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Generalized Tadmor Conditions and Structure-Preserving Numerical Fluxes for the Compressible Flow of Real Gases

This paper generalizes Tadmor's entropy-conservative flux condition to accommodate non-convex secondary structures in compressible real gas flows by introducing a "null-consistency" requirement, enabling the systematic construction of structure-preserving numerical fluxes for arbitrary equations of state and demonstrating their efficacy in complex transcritical flow simulations.

Original authors: Robin Klein, Benjamin Sanderse, Pedro Costa, Rene Pecnik, Ruud Henkes

Published 2026-03-17

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CC BY 4.0

Original authors: Robin Klein, Benjamin Sanderse, Pedro Costa, Rene Pecnik, Ruud Henkes

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 you are trying to simulate a storm, a jet engine, or the flow of liquid carbon dioxide in a rocket engine on a computer. To do this, you break the fluid down into a grid of tiny boxes (like a 3D chessboard) and calculate how the fluid moves from one box to the next.

The problem is that fluids are messy. When you use a computer to simulate them, tiny rounding errors can pile up, causing the simulation to explode, become unstable, or give you physically impossible results (like negative pressure or energy appearing out of nowhere).

This paper introduces a new set of "rules of the road" for these computer simulations, specifically designed for Real Gases (fluids that act weirdly under extreme heat and pressure, unlike the simple "Ideal Gases" taught in high school physics).

Here is the breakdown using simple analogies:

1. The Problem: The "Leaky Bucket"

In physics, certain things are always conserved. If you have a bucket of water, the amount of water stays the same unless you pour it in or out. In fluid dynamics, Mass, Momentum, and Energy are like that water.

However, there are other "hidden" quantities, like Entropy (a measure of disorder) and Kinetic Energy (movement energy). In a perfect, smooth world, these follow strict rules too. But when we chop the world into a computer grid, standard math methods often "leak" these hidden quantities.

The Analogy: Imagine trying to count the number of people in a stadium by asking them to pass a message down a line. If the message gets garbled at every step, your final count is wrong. Standard computer methods often garble the "message" of energy and entropy, leading to unstable simulations.

2. The Old Solution: Tadmor's "Magic Formula"

A mathematician named Tadmor figured out a special way to calculate the flow between grid boxes so that Entropy (disorder) is perfectly preserved. It's like a magic formula that ensures no information is lost as the message passes down the line. This works great for simple gases (Ideal Gases).

The Catch: This magic formula relies on the math being "convex" (shaped like a bowl). But Real Gases (like supercritical CO2 used in rockets) are weird. Their math is "non-convex" (shaped like a saddle or a crumpled paper). When you try to use Tadmor's old formula on these weird gases, the math breaks down. The "magic formula" hits a singularity (a division by zero) and the simulation crashes.

3. The New Solution: Generalized Tadmor Conditions

The authors of this paper asked: "Can we upgrade Tadmor's magic formula to work on these weird, crumpled shapes?"

They developed a Generalized Framework. Think of it as upgrading a GPS app. The old GPS only worked on flat, straight highways (Ideal Gases). The new GPS works on mountainous, winding roads with cliffs and tunnels (Real Gases).

Key Concept: The "Null-Consistency" Rule
When dealing with these weird gases, the math has "blind spots" (places where the gradient is zero or singular).

The Analogy: Imagine driving a car where the steering wheel sometimes locks up. If you try to turn the wheel when it's locked, you crash.
The Fix: The authors discovered a new rule called Null-Consistency. It's like a safety sensor that says, "Hey, the steering wheel is locked right now. Don't try to turn it; instead, just coast straight."
In math terms, this ensures that when the simulation hits a "blind spot" in the gas's behavior, the numerical flux (the calculation of flow) doesn't break. It forces the math to behave correctly even when the gas properties are acting up.

4. The Tool: "Discrete Gradients"

To build this new formula, they used a tool called Discrete Gradients.

The Analogy: Usually, to know how steep a hill is, you need a perfect map (a smooth equation). But for Real Gases, the map is often a black box—you can't see the equation, you can only measure the height at specific points.
The Solution: Discrete Gradients are like a smart surveyor. Even if you don't have the full map, the surveyor can look at two points, measure the difference, and estimate the slope in a way that guarantees the math stays consistent. This allows the authors to build their "magic flux" without needing to know the exact, messy equation of the gas beforehand.

5. The Result: The "KEEP-DG" Flux

They combined all this to create a new numerical flux (a method for calculating flow) called KEEP-DG (Kinetic-Energy and Entropy Preserving with Discrete Gradients).

What it does: It simulates super-hot, super-pressurized gases (like in a rocket engine) without crashing.
Why it's special:
1. No Leaks: It perfectly conserves entropy and kinetic energy, just like the real world does.
2. No Crashes: It avoids the "division by zero" errors that killed previous methods when gases got too hot or dense.
3. Robust: They tested it on a "Taylor-Green Vortex" (a complex, swirling turbulence test). While other methods crashed or became unstable, their new method kept chugging along, preserving the physics perfectly.

Summary

Think of this paper as inventing a new type of bridge for crossing a river.

Old bridges (standard methods) worked fine for calm, shallow water (Ideal Gases).
The new river (Real Gases) has rapids, whirlpools, and sudden drops. Old bridges collapsed.
Tadmor's old magic was a good design for calm water, but it failed in the rapids.
This paper designs a bridge with flexible joints (Generalized Conditions) and sensors (Null-Consistency) that automatically adjust to the turbulence. It ensures that even in the wildest, most chaotic fluid flows, the laws of physics (energy and entropy) are never broken.

This allows engineers to simulate next-generation rockets and engines with much higher confidence and accuracy.

1. Problem Statement

The paper addresses the challenge of designing stable, robust, and accurate numerical schemes for the compressible flow of real gases (non-ideal fluids), particularly in supercritical and transcritical regimes.

Context: Standard discretizations of conservation laws (like the Euler or Navier-Stokes equations) often suffer from stability issues in complex thermodynamic regimes where properties change rapidly (e.g., near pseudo-phase transitions).
Limitation of Existing Methods:
- Entropy Conservation: Tadmor's framework provides a systematic way to construct entropy-conserving fluxes, but it relies on the secondary quantity (entropy) being convex. Many physically important secondary quantities, such as kinetic energy and pressure, are non-convex.
- Non-Convexity Issues: For non-convex quantities, the Hessian matrix can be singular, and gradients may not be injective. This breaks standard construction methods and introduces compatibility constraints that are not automatically satisfied.
- General Equations of State (EoS): Real-gas simulations often rely on complex, implicit, or arbitrary EoS (e.g., Helmholtz energy-based models). Standard flux construction methods often require closed-form analytical expressions for thermodynamic potentials, which are unavailable for general EoS.
Goal: Develop a unified theoretical framework and a practical construction method for numerical fluxes that preserve both entropy and kinetic energy (and potentially other secondary structures) for compressible flows with arbitrary equations of state, without introducing singularities.

2. Methodology

The authors propose a two-pronged approach: a theoretical generalization of Tadmor's conditions and a methodological innovation using discrete gradients.

A. Theoretical Framework: Generalized Tadmor Conditions

The authors extend Tadmor's algebraic flux condition from convex entropies to general secondary structures (quantities evolving via both conservative and non-conservative terms).

Secondary Structures: Defined as a tuple $(q, F_q, G_q, c_q)$ where $q$ is the secondary quantity, $F_q$ is the conservative flux, and $c_q \nabla G_q$ represents non-conservative "work" terms.
Generalized Compatibility: They derive primal and dual compatibility conditions at the differential level. The dual condition relates the numerical flux, the gradient of the secondary quantity ( $\xi$ ), and the residual function ( $\psi_q$ ).
Null-Consistency (Key Innovation):
- For non-convex $q$ , the Hessian $H_q$ can be singular. The authors identify that when $\Delta \xi = 0$ (i.e., the gradient difference vanishes between two states), the numerical flux term drops out of the compatibility equation.
- To ensure the equation remains valid, the remaining terms (involving the work term) must satisfy a specific constraint. The authors term this Null-Consistency.
- They define Weak Discrete Null-Consistency as a necessary condition for the existence of a well-posed numerical flux. Without this, the flux condition becomes ill-posed when the gradient difference is zero.

B. Methodological Innovation: Discrete Gradient Operators

To solve the generalized Tadmor condition for arbitrary EoS (where functions are defined implicitly or partially), the authors employ Discrete Gradient Operators.

Jump Expansions: Instead of requiring closed-form expressions for thermodynamic potentials, they use discrete gradients to expand jumps ( $\Delta$ ) in the Tadmor condition.
Systematic Construction: By expressing jumps like $\Delta(p\beta)$ or $\Delta(g\beta)$ (where $\beta = 1/T$ ) using discrete gradients, they can construct linear systems for the numerical flux components without needing explicit algebraic forms of the EoS.
Handling Singularities: This approach naturally avoids the singularities found in previous methods (like the "KEEP-S" flux) that occur when neighboring cells have identical temperatures or densities.

C. Application: The KEEP-DG Flux

The framework is applied to the Euler equations with an arbitrary EoS to derive a new numerical flux, termed KEEP-DG (Kinetic-Energy- and Entropy-Preserving with Discrete Gradients).

Flux Formulation: The flux is constructed to satisfy the generalized Tadmor condition for kinetic energy and the standard Tadmor condition for entropy simultaneously.
Key Feature: It uses a density mean and internal energy mean derived from discrete gradients of thermodynamic potentials, ensuring positivity and consistency.

3. Key Contributions

Generalized Tadmor Framework: A rigorous extension of Tadmor's entropy-conserving framework to non-convex, non-conserved secondary quantities (like kinetic energy), introducing the concept of Null-Consistency as a fundamental structural requirement.
Discrete Null-Consistency: The identification that for non-convex structures, the numerical work term must satisfy specific constraints (null-consistency) to ensure the existence of a solution to the flux condition.
Arbitrary EoS Compatibility: The demonstration that Discrete Gradient Operators allow for the systematic construction of structure-preserving fluxes even when thermodynamic potentials are defined implicitly or via complex, arbitrary EoS.
Singularity-Free Flux (KEEP-DG): The derivation of a new numerical flux that is both entropy-conserving and kinetic-energy-preserving for general EoS, which removes the singularities present in recent state-of-the-art fluxes (specifically the KEEP-S flux) that occur at isothermal conditions.
Open-Source Solver: Development and release of HelmEOS2, a flexible, JAX-based, multi-GPU solver capable of handling Helmholtz-based equations of state for real-gas flows.

4. Results

The authors validated the KEEP-DG flux through a series of numerical experiments using the HelmEOS2 solver on supercritical/transcritical test cases (using CO2 as the working fluid).

Transcritical Density Wave (1D):
- Demonstrated second-order convergence ( $O(\Delta x^2)$ ) for mass, momentum, and energy.
- Showed machine-precision conservation of total entropy.
- Showed near-perfect conservation of total kinetic energy (deviations were due to numerical round-off in the work term, not the flux design).
- Confirmed the scheme handles real-gas effects without instability.
Inviscid Taylor-Green Vortex (TGV) - Stability Test:
- Compared KEEP-DG against the singular KEEP-S flux and the KEEP-Q flux.
- KEEP-S Failure: The singular flux crashed due to instability when many cells required switching to a non-singular replacement (triggered by isothermal conditions).
- KEEP-DG Success: The proposed flux remained stable throughout the simulation (up to $50 t_c$ ) and maintained entropy conservation to machine precision, proving the singularity-free nature of the design.
Viscous Transcritical TGV (3D):
- Simulated fully turbulent flow with state-of-the-art EoS (Kunz-Wagner) and transport models (viscosity, heat conductivity).
- Demonstrated that the scheme accurately reproduces the kinetic energy balance (dissipation vs. production terms) even with a modest grid ( $512^3$ ).
- Showed that the scheme is robust for both low-Mach (nearly incompressible) and higher-Mach ($Ma=0.3$) regimes in transcritical conditions.

5. Significance

Robustness in Extreme Conditions: The work provides a critical tool for simulating high-fidelity compressible flows in engineering applications involving supercritical fluids (e.g., rocket propulsion, cryogenic systems, and geothermal energy), where traditional ideal-gas assumptions fail and standard numerical schemes become unstable.
Theoretical Unification: It unifies the treatment of entropy and kinetic energy preservation under a single theoretical umbrella, clarifying the structural requirements (null-consistency) needed for non-convex quantities.
Practical Implementation: By leveraging discrete gradients, the method bypasses the need for complex, case-specific algebraic derivations for every new EoS, making it highly adaptable to industrial-grade thermodynamic models.
Elimination of Singularities: The removal of singularities in kinetic-energy-preserving fluxes for general EoS is a major step forward, eliminating the need for ad-hoc switching mechanisms that can compromise conservation properties.

In summary, this paper establishes a new standard for structure-preserving numerical methods in real-gas dynamics, offering a mathematically rigorous, singularity-free, and computationally efficient approach to simulating complex, high-speed, supercritical flows.