Energy conserving upwinded compatible finite element schemes for the rotating shallow water equations

TL;DR

This work develops an energy-conserving, upwinded, compatible finite element discretisation for the rotating shallow water equations within a Hamiltonian (Lie-Poisson) framework. It introduces upwinding in the depth field $D$ via a discontinuous Galerkin formulation and in the velocity field $\mathbf{u}$ using a velocity-recovering operator $\mathbb{U}$ to preserve bracket antisymmetry, ensuring exact energy conservation when paired with a Poisson time integrator. Numerical experiments on planar and spherical domains demonstrate energy conservation to machine precision and show that depth upwinding enhances stability and reduces small-scale oscillations, while field development remains physically consistent with non-upwinded references. The authors also outline plans to extend the framework to the compressible Euler equations and to incorporate additional dissipation mechanisms such as SUPG for density/temperature in future work.

Abstract

We present an energy conserving space discretisation of the rotating shallow water equations using compatible finite elements. It is based on an energy and enstrophy conserving Hamiltonian formulation as described in McRae and Cotter (2014), and extends it to include upwinding in the velocity and depth advection to increase stability. Upwinding for velocity in an energy conserving context was introduced for the incompressible Euler equations in Natale and Cotter (2017), while upwinding in the depth field in a Hamiltonian finite element context is newly described here. The energy conserving property is validated by coupling the spatial discretisation to an energy conserving time discretisation. Further, the discretisation is demonstrated to lead to an improved field development with respect to stability when upwinding in the depth field is included.

Figures (7)

• Figure 1: Field development after 1000 time steps for periodic unit square test case with energy conserving setup including upwinding in $\mathbf{u}$. Left two images: depth fields, including upwinding and not including upwinding in $D$, respectively. Right two images: velocity fields, including upwinding and not including upwinding in $D$, respectively. Spatial resolution $32\times32$, $\Delta t =0.001$, with 4 Picard iterations for each time step. Depth field values 0.75 to 1.5 with contours every 0.05, velocity field magnitude values 0 to 1 with contours every 0.05.
• Figure 2: Left: Relative energy error development for Williamson 2 test case, using fully energy conserving upwinded discretisation \ref{['discrete_u_eqn']} - \ref{['discrete_D_eqn']}, mesh refinement level 5, $\Delta t =50$s, with 4 Picard iterations for each time step (window: first 500 time steps). Right: L2 depth field error averaged over the last 1000 time steps for refinement levels 3 to 5.
• Figure 3: Relative energy error developments for Williamson 5 test case. Left: non-energy conserving setup. Right: Energy conserving setup with (cyan) and without (dashed purple) upwinding in $D$. Mesh refinement level 5, $\Delta t =50$s, with 8 Picard iterations per time step.
• Figure 4: Potential vorticity fields after 25 days for Williamson 5 test case. Left to right: Non-energy conserving setup, energy conserving setup with upwinding in $D$, energy conserving setup without upwinding in $D$. Mesh refinement level 5, $\Delta t =50$s, with 8 Picard iterations per time step. 30 contours, scale: $-3 \times 10^{-8}$ (blue) to $3 \times 10^{-8}$ (red).
• Figure 5: Relative energy error developments for the Galewsky test case. Left: non-energy conserving setup. Right: Energy conserving setup with (cyan) and without (dashed purple) upwinding in $D$. Mesh refinement level 6, $\Delta t =30$s, with 8 Picard iterations for each time step.

Theorems & Definitions (6)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Remark 6