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arxiv: 2605.15616 · v1 · pith:TAQBMLTKnew · submitted 2026-05-15 · 🧮 math.NA · cs.NA

Entropy stable finite difference schemes for One-Fluid Two-Temperature Euler Non-equilibrium Hydrodynamics

Chetan Singh , Harish Kumar This is my paper

Pith reviewed 2026-05-19 22:34 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords entropy-stable schemesfinite difference methodstwo-temperature Euler equationsnon-conservative productsentropy inequalityhyperbolic PDEsnumerical hydrodynamics
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The pith

Reformulating non-conservative terms enables entropy-stable finite difference schemes for the one-fluid two-temperature Euler equations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops numerical methods for the one-fluid two-temperature Euler equations that model non-equilibrium hydrodynamics with separate pressures for electrons and ions. These equations contain non-conservative products that normally complicate entropy stability. The authors reformulate the system so the non-conservative part contributes nothing to entropy production. Entropy-conservative schemes then follow from Tadmor's relation on the conservative fluxes and higher-order central differences on the remaining terms. Entropy-dissipative terms built from entropy-scaled eigenvectors complete the scheme, and one- and two-dimensional tests confirm accuracy and stability.

Core claim

By reformulating the OFTT-Euler equations such that the non-conservative part does not contribute to the entropy, higher-order entropy-conservative numerical schemes are designed by using Tadmor's relation for the conservative part and higher-order central differences for the non-conservative parts. Entropy-dissipation terms are then designed using the entropy-scaled right eigenvectors of the conservative part, thereby deriving the entropy inequality for the entire system.

What carries the argument

Reformulation of the non-conservative products so they contribute nothing to entropy production, allowing Tadmor's relation and central differences to produce entropy-conservative schemes before dissipation is added.

If this is right

  • The schemes satisfy the entropy inequality for the full system.
  • Higher-order accuracy is achieved while preserving entropy stability.
  • The approach applies to both one- and two-dimensional non-equilibrium flow problems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This reformulation approach may extend to other hyperbolic systems that include non-conservative products.
  • The schemes could support more reliable long-time integrations in applications involving distinct electron and ion temperatures.

Load-bearing premise

The reformulation of the non-conservative products ensures they do not contribute to the entropy production in the system.

What would settle it

A simulation in which the discrete entropy fails to satisfy the expected inequality or shows growth instead of dissipation despite the proposed dissipation terms.

Figures

Figures reproduced from arXiv: 2605.15616 by Chetan Singh, Harish Kumar.

Figure 1
Figure 1. Figure 1: Double rarefaction Riemann problem: Plots of density, electron pressure, and ion pressure at 𝑡 = 2.0 using 2000 cells. We have also plotted total entropy change with time. 6.5. Lax Riemann problem In another test case from [13], we consider a Riemann problem on the computational domain of [−5, 5] with Neumann boundary conditions. The initial conditions are given by, (𝜌, 𝒗, 𝑝𝑒 , 𝑝𝑖 ) = { (0.445, 0.689, 0, 1… view at source ↗
Figure 2
Figure 2. Figure 2: Sod Riemann problem: Plots of density, electron pressure, and ion pressure at 𝑡 = 2.0 using 2000 cells. We have also plotted total entropy change with time. a two-dimensional test problem. The problem is initialized as follows: 𝜌(𝑥, 𝑦, 0) = 1 + 0.2 sin (𝑥 + 𝑦), ( 𝒗, 𝑝𝑒 , 𝑝𝑖 ) = (0.5, 0.5, 2, 2) . with 𝛾𝑒 = 𝛾𝑖 = 1.4. We use periodic boundary conditions on the domain (𝑥, 𝑦) ∈ [0, 2𝜋] × [0, 2𝜋]. For this setu… view at source ↗
Figure 3
Figure 3. Figure 3: Lax Riemann problem: Plots of density, electron pressure, and ion pressure at 𝑡 = 2.0 using 2000 cells. We have also plotted total entropy change with time [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Two-dimensional Riemann problem: Plots of density, electron pressure, and ion pressure at time 𝑡 = 0.75 using 400 × 400 cells for 𝛾𝑒 = 𝛾𝑖 = 1.4. In Figure (5), we have presented the numerical results for the second case (𝛾𝑒 = 1.4, 𝛾𝑖 = 1.67) at time 𝑡 = 0.59 on 400 × 400 cells. We have plotted the density, electron pressure, and ion pressure for all three schemes. We note that the solution structure has ch… view at source ↗
Figure 5
Figure 5. Figure 5: Two-dimensional Riemann problem: Plots of density, electron pressure, and ion pressure at time 𝑡 = 0.59 using 400 × 400 cells for 𝛾𝑒 = 1.4, 𝛾𝑖 = 1.67. given by, ( 𝜌, 𝒗, 𝑝𝑒 , 𝑝𝑖 ) = (0.1819, 0, 0, 0.220458, 0.220458) In the rest of the domain, we consider the states, ( 𝜌, 𝒗, 𝑝𝑒 , 𝑝𝑖 ) = { (1, 0, 0, 0.3571425, 0.3571425), if 0 ≤ 𝑥 < 4.5 (1.3764,−0.3336, 0, 0.560643, 0.560643), if 4.5 ≤ 𝑥 < 6.5. The gas const… view at source ↗
Figure 6
Figure 6. Figure 6: Two-dimensional Riemann problem: Plots of total entropy change with time using 400 × 400 cells for both the cases using O2es exp , O3es exp and O4es exp schemes. In contrast, the O4es exp scheme is more accurate than the O3es exp scheme, and both are able to capture small-scale structures effectively. We have also plotted the total entropy change with time for all the schemes. We note that all the schemes … view at source ↗
Figure 7
Figure 7. Figure 7: Shock-bubble interaction problem: Plots of density and total entropy evolution at time 𝑡 = 7.1571 using 400×144 cells. We have also plotted the total entropy change with time. 7. Conclusion In this work, we have designed higher-order entropy-stable finite difference schemes for the One-Fluid Two￾Temperature Euler Non-equilibrium Hydrodynamics system, which are a set of hyperbolic PDEs with non-conservative… view at source ↗
Figure 8
Figure 8. Figure 8: Shock-bubble interaction problem: Plots of density for O4es exp scheme using 400 × 144 cells at different times. part is symmetrizable and the non-conservative part does not contribute to the entropy evolution. Finally, an entropy￾stable discretization for the new conservative part is proposed, which results in the entropy stability of the complete discretization. We have presented extensive numerical resu… view at source ↗
Figure 9
Figure 9. Figure 9: Richtmyer-Meshkov instability: Plots of density, electron pressure and ion pressure at time 𝑡 = 17.46 using 800 × 800 cells. [4] Karl-Heinz A Winkler and Michael L Norman. Astrophysical radiation hydrodynamics, volume 188. Springer Science & Business Media, 2012. [5] Stefano Atzeni and Jürgen Meyer-ter Vehn. The Physics of Inertial Fusion: BeamPlasma Interaction, Hydrodynamics, Hot Dense Matter. Number 125… view at source ↗
Figure 10
Figure 10. Figure 10: Richtmyer-Meshkov instability: Zoomed plots of density, electron pressure and ion pressure at time 𝑡 = 17.46 using 800 × 800 cells. [12] M Keith Matzen, MA Sweeney, RG Adams, JR Asay, JE Bailey, GR Bennett, DE Bliss, DD Bloomquist, TA Brunner, RB e Campbell, et al. Pulsed-power-driven high energy density physics and inertial confinement fusion research. Physics of Plasmas, 12(5), 2005. [13] Jian Cheng. A … view at source ↗
Figure 11
Figure 11. Figure 11: Richtmyer-Meshkov instability: Plots of total entropy change with time using 800 × 800 cells for O2es exp , O3es exp and O4es exp schemes. [19] Dinshaw S Balsara, Deepak Bhoriya, Chetan Singh, Harish Kumar, Roger Käppeli, and Federico Gatti. Physical constraint preserving higher￾order finite volume schemes for divergence-free astrophysical mhd and rmhd. The Astrophysical Journal, 988(1):134, 2025. [20] Br… view at source ↗
read the original abstract

In this work, we consider the One-Fluid Two-Temperature Euler (OFTT-Euler) equations used for modeling non-equilibrium hydrodynamics. The model comprises a system of nonlinear hyperbolic partial differential equations with non-conservative products. The model decomposed the total pressure into two scalar components: one for electrons and one for ions. Our aim in this work is to design entropy-stable finite difference numerical schemes for the model. This is achieved by reformulating the equations such that the reformulated non-conservative part does not contribute to the entropy. Then, we design higher-order entropy-conservative numerical schemes by using Tadmor's relation for the conservative part and higher-order central differences for the non-conservative parts. Finally, we design the entropy-dissipation terms using the entropy-scaled right eigenvectors of the conservative part, thereby deriving the entropy inequality for the entire system. We present several test cases in one and two dimensions to demonstrate the accuracy and stability of the proposed schemes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper develops entropy-stable finite difference schemes for the One-Fluid Two-Temperature Euler (OFTT-Euler) equations, a hyperbolic system with non-conservative products. The central construction reformulates the non-conservative terms so they contribute zero to the continuous entropy balance, applies Tadmor's two-point entropy-conservative flux to the conservative part, uses higher-order central differences on the non-conservative part, and augments the scheme with dissipation constructed from entropy-scaled right eigenvectors of the conservative flux Jacobian. One- and two-dimensional numerical tests are shown to illustrate accuracy and robustness.

Significance. If the discrete entropy-conservation property for the composite scheme is established, the work would supply a practical, provably stable discretization for two-temperature non-equilibrium hydrodynamics, a model relevant to plasma and high-energy flow simulations. The combination of Tadmor's flux with a reformulation that isolates entropy-neutral non-conservative terms is a natural extension of existing entropy-stable methodology; the eigenvector-based dissipation is also standard. The numerical examples provide initial evidence of utility, but the absence of explicit discrete entropy proofs or quantitative entropy-error tables limits the immediate impact.

major comments (2)
  1. [§3] §3 (Scheme construction): The reformulation ensures the non-conservative products vanish from the continuous entropy equation, but the subsequent use of higher-order central differences on these terms does not automatically guarantee that the same cancellation holds at the discrete level. For the composite scheme to be entropy-conservative before dissipation is added, the central-difference operator applied to the reformulated non-conservative term (when contracted with the entropy variables) must produce a telescoping sum. No such discrete identity is stated or proved; without it the entropy-conservative claim for the non-dissipative part rests on an unverified assumption.
  2. [§4] §4 (Entropy dissipation): The dissipation matrix is built from the entropy-scaled right eigenvectors of the conservative flux Jacobian alone. It is not shown that this choice remains entropy-dissipative once the non-conservative terms (even if entropy-neutral) are present; a short calculation verifying that the dissipation term produces a non-positive contribution to the discrete entropy balance for the full system would close the argument.
minor comments (2)
  1. [Abstract] The abstract states that 'higher-order central differences' are used for the non-conservative part but does not specify the exact stencil or the order; adding this detail in §3 would improve reproducibility.
  2. [§5] Figure captions for the 2-D tests should explicitly state the grid resolution and the CFL number used; several captions currently omit these parameters.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The comments highlight important points regarding the discrete entropy properties, and we have revised the paper to address them explicitly.

read point-by-point responses

  1. Referee: [§3] §3 (Scheme construction): The reformulation ensures the non-conservative products vanish from the continuous entropy equation, but the subsequent use of higher-order central differences on these terms does not automatically guarantee that the same cancellation holds at the discrete level. For the composite scheme to be entropy-conservative before dissipation is added, the central-difference operator applied to the reformulated non-conservative term (when contracted with the entropy variables) must produce a telescoping sum. No such discrete identity is stated or proved; without it the entropy-conservative claim for the non-dissipative part rests on an unverified assumption.

    Authors: We agree that an explicit discrete identity is required to rigorously establish entropy conservation for the non-dissipative part of the scheme. In the revised manuscript we have added Lemma 3.2, which proves that the higher-order central difference operator applied to the reformulated non-conservative product, when contracted with the entropy variables, yields a telescoping sum. The proof follows from the specific algebraic structure of the reformulation (the non-conservative term is written as a difference of entropy potentials) and the summation-by-parts property of central differences. This closes the gap and confirms that the composite non-dissipative scheme is entropy conservative. revision: yes

  2. Referee: [§4] §4 (Entropy dissipation): The dissipation matrix is built from the entropy-scaled right eigenvectors of the conservative flux Jacobian alone. It is not shown that this choice remains entropy-dissipative once the non-conservative terms (even if entropy-neutral) are present; a short calculation verifying that the dissipation term produces a non-positive contribution to the discrete entropy balance for the full system would close the argument.

    Authors: We thank the referee for this observation. Because the non-conservative terms have been shown (both continuously and, in the revision, discretely) to contribute zero to the entropy balance, the dissipation term—constructed solely from the entropy-scaled eigenvectors of the conservative flux Jacobian—acts only on the conservative part. We have inserted a short calculation immediately after the definition of the dissipation matrix (now Equation (4.8)) demonstrating that its inner product with the entropy variables is non-positive for the full system. The cross terms involving the non-conservative operator vanish identically due to the entropy-neutral reformulation, so the sign of the dissipation contribution is preserved. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses external Tadmor relation and standard central differencing on reformulated terms

full rationale

The paper reformulates the OFTT-Euler system so that non-conservative products contribute zero to the continuous entropy balance, then applies Tadmor's two-point entropy-conservative flux to the conservative fluxes and higher-order central differences to the non-conservative terms before adding dissipation via entropy-scaled eigenvectors. This construction depends on the external Tadmor relation and the algebraic properties of central differencing rather than any quantity defined by the paper itself or any self-citation chain. No step reduces a claimed prediction or uniqueness result to a fitted input or prior self-result by construction. The scheme design therefore remains self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the domain assumption that the OFTT-Euler system admits an entropy function and that a reformulation exists making non-conservative products entropy-neutral; no free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption The one-fluid two-temperature Euler equations admit an entropy function and can be reformulated so that non-conservative products do not contribute to entropy production.
    This premise is invoked to justify applying Tadmor's relation to the conservative part and central differences to the non-conservative part.

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    reformulating the equations such that the reformulated non-conservative part does not contribute to the entropy... Tadmor's relation for the conservative part and higher-order central differences for the non-conservative parts

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