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Entropy-Stable and Physical-Constraint-Preserving DGSEM for Symmetry-Reduced General-Relativistic Hydrodynamics on Stationary Spacetimes

Guosheng Fu, Jian-Guo Liu

arXiv:2608.29229v1math.NAgr-qcphysics.comp-ph

TL;DR

The paper addresses the challenge of controlling nonlinear stability, physical admissibility, and spatially varying geometry in high-order DG methods for general-relativistic hydrodynamics. It develops an entropy-stable, physical-constraint-preserving DGSEM using local orthonormal variables and compatible flux–source discretization, recovering high-order accuracy in smooth flows and robustly resolving shocks, jets, and curved-background accretion.

  • Problem

    High-order DG methods for general-relativistic hydrodynamics must control nonlinear stability, physical admissibility, and spatially varying spacetime geometry.

    Source: §1 · p. 2

  • Method

    The method uses local orthonormal variables with compatible entropy-conservative fluxes, geometric source discretization, and conservative physical-constraint-preserving scaling.

    Source: §1 · pp. 2–3

  • Results

    The method is conservative, satisfies a semidiscrete entropy inequality, and recovers high-order accuracy and robust performance across smooth, shocked, jet, and stationary accretion flows.

    Source: §1 · p. 3, §7 · p. 48

  • Takeaways & Limitations

    The framework provides a practical high-order method for relativistic flows on curved stationary backgrounds.

    Source: §7 · p. 48

    The analysis is limited to prescribed stationary spacetimes, conforming affine tensor-product elements, periodic entropy analysis, and two active coordinates.

    Source: §1 · p. 4

Abstract

from arXiv · show

We develop an entropy-stable and physical-constraint-preserving discontinuous Galerkin spectral element method for symmetry-reduced general-relativistic hydrodynamics on prescribed stationary spacetimes. Using a local orthonormal transformation, the fluid variables are expressed in a form for which the relativistic hydrodynamic algebra and the admissible set are independent of the spatial metric, while the spacetime geometry enters through stationary coefficients. This separation allows entropy-conservative special-relativistic fluxes to be combined with a compatible discretization of the geometric source terms. On affine tensor-product meshes, the resulting DGSEM is conservative and satisfies a semidiscrete entropy inequality, while the transformed variables provide a convex framework for physical-constraint preservation. For practical stabilization, we use a geometry-only causal speed that is sufficient for both classical local Lax--Friedrichs entropy dissipation and the physical-constraint-preserving Lax--Friedrichs splitting. The fully discrete method combines this stabilization with SSP Runge--Kutta time stepping, oscillation elimination, and conservative local-orthonormal-state scaling. Numerical experiments cover smooth and strongly shocked special-relativistic flows, an axisymmetric jet, stationary Michel accretion, Schwarzschild Bondi--Hoyle flow, and four Kerr accretion cases. The results demonstrate the designed high-order accuracy in smooth regimes and robust performance for demanding relativistic flows on curved stationary backgrounds.

1. Introduction

This work develops an entropy-stable, physical-constraint-preserving DGSEM for symmetry-reduced GRHD on prescribed stationary spacetimes. Local orthonormal variables separate metric-independent fluid algebra from geometric coefficients, enabling compatible entropy, PCP, stabilization, and high-order numerical treatments.

Motivation

  • The method addresses nonlinear stability, physical admissibility, and spatially varying geometry in high-order DG discretizations of GRHD.It targets applications including black-hole accretion, relativistic jets, compact objects, and stellar collapse.Source: §1 · p. 2

Formulation

  • Local orthonormal transformation converts the metric-dependent admissible set into a metric-independent Lorentz-type cone while retaining stationary geometry in coefficients.This places the fluid algebra in standard SRHD form and supports convex PCP analysis for the weighted reduced state.Source: §1 · pp. 2–3

Entropy stability

  • Matched geometry representations in entropy-conservative fluxes and SBP source discretization yield a conservative DGSEM with a semidiscrete entropy inequality on affine tensor-product elements.The shared nodal geometry produces the exact discrete flux–source contraction required by the entropy analysis.Source: §1 · p. 3

Physical-constraint preservation

  • The PCP formulation derives a forward-Euler admissibility condition and a conservative scaling limiter enforcing density, cone, and relative-margin constraints under spatially varying geometry.The limiter preserves conservative cell averages and is adapted to the weighted evolved state.Source: §1 · p. 3

Stabilization

  • A geometry-only causal speed supplies sufficient dissipation for both classical local Lax–Friedrichs entropy stability and the PCP Lax–Friedrichs splitting.The fully discrete scheme combines this stabilization with SSPRK(3,3), OE–PCP stabilization, and conservative mappings for weighted problems.Source: §1 · p. 4

Numerical validation and scope

  • Experiments recover designed high-order accuracy for smooth SRHD and Michel flow and robust behavior for shocks, jets, Schwarzschild Bondi–Hoyle accretion, and four Kerr accretion cases.The analysis is restricted to prescribed stationary geometries, conforming affine tensor-product elements, and periodic semidiscrete entropy results.Source: §1 · p. 4

2. GRHD model and symmetry reduction

The paper formulates GRHD on prescribed stationary spacetimes in Valencia and locally orthonormal W-forms, then derives weighted two-dimensional reductions that retain suppressed-direction geometry and entropy structure. These reductions specialize to planar, axisymmetric, Schwarzschild, and equatorial Kerr model problems.

2. GRHD model and symmetry reduction

  • GRHD is posed on prescribed stationary spacetimes using the Valencia balance law, its entropy pair, and a local orthonormal transformation to the W-form.The transformation separates the standard special-relativistic physical flux from stationary geometry coefficients and frame-variation source terms.Source: §2 · p. 5, §2.1 · pp. 6–7

2.2. Two-dimensional W-form reductions

  • The two-dimensional reduction retains the suppressed-direction metric scale in the volume measure, producing a weighted conservative system with an additional geometric source and corresponding entropy law.For true symmetry reductions, the suppressed coordinate is independent; the equatorial Kerr restriction instead represents an infinitesimally thin plane with no retained polar dependence.Source: §2.2 · pp. 8–10

2.3. Model geometries

  • The model geometries comprise true planar, cylindrical, and Schwarzschild symmetry reductions, alongside the thin equatorial Kerr restriction.These cases supply the active geometry and suppressed metric factors used in the numerical model problems.Source: §2.3 · p. 10

2.3.1. Planar special relativistic hydrodynamics.

  • Planar SRHD is recovered from Cartesian Minkowski space with unit lapse, zero shift, identity spatial metric, unit weights, and vanishing sources.The resulting system is the standard multidimensional SRHD benchmark setting for high-order physical-constraint-preserving schemes.Source: §2.3.1 · p. 11

2.3.2. Axisymmetric SRHD with zero azimuthal velocity.

  • Axisymmetric SRHD in cylindrical coordinates has suppressed-direction scale w = r, Euclidean active geometry, and a nonzero weighted suppressed-direction source.The symmetry axis r = 0 requires special numerical treatment.Source: §2.3.2 · p. 11

2.3.3. Axisymmetric Schwarzschild Bondi–Hoyle accretion.

  • Axisymmetric Schwarzschild Bondi–Hoyle accretion uses horizon-penetrating Eddington–Finkelstein slices, radial active geometry, and weight w = r sin θ.Its reduced equations include intrinsic and suppressed-direction sources, while polar axes require special numerical care; a supersonic wind produces a characteristic downstream shock cone.Source: §2.3.3 · p. 12

2.3.4. Equatorial Kerr–Schild restriction.

  • The equatorial Kerr–Schild model restricts the three-dimensional equations to θ = π/2 with zero polar velocity, radial geometric coefficients, and suppressed scale w = r.Its entropy condition follows from the weighted two-dimensional reduction, while the four-component formulation excludes flows with nonzero suppressed velocity.Source: §2.3.4 · pp. 13–14
  • The analysis applies where w > 0; zeros of symmetry-orbit scales are handled separately using a regular weighted boundary state and discrete l’Hopital reconstruction.This treatment covers axis singularities in the cylindrical and Schwarzschild examples.Source: §2.3.4 · p. 14

3. Entropy-stable DGSEM on affine tensor-product meshes

The section constructs an entropy-stable DGSEM for symmetry-reduced GRHD on affine tensor-product meshes with stationary geometry and nodal states satisfying w > 0. It combines a geometry-compatible entropy structure, entropy-conservative volume discretization, compatible sources, and LLF boundary dissipation to obtain discrete conservation and entropy control.

3. Entropy-stable DGSEM on affine tensor-product meshes

  • The formulation assumes stationary prescribed geometry and w > 0 at solution nodes, while axis cells with w = 0 require separate treatment.The entropy analysis is restricted to positive w; boundary-zero axis cells are handled separately in implementation.Source: §3 · p. 15

3.1. Numerical conservative form and entropy structure

  • The conservative reformulation yields a local SRHD entropy pair whose source contraction isolates geometry-coefficient derivatives, with residual geometric terms entropy-orthogonal.Only derivatives of the complete coefficients wA and wC contribute to the entropy contraction, defining the source form used by the DGSEM.Source: §3.1 · p. 16

3.2. Single-element SBP-DGSEM

  • The single-element SBP-DGSEM uses nodal GLL interpolation, entropy-conservative SRHD two-point fluxes in local orthonormal variables, and a compatible nodal source satisfying the discrete entropy contraction.The spatial flux is the entropy-conservative flux of Duan and Tang, while complete coefficients A = wA and C = wC are averaged directly.Source: §3.2 · p. 18
  • Classical local Lax–Friedrichs surface fluxes use a geometry-only causal speed and combine with the volume flux and compatible source in the semidiscrete scheme.The resulting construction is followed by a proposition establishing single-element conservation and an entropy balance.Source: §3.2 · p. 19

The scheme (94) satisfies the conservative balance

The scheme satisfies a conservative balance through weighted summation, nodal contraction, and cancellation of the interior source term. On conforming affine tensor-product meshes, causal LLF stabilization yields semidiscrete entropy stability and weighted rest-mass conservation under admissibility and geometry assumptions.

The scheme (94) satisfies the conservative balance

  • Weighted GLL summation, nodal contraction, and cancellation of the interior source term establish the scheme’s conservative balance.The proof uses Q + Q^T = B and the source cancellation in (88).Source: The scheme (94) satisfies the… · p. 20

3.3. Global entropy stability

  • For admissible states sharing prescribed face geometry, the LLF flux with the causal speed satisfies the required entropy inequality.The same causal bound also supplies the Lax–Friedrichs splitting used for physical-constraint preservation.Source: §3.3 · p. 20
  • On conforming affine tensor-product meshes, the scheme is semidiscretely entropy stable and conserves global weighted rest mass under periodic boundaries.The result assumes positive weights, admissible nodal states, and single-valued prescribed geometry at shared faces.Source: §3.3 · p. 21

4. Physical-constraint-preserving analysis

The analysis establishes metric-independent physical admissibility and proves positivity-preserving mechanisms for fluxes, sources, forward-Euler averages, conservative scaling, and timestep rejection. These results apply under positive reduction weights and retain conservation while controlling proximity to the admissible-set boundary.

4. Physical-constraint-preserving analysis

  • The transformed-state PCP framework is independent of the spatial metric, provided the reduction weight satisfies w > 0 at the nodes.The transformed cone, Lax–Friedrichs splitting, convex-decomposition argument, and conservative scaling limiter all use this metric-independent formulation.Source: §4 · p. 21

4.1. Metric-independent admissible cone

  • The admissible set G is an open convex cone whose closure is also convex, equivalent for Gamma-law gas to positive density and pressure with subluminal velocity.The local orthonormal, intrinsic W-state, and evolved state share this same metric-independent cone.Source: §4.1 · pp. 21–22

4.2. Coordinate-direction Lax–Friedrichs splitting and source control

  • A coordinate-direction causal bound yields Lax–Friedrichs splitting for admissible states, while a source escape rate controls source updates when the source state leaves G.The splitting follows from rotational invariance, special-relativistic splitting, subluminal characteristic speeds, and cone scaling.Source: §4.2 · pp. 22–23

4.3. Forward-Euler cell-average PCP property

  • Under positive nodal weights, admissible nodal states, single-valued face geometry, and a sufficient timestep condition, forward Euler preserves admissibility of the DGSEM cell average.The proof expresses the update as a convex combination using the splitting lemma and source treatment.Source: §4.3 · p. 24

4.4. Conservative weight-compatible scaling

  • The three-stage conservative scaling preserves the cell average and enforces density, cone, and relative-margin constraints, retaining at least ηPCP of the anchor’s normalized cone margin.The relative-margin stage improves primitive-recovery robustness near the admissible-set boundary; the extension also covers axis cells with a simple normal zero.Source: §4.4 · p. 27

4.5. Stage-average rejection

  • Rejected stage averages are recomputed after halving the timestep, and Theorem 4.3 guarantees that finitely many halvings reach an admissible positive-timestep update.If the cell average is admissible but below a numerical floor, scaling rather than rejection preserves that admissible average.Source: §4.5 · p. 28

5. Fully discrete scheme

The fully discrete DGSEM combines entropy-conservative volume fluxes, geometry-matched sources, causal LLF face stabilization, conservative oscillation elimination, PCP scaling, SSPRK(3,3) stepping, and certification-based retries. The implementation uses geometry-only causal rates for stabilization, timestep control, and OE propagation.

5.1. Residual evaluation and causal face stabilization

  • The residual evaluation recovers primitives through a scalar pressure solve, assembles weighted sources from cached geometry derivatives, and uses classical LLF interfaces with a geometry-only dissipation coefficient.That coefficient supplies dissipation for both the classical-LLF entropy inequality and the PCP Lax–Friedrichs splitting.Source: §5.1 · p. 29

5.2. Conservative local-state oscillation elimination

  • At each Runge–Kutta stage, OE acts on the regular orthonormal conservative state rather than the densitized state, avoiding spurious variation from the stationary geometric factor.The method uses direction-matched cross-line normalization to improve indicators near shocks and under-resolved structures.Source: §5.2 · p. 29

5.2.2. Cross-line normalization

  • The cross-line normalization localizes amplitude scales by coordinate direction, preventing large near-horizon excursions from masking discontinuities elsewhere or across transverse directions.It preserves separate normalizations for each direction, screens noise-level components, and is invariant under componentwise affine rescaling.Source: §5.2.2 · p. 31

5.2.3. Conservative OE map

  • The elementwise conservative OE map damps increments while exactly preserving the stored conservative cell average.For Q2 elements, 0.01 ≤ sOE ≤ 0.05 gives robust performance, and sOE = 0.02 is used unless stated otherwise.Source: §5.2.3 · p. 33

5.3. PCP stage stabilization

  • After OE, PCP scaling enforces density, cone, and relative cone-margin constraints through the same local orthonormal state while preserving the conservative element average.The combined stage map preserves the stored conservative cell average of W.Source: §5.3 · p. 33

5.4. SSPRK (3, 3) stage certification and retry

  • Each raw SSPRK(3,3) stage is certified by requiring every element’s conservative average to satisfy (W_K)_D > 0 and q(W_K) > 0; failure triggers whole-step retry.Certification occurs before applying the OE–PCP stage map.Source: §5.4 · p. 34

5.5. Causal timestep

  • Stationarity allows geometry-only PCP rates λ_PCP,K to be precomputed and reused, while source restrictions beyond the nominal hyperbolic timestep are handled by certification and retry.The source escape rate is excluded from the nominal CFL timestep.Source: §5.5 · p. 34

5. Fully discrete scheme

  • The complete algorithm applies entropy-conservative volume fluxes, matched geometry sources, causal LLF face fluxes, conservative OE, PCP scaling, SSPRK(3,3), and certification with whole-step retry.Failed stage-average certification rejects the entire step; the same causal speed field supports face dissipation, explicit CFL control, and OE propagation.Source: §5 · p. 28, §5.5 · p. 34, §5.4 · p. 34

6. Numerical experiments

The fully discrete scheme is tested from smooth special-relativistic waves through shocks, jets, stationary Michel accretion, and Schwarzschild and Kerr flows. Across these regimes, it retains high-order accuracy, preserves admissibility, and reproduces expected relativistic flow structures under the stated Q2, LLF, PCP, and OE configurations.

6. Numerical experiments

  • The experiments use the fully discrete scheme with Q2 affine meshes, geometry-only LLF/PCP dissipation, CFL = 0.8, enabled PCP limiting, and OE damping for nonsmooth tests.The implementation is written in C++ using MFEM.Source: §6 · p. 35, Table 1 · p. 35

6.1. Two-dimensional smooth-wave accuracy

  • The smooth-wave study retains third-order convergence for every tested OE strength, with finest-mesh observed orders of 3.04–3.34 in L1, 3.04–3.35 in L2, and 3.04–3.31 in L∞.At sOE = 0.02, added finest-mesh density error is only about 5–6%, although stronger damping increases finite-resolution dissipation.Source: §6.1 · pp. 36–37, Table 2 · p. 36

6.2. Two-dimensional Riemann problems

  • The method preserves admissibility while resolving vortex-sheet roll-up, curved shocks, and contacts in three ultra-relativistic two-dimensional Riemann problems, agreeing well with reference results.All cases use Q2 elements on 400 × 400 meshes with CFL = 0.8 and sOE = 0.02.Source: §6.2 · p. 38, Figure 2 · p. 38

6.3. Shock–bubble interaction

  • The shock–bubble calculations sharply resolve transmitted and reflected waves, material interfaces, wakes, and vortical structures for both light and heavy bubbles, agreeing well with reference results.Increasing sOE broadens interfaces and smooths spiral tips; sOE = 0.02 balances interface resolution and regularization.Source: Figure 3 · p. 39, §6.3 · p. 40, Figure 4 · p. 40

6.4. Axisymmetric relativistic jet

  • The axisymmetric jet sharply resolves its bow shock, expanding cocoon, and shear-driven vortices through t = 100, with morphology agreeing with prior simulations and high-order calculations.The computation uses Q2 elements on a 360 × 1080 affine mesh with CFL = 0.8 and sOE = 0.02.Source: §6.4 · p. 41, Figure 5 · p. 42

6.5. Steady Michel accretion

  • Michel accretion retains high-order convergence, while OE reduces all tested steady-state errors and leaves only approximately 9–15% relative spread between configurations on the finest mesh.Without OE, E1 and E2 approach third order and E∞ reaches 2.73 on the finest refinement.Source: Table 3 · p. 43, §6.5 · p. 44

6.6. Axisymmetric Schwarzschild Bondi–Hoyle accretion

  • The Schwarzschild Bondi–Hoyle calculation develops a dense downstream shock cone, post-shock deceleration, and near-horizon acceleration, matching the reported M1a morphology qualitatively.The final state is evolved to t = 500M on a 256 × 128 Q2 mesh with sOE = 0.02.Source: §6.6 · p. 44, Figure 6 · p. 45

6.7. Equatorial Kerr–Schild Bondi–Hoyle accretion

  • Four Kerr accretion cases form tail shocks with spin-dependent distortion concentrated near the black hole, while large-scale cones agree closely with prior Kerr–Schild results.Boyer–Lindquist visualizations likewise show increasingly pronounced rotational distortion and qualitatively agree with prior results.Source: Figure 7 · p. 46, §6.7 · pp. 47–48, Figure 8 · p. 47

7. Conclusions

The paper presents a compatible entropy-stable, physical-constraint-preserving DGSEM for symmetry-reduced GRHD on stationary spacetimes, combining metric-independent transformed variables with compatible flux–source discretization. Fully discrete experiments show high-order accuracy and robust performance across smooth, shocked, jet, and accretion flows.

Contributions

  • The proposed DGSEM combines entropy stability and physical-constraint preservation through local orthonormal variables, compatible flux–source discretization, and geometry-only causal-speed dissipation.The construction yields flux–source cancellation, a semidiscrete entropy inequality, a metric-independent admissible cone, conservative weighted PCP limiting, and regular treatment of simple-zero symmetry axes on conforming affine tensor-product meshes.Source: §7 · p. 48

Numerical results

  • SSPRK(3, 3) time stepping with OE and PCP scaling produces high-order accuracy for smooth SRHD and Michel flows and robust resolution of shocks, jets, and Schwarzschild and Kerr accretion.The experiments support the framework as a practical high-order method for relativistic flows on curved stationary backgrounds.Source: §7 · p. 48

Future work

  • Future work includes compatible curvilinear-mesh and adaptive-refinement methods, nonconforming interfaces, time-dependent metrics, three-dimensional discretizations, and extensions to other relativistic systems.Mentioned extensions include magnetohydrodynamics, radiation hydrodynamics, and coupled spacetime–fluid evolution.Source: §7 · p. 48

Appendix A. Explicit source terms for the black-hole geometries … Appendix B.1. Temporal EC state

The appendices explicitly specialize the geometric source terms for Schwarzschild and Kerr black-hole models, then define local SRHD entropy-conservative states and the symmetric temporal state. These constructions complete the source specifications and provide the algebraic ingredients for entropy-conservative fluxes.

Appendix A.1. Axisymmetric Schwarzschild Bondi–Hoyle GRHD

  • For axisymmetric Schwarzschild Bondi–Hoyle GRHD, the source combines the shared intrinsic form with active metric-derived terms and the suppressed-direction contribution S⊥.Substitution of the evaluated components and orthonormal-factor derivative into (45), together with (46), completely specifies the source in (47).Source: Appendix A.1 · p. 49
  • In the Schwarzschild reduction, the active θ-momentum source vanishes because the active spacetime block is independent of θ.Source: Appendix A.1 · p. 49

Appendix A.2. Equatorial Kerr–Schild GRHD

  • For equatorial Kerr–Schild GRHD, the source combines the active Valencia source derived from metric and frame derivatives with the already explicit suppressed-direction term S⊥.Substitution into (45), together with S⊥ from (59), completely specifies the source in system (60).Source: Appendix A.2 · pp. 50–51
  • In the equatorial Kerr reduction, the azimuthal momentum source vanishes because the metric is independent of eϕ.Source: Appendix A.2 · p. 50

Appendix B. Local SRHD entropy-conservative states and fluxes

  • Appendix B defines the local SRHD entropy-conservative construction using logarithmic and arithmetic means, including a stable near-equal evaluation of the logarithmic mean.It also introduces ζ = ρ/p, Ka = Lbva, and the coefficient a in terms of ζln and Γ.Source: Appendix B · p. 51

Appendix B.1. Temporal EC state

  • The temporal entropy-conservative state is symmetric, and its entropy-variable jump relation uses the identity L2 − KaKa = 1.Source: Appendix B.1 · p. 51

Appendix B.2. Directional spatial EC flux

Appendix B.2 constructs a directional spatial entropy-conservative flux from local orthonormal-direction quantities. The assembled vector is symmetric and consistent, with formulas shared across two and three dimensions under admissibility and stable logarithmic-mean evaluation.

Appendix B.2. Directional spatial EC flux

  • For local flux direction a, the construction defines v_a, K_a, and L as directional ingredients.Source: Appendix B.2 · p. 52
  • The vector assembled from these components is symmetric and consistent.Source: Appendix B.2 · p. 52
  • The formulas retain the same form in two and three dimensions because contractions occur only over the orthonormal index, requiring admissible endpoint primitives and stable logarithmic means.Source: Appendix B.2 · p. 52

Appendix C. Flux-adapted geometry variables

Appendix C defines flux-adapted geometry variables from the active ADM geometry and separate reduction weight, establishing a smooth local coordinate transformation and its associated flux coefficients and differentials.

Appendix C. Flux-adapted geometry variables

  • The flux-adapted geometry state uses the active ADM geometry together with a separately prescribed physical reduction weight w.In two active dimensions, the construction adopts an upper-triangular Cholesky convention.Source: Appendix C · p. 52
  • For j2 > 0, the map X ↔ Y is a smooth one-to-one local change of active geometry coordinates, with complete flux coefficients formed only forward.The forward-only construction is the stated direction for assembling the complete flux coefficients.Source: Appendix C · p. 53

Appendix D. Causal bound for classical-LLF entropy stability

Appendix D proves that the geometry-only causal speed in (92) suffices to make the classical LLF flux entropy stable. The proof uses the convex entropy-variable domain and bounds the entropy defect along an admissible entropy-variable path.

Proof of Proposition 3.3

  • The appendix establishes the classical-LLF entropy inequality using the geometry-only causal speed at fixed stationary geometry.The argument combines the entropy-defect estimate with a bound based on local orthonormal SRHD characteristic speeds.Source: Appendix D · pp. 53, 56

Convex entropy-variable domain

  • For 1 < Γ ≤ 2, admissible entropy variables form the convex set V = {V = (V_D, V_m, V_E) : V_D ∈ R, −V_E > |V_m|}, exactly characterizing the admissible states.The set is the product of R and the interior of a Lorentz cone, so straight entropy-variable segments remain admissible.Source: Appendix D · p. 54

Entropy-defect and characteristic-speed bounds

  • Along the straight entropy-variable path between two admissible states, Lemma D.2 bounds the entropy defect, while rotational invariance identifies the relevant characteristic speeds with local orthonormal SRHD speeds.The Gamma-law assumption is used when deriving the characteristic-speed bound for every state on the path.Source: Appendix D · pp. 55–56

Appendix E. Axis cells with vanishing reduction weight

Appendix E develops a discrete treatment for axis cells where the reduction weight vanishes but the underlying local orthonormal state remains regular. It preserves the exact zero weighted state, conservative averages, and physical admissibility during oscillation elimination and PCP scaling.

Axis trace treatment

  • At symmetry-axis nodes, the stored densitized state is exactly W⋆ = 0, while the finite local orthonormal state is reconstructed from a normal derivative ratio.This applies when the reduction weight has a simple zero and the metric factor is smooth and positive.Source: Appendix E · p. 57

OE on an axis cell

  • OE uses the reconstructed finite local state and weighted-projection conventions that avoid singular division while preserving the conservative element average and exact stored axis value.No axis Riemann problem is formed because the weighted flux coefficients vanish on the axis.Source: Appendix E · pp. 57–59

Conservative axis repair

  • Before stabilization, conservative axis repair restores W⋆ = 0 by redistributing the defect among positive-weight nodes on the same nodal line while preserving the linewise quadrature moment and element average.The repaired state supports subsequent regular-trace construction and OE or PCP scaling.Source: Appendix E · p. 58

Trace-aware PCP scaling

  • Under simple axis zeros and after repair, trace-aware three-stage PCP scaling preserves W⋆ = 0 and the conservative state while placing positive-weight nodal states and regular axis traces in Gcert_K.If the states already satisfy the certification conditions, all scaling factors equal one.Source: Appendix E · p. 59
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