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Structure-Preserving High-Order Methods for the Compressible Euler Equations in Potential Temperature Formulation for Atmospheric Flows

This paper develops and validates structure-preserving, high-order discontinuous Galerkin methods for the compressible Euler equations in potential temperature formulation, featuring new numerical fluxes and source term discretizations that ensure conservation of entropy, total energy, and kinetic energy on general curvilinear meshes while maintaining well-balanced properties for atmospheric flows.

Original authors: Marco Artiano, Oswald Knoth, Peter Spichtinger, Hendrik Ranocha

Published 2026-07-31

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CC BY 4.0

Original authors: Marco Artiano, Oswald Knoth, Peter Spichtinger, Hendrik Ranocha

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine trying to predict the weather by watching a giant, invisible ocean of air swirl around the Earth. This isn't just a gentle breeze; it's a chaotic, high-speed dance of gas that obeys strict rules of physics. Scientists use powerful computers to solve these rules, known as the Euler equations, to forecast storms, jet streams, and climate patterns. However, these equations are tricky. They involve a delicate balance between how fast the air moves (kinetic energy), how hot it is (temperature), and how heavy it is (density). If a computer simulation gets even slightly out of whack—like a dancer losing their rhythm—it can create fake storms or make the air disappear, ruining the forecast. To fix this, researchers have developed special "structure-preserving" methods. Think of these as a set of golden rules that force the computer to always keep the total energy and the "order" of the air (entropy) exactly right, no matter how wild the simulation gets. This paper dives into a specific, popular way of describing this air using a concept called "potential temperature," which is like a secret code meteorologists use to track how air parcels heat up or cool down as they rise and fall.

The authors of this paper, Marco Artiano and his team, have built a new set of mathematical tools to make these weather simulations more accurate and stable, specifically for the "potential temperature" approach. They designed three special "numerical fluxes"—which are like the rules the computer uses to swap information between neighboring grid cells. These rules are engineered to ensure that the total energy and the thermodynamic entropy (a measure of disorder) are perfectly conserved, even on complex, curved maps of the Earth. A major hurdle they tackled was gravity. In the real world, gravity pulls air down, creating a delicate balance called hydrostatic equilibrium. If a computer model can't perfectly mimic this balance, it creates fake winds that drown out real weather. The team proved that by treating the gravity term in a specific, clever way (as a "non-conservative product"), their new methods can maintain this balance on curved grids, whether the air is in a constant temperature state or a constant potential temperature state.

They didn't just stop at theory; they tested their ideas with a series of challenging simulations. They ran their new methods against old ones using classic atmospheric scenarios, like waves crashing over mountains and the formation of baroclinic instabilities (the engines behind mid-latitude storms). In these tests, their new "potential temperature" formulation performed just as well as the traditional "total energy" method, but with the added benefit of being easier to implement in existing weather models. For instance, in a test simulating a "density wave" where pressure and speed are constant, their new fluxes conserved energy and entropy perfectly, whereas older methods struggled unless specific mathematical tweaks were applied. In more complex tests, like the "Taylor-Green vortex" (a swirling fluid benchmark), their methods maintained conservation over long periods without the simulation blowing up.

Crucially, the paper highlights a trade-off. To achieve perfect conservation of both energy and entropy while keeping the air pressure stable, the new methods require a specific type of mathematical averaging (using a "logarithmic mean") for how density is calculated. If they used a simpler average, the pressure could theoretically become negative, which is physically impossible. The authors show that by carefully choosing these averages, they can preserve the "pressure equilibrium," meaning the air doesn't spontaneously start moving when it should be still. They also demonstrated that their methods work on "curvilinear meshes"—grids that are warped to follow the Earth's curvature or mountain shapes—without losing their stability. In simulations of air flowing over mountains, their "non-conservative" approach to gravity produced smooth, realistic results, while older "point-wise" methods created annoying, fake ripples in the data.

Ultimately, this work suggests that the potential temperature formulation is a robust and viable alternative for high-resolution atmospheric modeling. By proving that these new fluxes can preserve energy, entropy, and hydrostatic balance on curved grids, the authors provide a solid foundation for building better weather and climate models. Their simulations show that these methods are not just theoretically sound but practically effective, offering a way to run long-term, high-resolution atmospheric simulations that stay stable and accurate without needing extra "tuning" or artificial fixes. While the paper focuses on the mathematical construction and numerical testing of these schemes, the results indicate a promising path forward for making our digital weather forecasts more reliable and physically consistent.

Technical Summary: Structure-Preserving High-Order Methods for the Compressible Euler Equations in Potential Temperature Formulation

Problem Statement
The paper addresses the development of structure-preserving numerical methods for the compressible Euler equations, specifically tailored for atmospheric flows. While Discontinuous Galerkin (DG) methods are widely used for their high-order accuracy and parallel scalability, they often require additional stabilization (e.g., filters, artificial viscosity) to maintain stability in long-time simulations, which can degrade accuracy. A specific challenge in atmospheric modeling is the choice of prognostic variables. While the total energy formulation is standard in general computational fluid dynamics (CFD), the potential temperature ( $\theta$ ) formulation is preferred in atmospheric models due to its theoretical role in meteorology and its ability to reduce the thermodynamic equation to a simple transport equation. However, constructing structure-preserving schemes (conserving entropy, total energy, and kinetic energy) for the potential temperature formulation, particularly on general curvilinear meshes with gravitational source terms, remains a complex task. Furthermore, existing well-balanced schemes often struggle to preserve hydrostatic equilibrium for both isothermal and constant potential temperature background states on curved meshes.

Methodology
The authors develop a framework based on the Discontinuous Galerkin Spectral Element Method (DGSEM) using flux-differencing approaches. The core of the methodology involves:

Formulation: The study focuses on two sets of equations: the traditional total energy formulation and the potential temperature formulation. In the latter, the prognostic variables are density ( $\varrho$ ), momentum ( $\varrho \mathbf{V}$ ), and potential temperature ( $\varrho \theta$ ).
Entropy and Energy Conservation: The authors derive novel two-point numerical fluxes designed to satisfy specific conservation properties:
- Entropy Conservation (EC): Ensuring the conservation of thermodynamic entropy.
- Total Energy Conservation (TEC): Ensuring the conservation of total energy (internal + kinetic).
- Kinetic Energy Preservation (KEP): Preserving kinetic energy in the absence of pressure gradients.
- Pressure Equilibrium Preservation (PEP): Maintaining constant pressure and velocity states.
  The paper demonstrates that for the potential temperature formulation, achieving EC, TEC, KEP, and PEP simultaneously requires the density flux to depend on pressure, a necessary trade-off identified through rigorous analysis of the jump conditions and entropy variables.
Source Term Discretization: To handle the gravitational potential term ( $\phi$ ), the authors treat the source term as a non-conservative product. They adopt a flux-differencing approach where the source term is discretized using specific mean values (logarithmic mean for isothermal states, Stolarsky mean for constant potential temperature states) to ensure the scheme is well-balanced.
Curvilinear Extension: The methods are extended to three-dimensional curvilinear meshes. The authors prove that the derived fluxes and source term discretizations maintain their structure-preserving properties (TEC, KPEP, and well-balancedness) under arbitrary coordinate mappings, provided the discrete metric identities are satisfied.
Implementation: The schemes are implemented in the Julia language using the Trixi.jl framework, employing SSPRK43 time integration.

Key Contributions

New Fluxes for Potential Temperature: The paper derives and characterizes new numerical fluxes (TEC, EC, and a combined ETEC flux) specifically for the potential temperature formulation. It proves that the ETEC flux is the unique flux satisfying EC, TEC, KEP, and PEP for constant velocity, though it necessitates a pressure-dependent density flux.
Generalized KPEP and TEC with Gravity: The authors generalize the Kinetic and Potential Energy Preservation (KPEP) property and establish conditions for total energy conservation in the presence of a geopotential term for both the total energy and potential temperature formulations.
Generalized Well-Balanced Schemes: A significant contribution is the development of well-balanced schemes capable of preserving hydrostatic equilibrium for both isothermal ( $T=$ const) and constant potential temperature ( $\theta=$ const) background states on curvilinear meshes. Previous works often limited this capability to Cartesian meshes or specific background states.
Non-Conservative Source Term Treatment: The paper rigorously applies the non-conservative product discretization to the gravitational source term within the DGSEM framework, showing how to rewrite the semi-discrete equations into a flux-difference form that preserves the desired structural properties.

Results
The authors validate the theoretical findings through a series of numerical experiments:

Conservation Properties: In the absence of gravity, the new fluxes demonstrate exact conservation of entropy and total energy up to machine precision for smooth flows (density wave) and turbulent flows (Taylor-Green vortex).
Well-Balancedness: On warped curvilinear meshes, the schemes preserve hydrostatic balance for both isothermal and adiabatic (constant $\theta$ ) background states. The error remains at machine precision for long integration times (500,000 timesteps), confirming the robustness of the non-conservative source term discretization.
Convergence: Convergence analysis for inertia-gravity waves shows that the non-conservative discretization of the source term yields significantly lower errors (up to four orders of magnitude in the vertical velocity) compared to point-wise discretization, particularly on curvilinear meshes. The schemes achieve the expected high-order convergence rates.
Atmospheric Scenarios: The methods are tested on classical mountain-wave benchmarks (linear hydrostatic, linear non-hydrostatic, and Schär mountain). The results show that the non-conservative formulations are more robust, exhibiting fewer spurious oscillations compared to point-wise source term discretizations. The potential temperature formulation performs comparably to the total energy formulation in these scenarios.
Baroclinic Instability: The paper initiates testing on the baroclinic instability benchmark (DCMIP 2016) on a cubed-sphere mesh, further validating the applicability of the methods to complex global atmospheric flows.

Significance
The paper claims that these developments are of great interest to the atmospheric modeling community. By providing structure-preserving methods specifically for the potential temperature formulation, the work facilitates the integration of high-order, stable, and accurate numerical schemes into existing atmospheric models without requiring a switch to total energy variables. The ability to preserve hydrostatic equilibrium on general curvilinear meshes for multiple background states is highlighted as a critical advancement for realistic atmospheric simulations, where complex topography and long-time integration are standard. The work bridges the gap between the theoretical advantages of the potential temperature formulation and the practical requirements of structure-preserving numerical discretization.

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