hamiltonian description of the ideal fluid, Rev. Mod. Phys., Volume 70 (1998) no. 2, p. 467 | Article | MR 1627532 | Zbl 1205.37093

[29] P. J. Morrison; J. M. Greene Noncanonical Hamiltonian density formulation of hydrodynamics and ideal magnetohydrodynamics, Phys. Rev. Lett., Volume 45 (1980) no. 10, p. 790 | Article | MR 581650

[30] A. Natale; C. J. Cotter 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 | Article

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

[32] A. Natale; J. Shipton; C. J. Cotter Compatible finite element spaces for geophysical fluid dynamics, Dynamics and Statistics of the Climate System, Volume 1 (2016) no. 1 | Article

[33] F. Rathgeber; D. A. Ham; L. Mitchell; M. Lange; F. Luporini; A. T. T. McRae; G.-T. Bercea; G. R. Markall; P. H. J. Kelly Firedrake: automating the finite element method by composing abstractions, ACM Trans. Math. Softw., Volume 43 (2016) no. 3, p. 24 | MR 3615280 | Zbl 1396.65144

[34] P Ripa Conservation laws for primitive equations models with inhomogeneous layers, Geophys. Astrophys. Fluid Dyn., Volume 70 (1993) no. 1-4, pp. 85-111 | Article | MR 1278169

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

[36] J. Shipton; T. H. Gibson; C. J. Cotter 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 | Article | MR 3874575 | Zbl 1416.86014

[37] J. M. Straka; R. B. Wilhelmson; L. J. Wicker; J. R. Anderson; K. K. Droegemeier 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 | Article

[38] M. A. Taylor; O. Guba; A. Steyer; P. A. Ullrich; D. M. Hall; C. Eldred An energy consistent discretization of the nonhydrostatic equations in primitive variables, Journal of Advances in Modeling Earth Systems, Volume 12 (2020) no. 1 | Article

[39] D. L. Williamson; J. B. Drake; J. J. Hack; R. Jakob; P. N. Swarztrauber 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 | Article | MR 1177513 | Zbl 0756.76060

[40] G. A. Wimmer; C. J. Cotter; W. Bauer Energy conserving upwinded compatible finite element schemes for the rotating shallow water equations, J. Comput. Phys., Volume 401 (2020), p. 109016 | Article | MR 4038160 | Zbl 1453.65438

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

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