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Isothermal walls in the compressible Navier-Stokes equations: affine entropy, ballistic energy, and entropy-stable discrete formulations

Matteo Parsani, Rasha AlJahdali, Lisandro Dalcin

arXiv:2609.32249v1math.NAcs.NAmath-phmath.APmath.MPphysics.flu-dyn

TL;DR

For stationary no-slip walls at prescribed temperature, the canonical entropy has a heat-transfer boundary contribution that need not be nonpositive. The paper constructs an affine entropy and DG–SBP wall treatments that cancel or control that contribution. The reported analyses and calculations verify the continuous and semi-discrete balances under the stated assumptions.

  • Problem

    At a prescribed-temperature wall, the canonical entropy contribution qn/Tw can have either sign, leaving no homogeneous wall estimate from that entropy alone.

    Source: §1 · p. 2

  • Method

    The paper adds total-energy density divided by Tw to the canonical entropy and develops two DG–SBP SAT wall formulations for the adapted entropy.

    Source: §1 · p. 3

  • Results

    Both wall formulations give zero adapted-entropy wall contribution without penalty and a nonpositive contribution with penalty, without imposing an artificial relation between T− and Tw.

    Source: §4.3 · p. 10

  • Takeaways & Limitations

    The adapted entropy gives a homogeneous wall balance while retaining the canonical entropy structure and volume production at stationary no-slip isothermal walls.

    Source: §1 · p. 3

    The conditional L2-type estimate does not establish positivity preservation, global well-posedness, or regularity.

    Source: §4.4 · p. 10

Abstract

from arXiv · show

We develop entropy-stable isothermal-wall conditions for the compressible Navier--Stokes equations at continuous and semi-discrete levels. For a stationary, impermeable, no-slip wall at constant positive temperature, adding total-energy density divided by wall temperature to the canonical convex entropy cancels the wall contribution without suppressing heat transfer. This affine modification preserves the entropy Hessian, relative entropy, and volume entropy production. It is proportional to total ballistic-energy density and yields a conditional $L_2$-type bound based on the initial field under the stated assumptions. Two wall formulations use a common discontinuous Galerkin discretization with the summation-by-parts property and simultaneous approximation terms. Both give zero adapted wall contribution without a penalty and a nonpositive contribution with it. The complete-residual formulation penalizes the full wall-data residual and balances fluid energy through its complete numerical wall flux. The Fourier-energy-exact formulation combines an entropy-neutral skew correction with a momentum-projected penalty so that the complete numerical outward energy flux equals the one-sided numerical Fourier heat flux at every wall node and spatial resolution. Pointwise tests verify the local identities. Manufactured-solution studies with entropy-stable interior coupling demonstrate convergence rates consistent with order $p_s+1$ in the discrete $L_2$ norm, where $p_s$ is the solution polynomial degree. Thermal-relaxation calculations verify the semi-discrete energy and adapted-entropy balances. Three-dimensional calculations demonstrate balance closure in a closed domain, convergence of curved-wall traces, and applicability to a complex supersonic separated flow.

1. Introduction

Prescribed-temperature walls can contribute solution-dependent heat-entropy exchange, so canonical entropy alone does not provide a homogeneous wall estimate. The paper develops an affine entropy and compatible DG–SBP wall treatments to cancel this contribution while retaining heat transfer.

1. Introduction

  • At a prescribed-temperature wall, the canonical entropy contribution qn/Tw can have either sign, and a homogeneous estimate using that entropy has not previously been formulated with a compatible semi-discrete treatment.Earlier isothermal-boundary work reproduces the nonhomogeneous canonical heat-entropy exchange.Source: §1 · p. 2
  • The paper identifies a wall-temperature-adapted affine representative that preserves the canonical entropy Hessian, strict convexity, viscous symmetrizer, and volume entropy production while cancelling the stationary no-slip wall contribution.The homogeneous estimate therefore remains within standard convex-entropy theory.Source: §1 · p. 3
  • Two isothermal-wall SAT formulations in a common DG–SBP framework are boundary-entropy-conservative without wall dissipation and boundary-entropy-stable when wall dissipation is enabled.CR–SAT penalizes the complete wall-data residual; FEE–SAT uses an entropy-neutral skew correction and momentum-projected penalty to match the numerical outward energy flux to the one-sided Fourier heat flux.Source: §1 · p. 3
  • Matched one-dimensional manufactured-solution, thermal-relaxation, and three-dimensional studies test convergence, energy and adapted-entropy balances, curved-wall traces, and supersonic separated flow.The formulations require compatible entropy variables, numerical fluxes, viscous-gradient data, and SBP volume and face operators for adaptation to other discretizations.Source: §1 · p. 3

2. Governing equations and canonical entropy structure

This section sets out the compressible Navier–Stokes model, its canonical convex entropy structure, and the boundary contribution used in the entropy balance.

2.1. Compressible Navier–Stokes equations

  • The model assumes a fixed bounded domain and, for the stated initial-boundary-value problem, excludes body forces, volumetric heat sources, and radiative effects.Classical solutions are assumed to remain in the admissible state set; positivity preservation is not addressed.Source: §2 · p. 3, §2.1 · pp. 4–5

2.2. Outward-normal flux convention

  • The outward conductive heat flux is qn = −κ(T)∂nT, and at a stationary no-slip wall the outward total-energy flux equals qn.Positive qn denotes heat transported out of the fluid domain.Source: §2.2 · p. 5

2.3. Canonical mathematical entropy pair

  • The canonical mathematical entropy is strictly convex, with a symmetric positive-semidefinite viscous entropy structure and nonnegative viscous and thermal entropy production.The entropy Hessian is positive definite on the admissible set [4, 5, 11].Source: §2.1 · p. 5, §2.3 · p. 6

2.4. Canonical entropy balance

  • Contracting the equations with entropy variables and integrating over the domain yields an entropy balance whose boundary contribution must be controlled alongside the nonnegative volume dissipation.The entropy-pair compatibility identities support the entropy-flux potential used in the balance.Source: §2.4 · pp. 6–7, §2.3 · p. 6

3. Canonical entropy and total-energy balances at a stationary isothermal wall

At a stationary no-slip isothermal wall, the canonical entropy exchange and total-energy flux are both determined by conductive heat transfer. Their combination identifies a wall-balance quantity in which that exchange cancels.

3. Canonical entropy and total-energy balances at a stationary isothermal wall

  • At the stationary no-slip wall, the canonical entropy boundary contribution is qn/Tw, while the outward total-energy flux is qn.No penetration removes advective energy flux, and no slip removes viscous mechanical-work flux.Source: §3 · p. 7
  • Because qn can have either sign, the canonical entropy balance alone does not yield a homogeneous boundary-entropy estimate for prescribed isothermal wall data.The outward total-energy contribution to the rate of ∫Ω ρE/Tw dΩ is −qn/Tw, cancelling the canonical wall exchange when combined.Source: §3 · p. 7
  • Adding total energy divided by Tw to the canonical entropy balance cancels the solution-dependent wall heat flux exactly.This balance combination motivates the wall-temperature-adapted affine entropy representative introduced in Section 4.Source: §3 · p. 7

4. Wall-temperature-adapted affine entropy and entropy stability

The paper chooses a wall-temperature-adapted affine entropy by adding total-energy density divided by the constant wall temperature to canonical entropy. This yields homogeneous wall entropy control while preserving the canonical entropy structure and allowing conductive heat transfer.

4.1. Affine modification by conserved variables

  • Adding a constant multiple of conserved variables changes the entropy representative without changing its Hessian or volume entropy production.The affine invariance preserves strict convexity and viscous symmetrization.Source: §4.1 · p. 8

4.2. Entropy pair associated with the wall temperature

  • The affine representative preserves strict convexity, the entropy Hessian, viscous coefficients, and volume entropy production.Its energy entropy-variable component vanishes when T = Tw.Source: §4.2 · p. 9

4.3. Boundary entropy conservation at an isothermal wall

  • At a stationary no-slip isothermal wall, the adapted entropy contribution vanishes even when conductive heat crosses the wall.The heat exchange in the canonical entropy balance is cancelled by the total-energy flux divided by Tw.Source: §4.3 · pp. 9–10

4.4. Entropy stability with respect to STw and a conditional L2-type a priori bound

  • Under the stated assumptions, the global adapted-entropy estimate yields a conditional L2-type bound based on the prescribed initial field.The estimate does not establish positivity preservation, global well-posedness, or regularity.Source: §4.4 · p. 10

4.5. Ballistic free-energy interpretation

  • Scaling the adapted entropy by Tw gives the total ballistic-energy density, whose wall balance cancels the conductive heat-flux terms at T = Tw.The total ballistic energy inherits strict convexity on the admissible set.Source: §4.5 · p. 11

5. Semi-discrete DG–SBP formulation and two isothermal-wall SAT formulations

This section presents two isothermal-wall SAT formulations in a common semi-discrete nodal DG–SBP discretization and defines their shared stationary wall data. Both yield a zero adapted wall-entropy contribution without dissipation and a nonpositive one with dissipation, while CR–SAT uses the complete wall-data residual and FEE–SAT targets exact Fourier-energy flux through an entropy-neutral correction and momentum-only dissipation.

5. Semi-discrete DG–SBP formulation and two isothermal-wall SAT formulations

  • Both wall formulations satisfy the adapted wall balance algebraically: the wall contribution is zero without wall dissipation and nonpositive when it is enabled.The identities are also verified numerically up to floating-point roundoff in Section 6.Source: §5 · p. 11
  • CR–SAT penalizes the complete wall-data residual, while FEE–SAT additionally equates the complete outward energy flux to the one-sided numerical Fourier heat flux at finite resolution.Both corrections are constructed within the shared wall-data and SAT architecture.Source: §5 · p. 11

5.1. Semi-discrete DG–SBP setting

  • The nodal DG–SBP formulation is developed on conforming Cartesian affine elements; extensions require entropy-compatible metric approximations or SBP-compatible conservative entropy-stable interface operators.Under those conditions, the local CR–SAT and FEE–SAT wall identities retain the same algebraic form.Source: §5.1 · p. 12

5.2. Inviscid entropy-conservative flux differencing

  • The entropy-conservative SBP flux-differencing volume operator telescopes in the adapted entropy contraction to element-face terms, with no inviscid volume entropy production.Any two-point flux satisfying the stated symmetry, consistency, and entropy-conservation conditions may be used.Source: §5.2 · p. 13

5.3. Viscous auxiliary-gradient discretization and semi-discrete element equations

  • The LDG auxiliary equations approximate entropy-variable gradients, and the constant wall-temperature affine shift leaves those gradients unchanged.The same discrete gradient therefore approximates derivatives of both canonical and adapted entropy variables.Source: §5.3 · pp. 13–14

5.4. Conforming interior-interface SAT coupling

  • Paired LDG coupling is entropy-conservative, while optional inviscid dissipation and interior penalty yield a nonpositive interface contribution that vanishes when both are off.For constant Tw, the adapted and canonical interior SAT differences coincide, so the same interior dissipation applies.Source: §5.4 · p. 16

5.5. Common stationary isothermal-wall data and wall residual

  • The shared wall data use an inviscid mirror state that reverses only normal velocity, and the wall residual vanishes exactly when velocity is zero and T=Tw.The residual encodes no-slip and prescribed-temperature mismatch in entropy variables.Source: §5.5 · pp. 16–17
  • The common wall treatment uses a weak auxiliary-gradient correction rather than imposing zero wall residual directly; zero adapted entropy contribution does not require zero numerical energy flux.The common inviscid wall flux contributes zero to the pointwise adapted-entropy balance without requiring the residual to vanish.Source: §5.5 · pp. 17–18

5.6. Complete-residual SAT formulation

  • CR–SAT penalizes the complete wall-data residual, and its finite-resolution energy flux can include mechanical work and a penalty contribution.When exact wall data are attained, the flux reduces to the one-sided Fourier heat flux, which converges to the physical Fourier heat flux.Source: §5.6 · p. 19
  • CR–SAT is boundary-entropy-stable with respect to STw for every βw ≥ 0, and its βw = 0 limit is boundary-entropy-conservative.The n,w formulation is consistent with stationary no-slip isothermal boundary conditions.Source: §5.6 · pp. 19–20

5.7. Fourier-energy-exact SAT formulation

  • FEE–SAT requires the complete numerical outward energy flux to equal the one-sided numerical Fourier heat flux at every wall node and spatial resolution.Its viscous-flux correction cancels the one-sided mechanical-work term in the energy component.Source: §5.7 · p. 20
  • FEE–SAT uses an entropy-neutral skew correction for the energy adjustment and a momentum-projected penalty that adds dissipation without changing mass or total-energy flux.The energy adjustment contributes zero to the pointwise adapted-entropy balance.Source: §5.7 · pp. 20–21
  • FEE–SAT is boundary-entropy-stable for every βw ≥0 and boundary-entropy-conservative at βw = 0.The projected penalty acts only on momentum components.Source: §5.7 · p. 21
  • The FEE–SAT penalty does not directly dissipate a pure temperature mismatch when wall velocity is zero; temperature is weakly imposed through common wall data and the auxiliary-gradient SAT.When u− = 0, the projected residual vanishes even if T− ≠ Tw.Source: §5.7 · p. 21

E bF (V ),FEE

FEE–SAT makes the complete numerical outward energy flux equal the one-sided numerical Fourier heat flux pointwise, at every spatial resolution. Under the stated source-free assumptions, both wall formulations yield a nonincreasing adapted-entropy balance with dissipation, while fluid total energy may change through the open isothermal boundary.

E bF (V ),FEE

  • At every spatial resolution and for every βw ≥0, FEE–SAT equates the complete numerical outward total-energy flux with the one-sided numerical Fourier heat flux, without requiring exact wall traces.Fourier-energy exactness concerns this numerical flux identity, not exact evaluation of the continuum heat flux.Source: E bF (V ),FEE · p. 21
  • The FEE–SAT adapted wall contribution is zero when βw = 0 and nonpositive when βw ≥0.Its entropy-neutral skew correction adjusts the viscous flux, while a momentum-projected penalty supplies dissipation.Source: E bF (V ),FEE · p. 21

5.8. Global source-free balances and structural comparison

  • Under the stated source-free assumptions, either wall formulation gives a nonincreasing discrete adapted entropy when the remaining boundary contribution vanishes; multiplying by Tw gives the total-ballistic-energy balance.The theorem assumes admissible nodal states, constitutive conditions, a constant positive wall temperature, and βw ≥0.Source: §5.8 · pp. 22–23
  • CR–SAT balances fluid energy through its complete numerical outward flux, whereas FEE–SAT sets that flux to the one-sided numerical Fourier heat flux at every wall node.Fluid total energy need not remain constant because the isothermal wall is an open energy boundary.Source: §5.8 · p. 23
  • Neither global balance makes canonical fluid entropy alone satisfy a homogeneous non-increase statement.The balances include the wall exchange rather than eliminating it from canonical entropy.Source: §5.8 · p. 23

6. Numerical verification

Numerical verification tests the wall identities locally, assesses manufactured-solution accuracy and assembled energy and adapted-entropy balances, and extends the assessment to three-dimensional curved-wall and supersonic-flow cases. The multidimensional tests examine balance closure, curved-wall trace convergence, and operation in a shock-dominated separated flow; the inclined-cube case is an operational stress test, not an accuracy study.

6. Numerical verification

  • Randomized matched wall tests in two and three dimensions satisfy the scaled equality tolerance of 10^-11, and every entropy-stable test has a nonpositive wall contribution.The tests check local energy and entropy identities independently of mesh assembly and time integration.Source: §6 · p. 24

6.1. Common semi-discrete balance diagnostics

  • The instantaneous diagnostics test assembled semi-discrete cancellations: adapted entropy decreases with positive dissipation, while a CR–SAT Fourier-only mismatch is an intended finite-resolution flux difference, not an energy-balance failure.These instantaneous checks do not establish a fully discrete entropy identity for the time integrator.Source: §6.1 · pp. 25–26

6.2.1. Test problem and error measures

  • For both wall formulations in the complete entropy-stable configuration, halving ∆t at p_s = 3 and 64 cells changed temperature at t_f = 0.5 by at most 5.18 × 10−8 times nominal-step spatial error.Source: §6.2.1 · p. 27

6.2.2. Convergence comparison

  • In the entropy-stable manufactured-solution tests, both formulations show the p_s + 1 rate trend; the entropy-conservative configuration instead gives approximately p_s for odd degrees and p_s + 1 for even degrees.With an entropy-conservative wall and entropy-stable interior, both recover the p_s + 1 trend while retaining βw = 0; neither formulation is consistently more accurate across variables and degrees.Source: §6.2.2 · pp. 27–28, Figure 1 · p. 28, Table 2 · p. 29

6.3. One-dimensional thermal-relaxation energy and adapted-entropy balances

  • Across four thermal-relaxation cases, complete-energy and adapted-entropy residuals stay below 6.94 × 10^-18 and 7.98 × 10^-18, respectively; the CR–SAT Fourier-only mismatch reaches 1.78 × 10^-4, while the FEE–SAT mismatch stays at roundoff scale.The FEE–SAT complete-energy residual and Fourier-only mismatch coincide; the CR–SAT mismatch reflects finite-resolution mechanical-work and complete-residual penalty contributions.Source: §6.3 · p. 29, Figure 2 · p. 30

6.4. Three-dimensional test cases

  • The three-dimensional calculations assess adapted-entropy balance closure, curved-wall trace convergence, and operation in a supersonic separated flow.The wall identities remain pointwise on curvilinear and hp-nonconforming meshes; global balance extension requires entropy-compatible metric identities and suitable SBP-compatible, conservative, entropy-stable interface operators.Source: §6.4 · pp. 29–30

6.4.1. Closed-domain adapted-entropy balance

  • The closed-domain three-dimensional test uses stationary impermeable, no-slip isothermal walls at T_w = 1 on a curvilinear cube-and-spherical-cavity domain, with both formulations in boundary-entropy-conservative settings and optional numerical entropy dissipation disabled.Here, “closed domain” means no penetration, not thermal insulation; the test uses a cubic-geometry mesh with 384 curvilinear hexahedra and p_s = 3.Source: §6.4.1 · p. 31
  • In both closed-domain calculations, the adapted-entropy rate and viscous-volume dissipation remain nonzero with opposite signs, while their signed sum fluctuates around zero at a level consistent with roundoff.The reported maxima of absolute and scaled residuals provide numerical evidence of balance closure without added numerical entropy dissipation.Source: §6.4.1 · p. 32, Figure 4 · p. 33, Table 3 · p. 33

6.4.2. Curved-wall trace convergence in an annular pipe

  • Curved-wall errors decrease under refinement, with fine-grid rates close to orders two through five; CR–SAT errors are smaller than FEE–SAT errors at every tested degree and level.The first degree-two mesh pair is preasymptotic for both formulations; finer-grid trends are consistent with O(h^(p_s+1)).Source: Table 4 · p. 35, §6.4.2 · p. 35, Figure 6 · p. 36

6.4.3. Supersonic inclined-cube stress test

  • The inclined-cube calculation at t = 50 shows a detached bow shock, a heated windward region, and a separated wake; it uses no shock-specific treatment.Most velocity and temperature variation occurs before the first marked mesh interface in the wall-adjacent p_s = 4 region. The reported pointwise wall-trace discrepancies do not establish convergence of wall stress or heat flux.Source: §6.4.3 · pp. 37–39, Figure 8 · p. 38

7. Conclusions

The affine wall-adapted entropy cancels the continuous isothermal-wall contribution and preserves key entropy structures; two DG–SBP wall formulations reproduce the balance semi-discretely. Tests report the formulations’ balance properties, convergence, and three-dimensional behavior.

7. Conclusions

  • For a stationary, impermeable, no-slip wall at constant positive temperature, S_Tw = S + ρE/Tw cancels the canonical wall exchange and yields a conditional L2-type bound based on Q0.The affine representative preserves the entropy Hessian and relative entropy; scaling it by Tw gives the total ballistic-energy balance.Source: §7 · p. 39
  • CR–SAT and FEE–SAT reproduce the continuous adapted-entropy wall balance semi-discretely, with zero wall contribution without a penalty and a nonpositive quadratic contribution with one.CR–SAT penalizes the complete wall-data residual; FEE–SAT uses an entropy-neutral skew correction and momentum-projected penalty to match outward energy flux to one-sided Fourier heat flux at every wall node.Source: §7 · p. 39
  • Roundoff-level semi-discrete energy and adapted-entropy closure, plus rates consistent with p_s + 1 under entropy-stable interior coupling, are reported for both formulations.Thermal-relaxation tests also show that only FEE–SAT closes the Fourier-only energy balance at roundoff.Source: §7 · p. 39
  • Three-dimensional tests report closed-domain adapted-entropy balance closure, curved-wall trace convergence consistent with p_s + 1, and operation in a supersonic separated-flow configuration.The formulations’ principal distinction is their finite-resolution energy structure, not their observed convergence order.Source: §7 · p. 39

Appendix A.1. Thermodynamic reference state and admissibility

  • Changing the positive thermodynamic reference state leaves s(ρ,T) unchanged, while shifting the entropy zero leaves the entropy Hessian, relative entropy, and entropy production unchanged.For the stated gas, admissibility requires ρ > 0 and ρe > 0, equivalent to T > 0.Source: Appendix A.1 · p. 40

Appendix A.2. Constitutive and entropy-production identities

  • The appendix records the local entropy-production identity from the Newtonian stress, Fourier heat flux, and Gibbs relation.It defines outward heat flux as qn = −κ∂nT.Source: Appendix A.2 · p. 40

Appendix A.3. Canonical entropy balance

  • The canonical entropy is strictly convex on the admissible set, and the viscous contribution to its balance is nonpositive.The appendix identifies this as the canonical-balance result used later.Source: Appendix A.3 · p. 40

Appendix B. Stationary isothermal-wall balance

  • At a stationary isothermal wall, the adapted-entropy contribution cancels pointwise through the total-energy balance, reproducing the continuous mechanism used by the discrete treatments.No penetration removes inviscid entropy and advective energy fluxes, while no-slip removes viscous mechanical work.Source: Appendix B · p. 40

Appendix C.2. Derivation of the conditional L2-type bound

The conditional L2-type estimate follows from a lower bound on the adapted-entropy Hessian along admissible state segments. The appendix also gives the associated ballistic-energy balance and shows that the affine modification leaves relative entropy unchanged.

Appendix C.2. Derivation of the conditional L2-type bound

  • A positive lower bound on the adapted-entropy Hessian yields a conditional L2-type bound relative to Q0, and a uniform-in-time estimate requires that lower bound to remain uniform.Strict convexity alone does not guarantee a uniform lower bound.Source: Appendix C.2 · p. 41

Appendix C.3. Ballistic-energy identities

  • Scaling the adapted-entropy balance by Tw gives the ballistic-energy balance, whose wall contribution vanishes when T = Tw.The displayed wall identity is stated for a stationary no-slip wall.Source: Appendix C.3 · p. 42

Appendix C.4. Relative-entropy invariance under the affine modification

  • For every pair of admissible states, the affine contribution cancels between the entropy difference and tangent-plane term, leaving relative entropy unchanged.This identity is not needed for the conditional L2-type bound.Source: Appendix C.4 · p. 42

Appendix D. Discrete operators, viscous matrices, and implementation details

The appendix specifies Cartesian tensor-product SBP operators, viscous coefficient matrices, and an admissible numerical wall state. These constructions support the discrete entropy identities under the stated compatibility conditions.

Appendix D.1. Explicit Cartesian affine tensor-product SBP construction

  • The Cartesian affine construction defines physical volume and face quadrature, normals, and tensor-product derivative operators, with flux-differencing sums evaluated along coordinate pencils.The appendix notes that curvilinear and nonconforming extensions require additional metric and interface conditions.Source: Appendix D.1 · pp. 42–43

Appendix D.2. Viscous coefficient matrices

  • The dimension-independent viscous coefficient matrix is symmetric positive semidefinite under the stated condition and is equivalent to explicit three-dimensional matrices cited in the appendix.Nodal matrices are formed by evaluating the matrix at each element node and assembling block diagonally.Source: Appendix D.2 · p. 44

Appendix D.3. Admissible wall numerical state

  • The admissible wall state is constructed from the interior trace and prescribed Tw; the inverse entropy map is applied only to that admissible wall value.No reflected exterior entropy-variable state is used to evaluate the viscous coefficient matrices.Source: Appendix D.3 · p. 44

Appendix D.4. Fourier-energy-exact skew correction and projected penalty

FEE–SAT’s skew correction cancels the mechanical-work component of the one-sided viscous flux, while its projected penalty supplies the desired entropy and energy identities. The discrete entropy contraction combines SBP identities with interface and wall flux choices.

Appendix D.4. Fourier-energy-exact skew correction and projected penalty

  • The FEE–SAT skew correction cancels one-sided viscous mechanical work, and its projected penalty is positive semidefinite while having a zero energy row.For a pure temperature mismatch, the projected penalty vanishes; together these properties yield the local entropy and energy identities.Source: Appendix D.4 · pp. 44–45

Appendix D.5. Discrete entropy contraction

  • In the discrete entropy contraction, the wall flux choices yield the CR–SAT or FEE–SAT wall identities, while complementary LDG trace weights cancel and leave only the IP term.The interior inviscid terms use Tadmor’s identity, and the characteristic dissipation is positive semidefinite.Source: Appendix D.5 · pp. 45–46

Appendix D.6. Nodewise wall implementation sequences

The nodewise sequences construct shared wall data first, then branch into the complete-residual and Fourier-energy-exact formulations. Both finish by assembling SAT data and monitoring formulation-specific wall identities.

Appendix D.6. Nodewise wall implementation sequences

  • Both formulations begin with the outward normal, prescribed temperature, and interior trace, recover primitive and entropy variables, and check positive density and temperature.They then construct the inviscid mirror state and flux, wall entropy state, residual, and wall-adjacent coefficient data.Source: Appendix D.6 · p. 46
  • The shared auxiliary-gradient SAT uses −n_i r_w, after which its one-sided trace is used to form the conservative wall data.The penalty parameter selects the boundary entropy-conservative variant when βw = 0 and the entropy-stable variant when βw > 0.Source: Appendix D.6 · p. 46
  • CR–SAT constructs its viscous wall flux and conservative SAT data, then monitors numerical outward energy flux and the pointwise adapted-entropy wall contribution.FEE–SAT instead forms a skew entropy-neutral correction and a momentum-projected penalty before constructing its viscous flux and conservative SAT data.Source: Appendix D.6 · p. 46
  • The FEE–SAT sequence stacks its conservative SAT data and monitors the pointwise Fourier-energy identity residual.The implementation instructions also call for stacking the shared face data into the assembled face vectors.Source: Appendix D.6 · p. 46

Appendix E. Closed-form fluxes and detailed pointwise verification

The appendix provides closed-form entropy-conservative fluxes and verifies their algebraic properties. Randomized tests report residuals consistent with double-precision roundoff.

Appendix E. Closed-form fluxes and detailed pointwise verification

  • The appendix presents closed-form entropy-conservative fluxes and randomized verification procedures, while its SBP identity checks cover solution degrees 1 through 12.The closed-form flux tests do not use a tensor-product cell, LGL grid, or solution degree.Source: Appendix E · p. 47

Appendix E.1. Closed-form entropy-conservative fluxes

  • Positive scalar inputs define arithmetic and logarithmic means, with the equal-input logarithmic mean continuously extended to that input.Vector arithmetic means are taken componentwise.Source: Appendix E.1 · p. 47
  • The Ismail–Roe and Chandrashekar flux constructions use two-state auxiliaries, including pressure averages and squared-sound-speed and enthalpy auxiliaries.These auxiliaries need not describe a single common thermodynamic state; β in the Chandrashekar flux is an inverse-temperature variable, not a penalty parameter.Source: Appendix E.1 · p. 47
  • Both fluxes are symmetric, consistent, and satisfy the Tadmor identity for the canonical entropy pair, with the same identity holding for the affine representative STw.The subscript m labels momentum, while m = d + 2 denotes state-vector dimension.Source: Appendix E.1 · p. 47

Appendix E.2. Closed-form flux test

  • The randomized test samples positive-density and positive-temperature primitive states across three dimension–gas-constant cases, evaluating 10,000 state pairs per case and flux.The tests use double precision; the scaled identity tolerance is 2 × 10^-12.Source: Appendix E.2 · p. 48, Table E.5 · p. 48
  • The maximum scaled Tadmor residual lies between 4.14 × 10^-16 and 1.17 × 10^-15, and symmetry error is zero in all six cases.For R = 287, unscaled consistency and reflected-wall errors rise to order 10^-11, while scaled residuals remain at roundoff.Source: Appendix E.2 · pp. 48–49

Appendix F. Thermal-relaxation supplementary tests

The appendix checks instantaneous and stage-integrated balances for four thermal-relaxation wall settings. The complete energy balance closes when the spatial balance closes at each Runge–Kutta stage, while the Fourier-only diagnostic differs by formulation.

Appendix F. Thermal-relaxation supplementary tests

  • The complete stage-integrated energy balance closes when the spatial energy balance closes at every Runge–Kutta stage.This uses the same stage fluxes and weights as the solution update and does not require exact time integration of the continuous wall flux.Source: Appendix F · p. 52
  • The Fourier-only quantity is a balance residual for FEE–SAT and a finite-resolution mismatch for CR–SAT.For FEE–SAT, the complete energy residual and Fourier-only mismatch coincide; CR–SAT can have a nonzero Fourier-only mismatch.Source: Appendix F · pp. 52, 54
  • The RK4 update does not automatically satisfy an analogous stage-integrated identity for nonlinear canonical or adapted entropy.No relaxation correction is used in these thermal-relaxation calculations.Source: Appendix F · p. 52
  • Across four wall settings, Table F.8 reports maximum absolute instantaneous rates, maximum absolute stage-integrated increments, and signed final fluid-energy change.Instantaneous quantities are rates; stage-integrated quantities are energy increments formed with the solution update’s Runge–Kutta stage weights.Source: Table F.8 · p. 52, Appendix F · p. 52
  • The reported histories cover both wall formulations with βw = 0 and βw = 0.25, including energy rates, energy residuals, Fourier-only mismatch, and adapted-entropy rates.Figure F.9 compares the energy rate with reversed-sign Fourier and complete outward-flux sums; its overlap does not resolve residuals or Fourier-only mismatch.Source: Appendix F · pp. 53–54, Figure F.9 · p. 53, Figure F.10 · p. 54

Appendix G.1. SBP-quadrature error norms and observed rates

The appendices define the SBP-quadrature error and rate conventions, point to supplementary convergence data, and specify reference conventions and data availability for multidimensional tests. The closing reflection describes the authors’ research process and views on scientific practice.

Appendix G.1. SBP-quadrature error norms and observed rates

  • The reported errors are evaluated at final time tf = 0.5, and the normalized SBP-quadrature errors use a discrete domain measure equal to the domain measure for the affine meshes, to roundoff.The nodal L∞ norm includes element-local LGL nodes, with duplicated interface nodes treated as distinct DG degrees of freedom.Source: Appendix G.1 · p. 56

Appendix G.2. Supplementary convergence data

  • Observed rates use successive errors for each variable, norm, degree, and wall formulation, are assigned to the finer level, and are omitted at ℓ = 0.Supplementary Material S1 contains paired volume-error and rate tables, density and velocity L2 plots, and annular-pipe wall-error tables.Source: Appendix G.1 · p. 56, Appendix G.2 · p. 56

Appendix H. Reference conventions for multidimensional tests

  • The three-dimensional test descriptions use nondimensional variables and coordinates unless explicitly identified as dimensional.The appendix states that its reference conventions apply to the tests in Subsection 6.4.Source: Appendix H · p. 56, Appendix H.1 · p. 56

Appendix H.1. Reference scales

  • The reference scales define dimensional variables using reference length, density, velocity, and temperature, with gas and transport parameters specified from dimensional quantities.The reference sound speed is a0 = √γRT0.Source: Appendix H.1 · p. 56

Data availability

  • Numerical data, verification and post-processing tools, figure and table generators, and reproducibility instructions will be made available upon request.The simulations use in-house Fortran 2023 and mixed-language Fortran, C, and C++ codes.Source: Data availability · p. 57

A decade of work and a changing landscape of discovery

  • The authors describe mentorship, repeated derivation and checking, and human discussion as shaping their understanding and standards for scientific work.They report not consulting generative AI during the work and argue for accessible data, checkable arguments, and transparent AI-use descriptions.Source: A decade of work and a changing… · pp. 57–58
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