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.

Keywords: Compatible finite element methods; Hamiltonian mechanics; Poisson bracket; SUPG method

Author's affiliations:

^{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 DA - 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/ UR - https://doi.org/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://doi.org/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/

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