Energy conserving SUPG methods for compatible finite element schemes in numerical weather prediction

Abstract

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.

A comparison of variational upwinding schemes for geophysical fluids, and their application to potential enstrophy conserving discretisations

“…We end by noting that while the upwind stabilisation of the potential vorticity has been the main subject of this article, these approaches are equally applicable to other material variables that appear in the skew-symmetric operators of Hamiltonian systems, as has been previously demonstrated for the potential temperature in the case of the 3D compressible Euler equations [7,9].…”

“…We discretise the prognostic equations (6) in time using a discrete gradient method [14] for which the variational derivatives (7) are exactly integrated (to second order) between time levels n and n + 1 [6,13,15]. The resulting nonlinear problem is then solved using a Newton method with a constant in time approximate Jacobian [6,16], which at each nonlinear iteration k is solved for the updates…”

“…While the APVM results in an inconsistent symmetric positive definite correction to the potential enstrophy conservation equation that is always dissipative, the SUPG scheme results in a consistent correction that may act to either dissipate or inject potential enstrophy, resulting in a richer and more accurate turbulent profile for long time integrations. Both the APVM [4] and SUPG methods have previously been used in finite element models of geophysical systems for the energetically consistent transport of material quantities (potential vorticity [5,6] and potential temperature [7]).…”

Compatible finite element methods for geophysical fluid dynamics

“…Inspired by the observation of Gassmann and Herzog (2008) that upwinding for advected quantities (such as layer depth, density, temperature etc.) can be incorporated into an energy conserving scheme by simply ensuring that the antisymmetry is maintained in the bracket, Wimmer, Cotter and Bauer (2020) and Wimmer et al (2021) examined energy conserving tracer upwinding using upwind discontinuous Galerkin schemes and SUPG schemes, respectively. Wimmer et al (2020) considered upwind discontinuous Galerkin schemes for the rotating shallow water equations, applied to both the layer depth (which is in V 2 ℎ which allows arbitrary discontinuities between cells) and the velocity (which is in V 1 ℎ , and hence allows…”

“…This is an example where the reduction in energy error through space and time discretisation tricks is lost due to the energy error from doing a finite number of nonlinear iterations . Wimmer et al (2021) examined similar approaches when SUPG schemes are applied to the advected quantities. The motivation for this is that an SUPG scheme is required to stabilise the vertical transport of temperature when it is approximated in the W ℎ space proposed in Section 2.…”