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Entropy-Stable and Well-Balanced Discontinuous Galerkin Methods for the Compressible Euler Equations in Vector-Invariant Form

Marco Artiano, Kieran Ricardo, Oswald Knoth, Peter Spichtinger, Hendrik Ranocha

arXiv:2610.05820v1math.NAcs.NAphysics.ao-phphysics.comp-ph

TL;DR

The vector-invariant Euler equations with potential temperature are used in modern atmospheric dynamical cores, but the paper notes that a provably entropy-stable scheme for this formulation had not been proposed. It constructs energy- and entropy-conserving fluxes and an entropy-stable, well-balanced flux, with the properties extending to curvilinear DGSEM meshes.

  • Problem

    A provably entropy-stable scheme had not been proposed for the vector-invariant Euler equations with potential temperature, a formulation used in modern dynamical cores.

    Source: §1 · p. 2

  • Method

    The method discretizes nonconservative terms to construct two-point fluxes that conserve total energy and thermodynamic entropy, plus an entropy-stable flux for the DGSEM.

    Source: §9 · pp. 28–29

  • Results

    The entropy-stable semidiscretization is well-balanced on general curvilinear meshes for isothermal and isentropic backgrounds, with slip-wall or periodic boundary conditions.

    Source: §7 · p. 15, §9 · pp. 28–29

  • Takeaways & Limitations

    The conservation, well-balancing, and entropy-stability properties carry over to the high-order DGSEM on general curvilinear meshes.

    Source: §7 · p. 15, §9 · p. 29

    The positivity analysis covers fully discrete explicit schemes; positivity for vertically implicit schemes, typical of operational atmospheric models, is left for future work.

    Source: §4 · p. 9

Abstract

from arXiv · show

We develop structure-preserving methods for the compressible Euler equations with gravity in vector-invariant form and potential temperature as a prognostic variable within the flux-differencing discontinuous Galerkin spectral element method (DGSEM) framework. By discretizing the nonconservative terms as symmetric and antisymmetric products, we derive two-point numerical fluxes that conserve both the thermodynamic entropy and the total energy. Moreover, we design an entropy-stable numerical flux that is well-balanced for both isothermal and isentropic background states. All properties are shown to carry over to the high-order DGSEM on general curvilinear meshes. Several numerical examples confirm the theoretical findings and show the robustness and accuracy of the scheme for use in modern dynamical cores for atmospheric flows.

1 Introduction

Modern atmospheric dynamical cores need accurate, efficient, and robust methods, and DGSEM offers high-order accuracy and scalability for this setting. This work develops energy- and entropy-preserving, well-balanced fluxes for the vector-invariant Euler equations with potential temperature, then extends them to curvilinear DGSEM.

1 Introduction

  • Modern dynamical cores seek accurate, efficient, and robust atmospheric-flow methods, while spectral-element methods offer high-order accuracy and good scalability on modern hardware.Many operational and research cores use spectral-element methods.Source: §1 · p. 2
  • DG methods combine high-order accuracy and scalability with the ability to handle arbitrary geometries, motivating their use for compressible Euler equations.Flux-differencing DGSEM with suitable volume and interface fluxes can yield entropy-stable or entropy-conservative, well-balanced schemes.Source: §1 · p. 2
  • The choice of thermodynamic variables and momentum formulation for atmospheric dynamical cores remains open, with no established advantage for one formulation over the others.The paper focuses on the vector-invariant form with potential temperature as the thermodynamic variable.Source: §1 · p. 2
  • Prior DG work addressed entropy and energy properties or well-balancing for selected formulations and background states, but the present work targets both isothermal and isentropic states in its chosen formulation.Related constructions include schemes for isothermal backgrounds and methods extended to curvilinear meshes.Source: §1 · p. 2
  • The paper derives two-point fluxes for the vector-invariant Euler equations with potential temperature, including an entropy-stable flux that preserves isothermal and isentropic backgrounds.The work extends the construction to DGSEM on arbitrary curvilinear meshes.Source: §1 · pp. 2–3

2 Vector-invariant form of the compressible Euler equations

This section sets out the vector-invariant Euler system with density, velocity, and density-weighted potential temperature as prognostic variables, and develops fluxes using a generalized entropy condition. Its central construction uses flux freedom to obtain total-energy and thermodynamic-entropy conservation, with well-balanced properties developed subsequently.

2 Vector-invariant form of the compressible Euler equations

  • The vector-invariant Euler system uses density, velocity, and density-weighted potential temperature as prognostic variables, with an ideal-gas equation of state and geopotential.For smooth solutions, total energy and thermodynamic entropy are mathematical entropies for the system.Source: §2 · p. 3

2.1 Entropy analysis

  • The nonconservative products include vorticity, the pressure term, and the geopotential gradient; the derivation uses a specialized fluctuation form to construct numerical fluxes.The chosen third form is described as practical and sufficiently flexible for deriving entropy-conservative fluxes.Source: §2.1 · pp. 4–5
  • In these prognostic variables, thermodynamic entropy is convex but not strictly convex, while total energy is nonconvex for nonzero velocity.This loss of strict convexity prevents applying Barth’s theorem to uniquely determine an entropy-stability dissipation operator; the generalized Tadmor condition still characterizes entropy conservation or stability.Source: §2.1 · p. 5
  • The generalized Tadmor condition supplies the criterion used to derive total-energy- and thermodynamic-entropy-conserving fluxes despite the convexity issue.The authors apply the condition with equality for total-energy conservation.Source: §2.1 · p. 5

2.2 Total energy conservation

  • A total-energy-conservative flux allows any consistent discretization of potential temperature in the nonconservative pressure product.The theorem characterizes the flux subject to its stated condition; the density flux is chosen as the arithmetic mean for simplicity.Source: §2.2 · p. 6
  • The potential-temperature flux freedom distinguishes this construction from Ricardo et al.’s scheme and is used in the later joint conservation and well-balancing construction.The present formulation uses a split-form nonconservative vorticity term; Ricardo et al.’s scheme is recovered with pointwise vorticity and a central potential-temperature flux.Source: §2.2 · p. 6

2.3 Thermodynamic entropy conservation

  • The thermodynamic-entropy conservation condition involves density and potential-temperature fluxes, with the density flux remaining a degree of freedom.The potential-temperature flux is solved for to avoid pressure influencing the density flux, yielding an entropy-conservative flux.Source: §2.3 · p. 7
  • Degrees of freedom in the energy- and entropy-conservative fluxes can be combined to conserve both total energy and thermodynamic entropy.The combined characterization is presented alongside well-balancedness properties.Source: §2.3 · p. 7

3 Entropy stability

The section modifies an entropy-conservative flux to obtain entropy stability and pressure positivity, with a bounded potential-temperature flux. It also shows that upwinding this flux can yield entropy stability for either conservative or vector-invariant formulations.

3 Entropy stability

  • The proposed modification of the entropy-conservative flux yields an entropy-stable flux and guarantees pressure positivity.The usual approach adds dissipation to an entropy-conservative flux to satisfy the entropy condition as an inequality.Source: §3 · p. 7
  • Theorem 3.1 establishes entropy stability for the specified flux with any consistent density flux, while bounding the potential-temperature flux between interface values.The simpler potential-temperature flux is also entropy-stable, but does not guarantee that its value remains within that interval.Source: §3 · pp. 7–8
  • The proof uses bounds on the inverse logarithmic mean and a monotonicity property of integral-generated means.The section also notes that the simpler potential-temperature flux is entropy-stable but lacks the stated interval bound.Source: §3 · p. 8
  • When density and conservative potential temperature are prognostic variables, upwinding the potential-temperature flux by the inverse logarithmic mean achieves entropy stability in either formulation.The result applies to both conservative and vector-invariant forms of the compressible Euler equations.Source: §3 · p. 8

4 On the positivity of the density and pressure

The positivity analysis treats a fully discrete explicit scheme and uses the convexity of the admissible set to reduce SSP Runge–Kutta analysis to one explicit Euler step. Under a CFL condition, the scheme preserves density and pressure positivity; vertically implicit schemes are left for future work.

4 On the positivity of the density and pressure

  • The positivity analysis covers fully discrete explicit schemes; the corresponding analysis for vertically implicit schemes is left for future work.The paper notes that operational atmospheric models typically require vertically implicit time integrators.Source: §4 · p. 9
  • SSP Runge–Kutta positivity follows from analyzing one explicit Euler step because the method is a convex combination of such steps and the admissible set is convex.The analysis is presented for the first-order finite-volume scheme and can be extended to DG methods with similar arguments.Source: §4 · p. 9
  • Under the stated CFL condition, the explicit Euler finite-volume scheme preserves positivity of density times potential temperature, and hence pressure.The interface potential-temperature flux must lie between the neighboring values.Source: §4 · p. 9
  • The density positivity result follows by the same argument with potential temperature set to one.The theorem states that density remains positive under its CFL condition.Source: §4 · p. 10

5 Well-balanced schemes

The section derives flux conditions for preserving hydrostatic balance and identifies choices that preserve isothermal or isentropic states. A single flux achieves total-energy and entropy conservation while being well-balanced for both backgrounds.

5 Well-balanced schemes

  • Well-balancedness requires no spurious density or potential-temperature contributions at rest and a discrete analogue of hydrostatic balance in momentum.The condition is derived for nonconservative hyperbolic systems and recast under the assumption that kinetic-energy and vorticity terms vanish at steady state.Source: §5 · p. 10
  • The choice of density flux can use the Stolarsky mean for isentropic well-balancing or the logarithmic mean for isothermal well-balancing.The potential-temperature flux is the remaining degree of freedom used to enforce well-balancedness.Source: §5 · p. 11
  • Any consistent potential-temperature flux preserves the isentropic background, while preserving the isothermal background requires the inverse logarithmic-mean flux.The scheme of Ricardo et al. preserves the isentropic state because its potential-temperature flux is central and consistent.Source: §5 · pp. 11–12
  • Theorem 5.6 gives a flux that is total-energy-conservative, entropy-conservative, and well-balanced for both isothermal and isentropic background states.The paper states that this combination cannot be achieved with the conservative Euler formulation using potential temperature.Source: §5 · p. 12

6 An entropy-stable and well-balanced scheme

The section fixes the remaining numerical-flux choices to obtain an entropy-stable, well-balanced interface flux. The resulting flux supports both isothermal and isentropic background states.

6 An entropy-stable and well-balanced scheme

  • The selected numerical flux is entropy-stable and well-balanced for both isothermal and isentropic background states.The proof uses the established entropy-stability result and shows that only the discrete hydrostatic balance terms remain at rest.Source: §6 · p. 13
  • The entropy-stable Riemann solver also applies to the conservative-form equations, with the density mean chosen for the desired equilibrium: logarithmic for isothermal and Stolarsky for isentropic states.Source: §6 · p. 13

7 Extension to discontinuous Galerkin spectral element methods and curvilinear meshes

The finite-volume two-point fluxes are extended to a high-order flux-differencing DGSEM using SBP operators. Their structure-preserving properties hold on general curvilinear meshes under the stated metric and boundary conditions.

7 Extension to discontinuous Galerkin spectral element methods and curvilinear meshes

  • The two-point fluxes serve as building blocks for high-order flux differencing, and the fluctuation-form extension carries their structure-preserving properties to diagonal-norm SBP discretizations.Source: §7 · p. 14
  • The mapped equations use contravariant fluxes built from metric terms, with discrete metric identities required for the curvilinear-mesh result.Source: §7 · pp. 14–15
  • With the specified volume and surface fluxes, the DGSEM is entropy-stable and well-balanced on general curvilinear meshes, given discrete metric identities and slip-wall or periodic boundaries.Source: §7 · p. 15
  • For slip walls, a reflected exterior state makes the boundary entropy contribution vanish, regardless of the momentum flux.Source: §7 · pp. 15, 17
  • The scheme preserves two background states while requiring one nonlinear mean value; its antisymmetric nonconservative terms also permit computing only one triangular part of the derivative operator.Source: §7 · p. 17

8 Numerical results

The numerical results assess conservation and well-balancing alongside wave and atmospheric-flow tests. In the Taylor–Green vortex test, thermodynamic entropy and total energy are conserved to machine precision across the tested polynomial degrees.

8 Numerical results

  • The evaluation includes conservation tests and proceeds through well-balancing, inertia-gravity waves, mountain waves, breaking waves, critical-layer flow, and a cubed-sphere baroclinic-instability benchmark.The methods are implemented in Julia using Trixi.jl, with third-order, four-stage SSP Runge–Kutta time integration unless stated otherwise.Source: §8 · p. 17, §8.1 · pp. 17–18

8.1 Conservation of entropy and total energy

  • Both thermodynamic entropy and total energy are conserved up to machine precision for all polynomial degrees; their semidiscrete entropy rates remain at round-off level, in agreement with theoretical results.Source: §8.1 · p. 18

8.2 Well-balancedness

The tests examine well-balancing on a warped mesh and convergence for inertia-gravity waves. The isothermal rest state remains at machine precision, while the neutrally stratified case is sensitive to floating-point round-off.

8.2 Well-balancedness

  • On the warped mesh, the isothermal rest state stays at machine precision; neutral-state perturbations reach O(10^-4) in double precision, while double-double precision restores machine-precision well-balancing.The neutral-state growth is attributed to floating-point round-off; lowering its floor by more than ten orders of magnitude confirms this interpretation.Source: §8.2 · pp. 18–19

8.3 Inertia-gravity waves

  • For inertia-gravity waves, the scheme attains expected high-order convergence on regular and curvilinear meshes; nonlinear saturation appears at degree 4, especially in horizontal velocity.The reported rates on the curvilinear mesh agree with those obtained in prior studies.Source: §8.3 · p. 19

8.4 Numerical results with orography

Four idealized orographic-flow tests assess the vector-invariant formulation, spanning linear mountain waves, wave breaking, and critical-layer mixing.

8.4.1 Linear hydrostatic mountain

  • The hydrostatic mountain-wave normalized momentum flux evolves toward the expected value of 1, with velocity contours agreeing with reference results.The reported flux and velocity fields are compared with prior results and, respectively.Source: §8.4.1 · p. 22, Figure 6 · p. 23, Figure 7 · p. 23

8.4.2 Linear non-hydrostatic mountain

  • For the non-hydrostatic mountain wave, normalized momentum flux progressively approaches 1, and final-time velocity components agree with reference solutions.The momentum flux is normalized by 𝑚𝑁𝐻= 0.457 𝑚𝐻.Source: §8.4.2 · p. 22, Figure 8 · p. 24, Figure 9 · p. 24

8.4.3 Breaking gravity waves

  • The breaking lee wave steepens upward by 2.5 h and begins breaking by 3.0 h, generating small-scale instabilities.Beyond breaking, numerical-scheme dissipation governs the flow, so quantitative comparison is no longer meaningful.Source: §8.4.3 · p. 25, Figure 10 · p. 25

8.4.4 Critical layer

  • In the critical-layer test, the upward-propagating wave is absorbed below the critical level; small-scale structures and turbulence develop beneath it from 7.5 h onward.As the wave approaches the critical level, its vertical wavelength and vertical group velocity tend to zero, preventing it from reaching that level.Source: §8.4.4 · p. 26, Figure 11 · p. 26
  • The passive tracer follows the wave until breaking, then turbulence stirs it and transports tracer from the layer center toward its edges, broadening the layer through 12 h.The horizontally averaged tracer profile shows that mixing is still developing at the end of the simulation.Source: §8.4.4 · p. 26, Figure 12 · p. 27, Figure 13 · p. 27

8.5 Baroclinic instability

The baroclinic-wave simulations remain close between equal-degree-of-freedom discretizations and reference solutions through day 15. The high-order discretization dissipates less entropy, while the scheme remains robust and accurate in long-time simulations.

8.5 Baroclinic instability

  • The benchmark uses an axisymmetric deep-atmosphere balanced state perturbed by a localized northern-midlatitude zonal-wind perturbation, with three cubed-sphere configurations.Two configurations use equal degrees of freedom with different polynomial degrees.Source: §8.5 · p. 28
  • The coarse-grid simulation gives good results against refined-grid simulations, and plots generally agree with MCore, ENDGame, MPAS, and the entropy-stable DG scheme.Relative-vorticity fields also exhibit grid oscillations in the cited comparisons, attributed to their greater sensitivity to numerical dissipation than pressure and temperature.Source: §8.5 · p. 28
  • At equal degrees of freedom, the p=7 discretization dissipates less entropy than p=3; minimum surface pressure and maximum wind speed agree closely through day 15 with each other and reference solutions.Entropy decreases after day 8 as instability generates underresolved scales.Source: §8.5 · p. 28

9 Summary and conclusions

The method combines a potential-temperature formulation of the gravity-driven, vector-invariant Euler equations with flux-differencing DGSEM, deriving energy- and entropy-conserving fluxes and an entropy-stable flux. Its central well-balanced result is preservation of both isothermal and isentropic hydrostatic states on general curvilinear meshes.

9 Summary and conclusions

  • Antisymmetric discretization of nonconservative vorticity, pressure, and geopotential terms yields total-energy-conservative and entropy-conservative fluxes alongside an entropy-stable flux.Potential temperature is the prognostic thermodynamic variable and is conserved by construction.Source: §9 · p. 28
  • The same entropy-stable flux preserves both isothermal and isentropic hydrostatic backgrounds, locally and globally, without prescribing the background state a priori.The method is reported as the first to preserve both families simultaneously.Source: §9 · pp. 28–29
  • Any consistent potential-temperature flux preserves the isentropic background, whereas the isothermal background requires an inverse logarithmic-mean flux.The entropy-stable construction also guarantees density and pressure positivity for the first-order finite-volume scheme with SSP Runge–Kutta methods under a suitable CFL condition.Source: §9 · pp. 28–29
  • Slip-wall boundary conditions are entropy stable, and the properties extend to arbitrary polynomial degree on general curvilinear meshes.Entropy stability uses a single nonlinear logarithmic mean, at lower cost than total-energy- or potential-temperature-based conservative formulations.Source: §9 · p. 29
  • Numerical experiments spanning wave, mountain-flow, breaking-wave, critical-layer, and baroclinic-instability cases confirm the theoretical properties and the method’s robustness and accuracy.Figures 14 and 15 show surface pressure, 850 hPa temperature, and 850 hPa relative vorticity at days 8 and 10 for two baroclinic-instability configurations.Source: §9 · p. 29, Figure 14 · p. 29, Figure 15 · p. 30
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