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Energy and entropy conserving compatible finite elements with upwinding for the thermal shallow water equations

Tamara A. Tambyah, David Lee, Santiago Badia

TL;DR

This paper introduces a compatible finite element discretisation for the thermal shallow water equations that preserves energy and quadratic tracer entropies through a reformulation enabling discontinuous thermodynamic variables and a Poisson time integrator. It distinguishes centred fluxes, which conserve both energy and entropy, from upwinded fluxes, which conserve energy and damp entropy, and demonstrates long-time stability in turbulent regimes via a new linearised Jacobian. To address the intrinsic temporal entropy drift of cubic Casimir invariants under the base Poisson integrator, the authors implement a constrained Lagrange-multiplier approach to enforce exact entropy conservation when needed. Numerical experiments confirm convergence under h-p refinement and show robust long-time behavior for both flux types, with the constrained formulation enabling precise entropy conservation in time. The work highlights the need for Poisson integrators capable of preserving higher-order Casimirs in non-canonical Hamiltonian systems and lays groundwork for extensions to spherical geometries and more complex atmospheric models.

Abstract

In this work, we develop a new compatible finite element formulation of the thermal shallow water equations that conserves energy and mathematical entropies given by buoyancy-related quadratic tracer variances. Our approach relies on restating the governing equations to enable discontinuous approximations of thermodynamic variables and a variational continuous time integration. A key novelty is the inclusion of centred and upwinded fluxes. The proposed semi-discrete system conserves discrete entropy for centred fluxes, monotonically damps entropy for upwinded fluxes, and conserves energy. The fully discrete scheme reflects entropy conservation at the continuous level. The ability of a new linearised Jacobian, which accounts for both centred and upwinded fluxes, to capture large variations in buoyancy and simulate thermally unstable flows for long periods of time is demonstrated for two different transient case studies. The first involves a thermogeostrophic instability where including upwinded fluxes is shown to suppress spurious oscillations while successfully conserving energy and monotonically damping entropy. The second is a double vortex where a constrained fully discrete formulation is shown to achieve exact entropy conservation in time.

Energy and entropy conserving compatible finite elements with upwinding for the thermal shallow water equations

TL;DR

This paper introduces a compatible finite element discretisation for the thermal shallow water equations that preserves energy and quadratic tracer entropies through a reformulation enabling discontinuous thermodynamic variables and a Poisson time integrator. It distinguishes centred fluxes, which conserve both energy and entropy, from upwinded fluxes, which conserve energy and damp entropy, and demonstrates long-time stability in turbulent regimes via a new linearised Jacobian. To address the intrinsic temporal entropy drift of cubic Casimir invariants under the base Poisson integrator, the authors implement a constrained Lagrange-multiplier approach to enforce exact entropy conservation when needed. Numerical experiments confirm convergence under h-p refinement and show robust long-time behavior for both flux types, with the constrained formulation enabling precise entropy conservation in time. The work highlights the need for Poisson integrators capable of preserving higher-order Casimirs in non-canonical Hamiltonian systems and lays groundwork for extensions to spherical geometries and more complex atmospheric models.

Abstract

In this work, we develop a new compatible finite element formulation of the thermal shallow water equations that conserves energy and mathematical entropies given by buoyancy-related quadratic tracer variances. Our approach relies on restating the governing equations to enable discontinuous approximations of thermodynamic variables and a variational continuous time integration. A key novelty is the inclusion of centred and upwinded fluxes. The proposed semi-discrete system conserves discrete entropy for centred fluxes, monotonically damps entropy for upwinded fluxes, and conserves energy. The fully discrete scheme reflects entropy conservation at the continuous level. The ability of a new linearised Jacobian, which accounts for both centred and upwinded fluxes, to capture large variations in buoyancy and simulate thermally unstable flows for long periods of time is demonstrated for two different transient case studies. The first involves a thermogeostrophic instability where including upwinded fluxes is shown to suppress spurious oscillations while successfully conserving energy and monotonically damping entropy. The second is a double vortex where a constrained fully discrete formulation is shown to achieve exact entropy conservation in time.

Paper Structure

This paper contains 21 sections, 6 theorems, 43 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Thermal shallow water equations
  3. Continuous system
  4. System invariants
  5. Semi-discrete formulation
  6. Notation
  7. Finite element spaces
  8. Finite element approximation
  9. Semi-discrete conservation
  10. Fully discrete formulation
  11. Poisson integrator
  12. Temporal discretisation
  13. Temporal conservation
  14. Constrained formulation
  15. Quasi-Newton approach
  16. ...and 6 more sections

Key Result

Proposition 3.1

Solutions of the continuous system are consistent with the semi-discrete formulation.

Figures (5)

  • Figure 1: Convergence of the zonal thermogeostrophic test case showing relative $L^2$ error between initial and final solutions for (a) the centred scheme and (b) the upwinded scheme (hard signum, $\epsilon=10^{-4}$), with 16, 32, 64, 128 spatial elements in each case.
  • Figure 2: Evolution of buoyancy $b_h$ for the thermal instability case study where (a,c,e,g) relate to the centred scheme and (b,d,f,h) correspond to the upwinded scheme (hard signum, $\epsilon = 10^{-4}$). Snapshots are shown at $t=25,50,75,100$ where time is unit-less. Parameters: $n_s=192$ spatial elements, $p=1$ spatial finite elements, $\mathrm{CFL} = 0.2$.
  • Figure 3: Conservation errors for the thermal instability case study. (a,b,c) show normalised values of energy, mass and entropy, and (d) illustrates the change in entropy due to forcing terms \ref{['eq: dsdt semi 2']}. Each figure compares the centred and upwinded schemes for different approximations of signum and $\epsilon$ values. The vertical axis is logarithmic and the horizontal axis shows unit-less time. Parameters: $n_s=192$ spatial elements, $p=1$ spatial finite elements, $\mathrm{CFL} = 0.2$.
  • Figure 4: Evolution of buoyancy $b_h$ for the double vortex case study, simulated with the centred scheme. Snapshots are shown at $t=0.75,1.5,2,3,4,5$ where $t$ is unit-less. Parameters: $n_s=192$ spatial elements, $p=1$ spatial finite elements, $\mathrm{CFL} = 0.2$.
  • Figure 5: Conservation errors for the double vortex case study. (a,b,c) show normalised values of energy, mass and entropy, and (d) illustrates the change in entropy due to forcing terms \ref{['eq: dsdt semi 2']}. Each figure compares the centred (blue) and constrained schemes (orange). The vertical axis is logarithmic and the horizontal axis shows unit-less time. Parameters: $n_s=64$ spatial elements, $p=1$ spatial finite elements, $\mathrm{CFL} = 0.2$.

Theorems & Definitions (12)

  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Proposition 4.1
  • proof
  • ...and 2 more