Invariant domain preservation for hybrid point-value and cell-average discretizations of hyperbolic equations on general meshes

Shengrong Ding; Shumo Cui; Remi Abgrall; Kailiang Wu

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Invariant domain preservation for hybrid point-value and cell-average discretizations of hyperbolic equations on general meshes

Shengrong Ding, Shumo Cui, Remi Abgrall, Kailiang Wu

arXiv:2609.10476v1Sep 9, 2026math.NAcs.NAphysics.comp-ph

TL;DR

Hybrid discretizations must preserve admissibility for conservative cell averages and possibly non-conservative point values, but continuous traces alone do not control the conservative update. The paper combines a transformed point-value evolution with CAD-based admissible flux traces and an explicit local stability condition. It proves a continuous-trace obstruction and reports third-order accuracy on smooth tests alongside admissibility preservation in scalar and compressible Euler benchmarks.

Problem
Hybrid methods couple conservative cell averages and non-conservative point values, so admissibility cannot be reduced to a single convex update or standard finite-volume argument.
Source: §1 · p. 2
Method
The framework uses a barrier–Legendre point transform and CAD/GQL with local a priori scaling to construct admissible trace states for the conservative update.
Source: §1 · p. 3, §2 · p. 4, §2.2 · p. 6
Results
The paper proves that continuous boundary traces used with a single-state physical flux cannot ensure a conservative IDP update for any prescribed CFL number, and reports third-order smooth-solution accuracy and admissibility preservation in scalar and Euler tests.
Source: §1 · p. 3, §7 · p. 44
Takeaways & Limitations
The obstruction identifies the need for an additional local IDP flux mechanism to control the conservative cell-average update.
Source: §7 · p. 44
For strict interior admissibility in compressible Euler, the theorem requires finite wave-speed bounds, which entails avoiding vacuum states.
Source: §2.6 · p. 11

Abstract

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This paper presents a unified invariant-domain-preserving (IDP) framework for hybrid discretizations of hyperbolic conservation laws, including active flux and PAMPA methods, in which cell averages are updated conservatively while cell-boundary point values evolve under a possibly non-conservative operator. The main challenge is to preserve admissibility for these two coupled sets of states under a single local stability condition, without adding evolved degrees of freedom or relying on post-update repairs. For the point-value update, we introduce an admissibility transform based on a barrier--Legendre map for convex admissible interiors described by concave constraints, and prove that the inverse map is globally defined and Lipschitz continuous on the relevant sets. For the conservative cell-average update, we establish a structural obstruction theorem: the single-state continuous physical flux built from admissible boundary traces alone cannot provide a conservative IDP guarantee, for any prescribed CFL number, when internal reconstruction values are uncontrolled. This identifies the missing local control that must be supplied by an additional IDP flux mechanism. To realize this mechanism explicitly, we combine cell average decompositions (CAD), geometric quasilinearization, and local a priori scaling to construct admissible, generally discontinuous trace states before flux evaluation. Under an explicit trace-based CFL condition, with constants determined by local CAD weights and trace-state wave-speed bounds, the coupled hybrid update preserves the prescribed invariant domain. Concrete third-order schemes are developed on triangular, Cartesian, convex quadrilateral, and general convex polygonal meshes. Numerical results demonstrate the designed order of accuracy for smooth solutions and the strict preservation of physical admissibility.

1 Introduction

Hybrid discretizations couple conservative cell averages with generally non-conservative skeleton point values, creating distinct admissibility requirements that continuous traces alone cannot resolve for the conservative update. The paper develops separate but compatible controls for these states, including an admissibility transform for point values and a local admissible-trace flux mechanism for cell averages.

1 Introduction

Hybrid schemes couple conservative cell-average updates with possibly non-conservative evolution of shared mesh-skeleton point values, whose continuity complicates local control of the conservative flux.These schemes include Active Flux, PAMPA, and VEM formulations.Source: §1 · p. 2
Earlier AF and PAMPA stabilization strategies include a posteriori or model- and mesh-specific limiters, while this work seeks an a priori mechanism for controlling both update components.The paper frames limiting or blending as ways to introduce an IDP numerical-flux contribution.Source: §1 · p. 2
Admissible continuous traces used with the single-state physical flux cannot guarantee conservative invariant-domain updates for any prescribed CFL number when internal reconstruction values are uncontrolled.The proposed remedy uses locally admissible, generally discontinuous trace inputs and a numerical flux mechanism.Source: §1 · p. 3
An a priori hybrid admissibility principle controls cell averages and point states through distinct but compatible mechanisms under an explicit forward-Euler CFL condition.The guarantee extends to SSP Runge–Kutta methods through convex-stage embedding.Source: §1 · p. 3
The paper introduces a barrier–Legendre map for convex admissible interiors defined by concave constraints, with a globally defined, Lipschitz-continuous inverse.Point evolution uses transformed variables and evaluates directional derivatives from admissible current point states.Source: §1 · p. 3
CAD-compatible a priori scaling constructs admissible, generally discontinuous trace states before flux evaluation, with CFL constants depending on local CAD weights and trace-state wave-speed bounds.The resulting schemes are third order on triangular, Cartesian, convex quadrilateral, and general convex polygonal meshes.Source: §1 · p. 3

2 Framework and main results

The section presents a hybrid framework coupling point-value and conservative cell-average updates under model- and mesh-dependent admissibility, flux, CAD, and GQL conditions. Its theorems establish invariant-domain preservation for the coupled forward Euler stage and SSP Runge–Kutta stages under specified conditions, including a vacuum-avoidance qualification for open-interior positivity.

2.1 Geometry and degrees of freedom

The hybrid state comprises conservative cell averages and point values at mesh-skeleton evolution nodes, with reconstruction acting on both sets of degrees of freedom.The analysis requires reconstruction compatibility and smooth-data accuracy, not a particular reconstruction space.Source: §2.1 · p. 5

2.2 Invariant domains and the GQL half-space representation

Geometric quasilinearization represents convex admissible sets defined by concave constraints as an equivalent family of linear half-space constraints.The framework uses this representation to address nonlinear admissibility constraints, including positivity of internal energy for compressible Euler flows.Source: §2.2 · p. 6

2.3 Admissibility transform

A barrier–Legendre transform for convex interiors defined by finitely many concave constraints has a globally defined inverse with a global Lipschitz bound.The transform satisfies the stated bijection, regularity, and PDE-compatibility conditions for exact point evolution.Source: §2.3 · p. 7

2.4 Hybrid update operator

The hybrid stage evolves point values in transformed variables before mapping them back, while cell averages use a conservative update with fixed CAD boundary weights.The auxiliary CAD control state need not be a reconstruction point value; the conservative analysis requires only that it be admissible.Source: §2.4 · pp. 7–8

2.5 Standing assumptions

Standing assumptions combine mesh regularity, feasible CAD, convex GQL constraints, exact point evolution, and a weak Lax–Friedrichs splitting condition for conservative flux admissibility.Boundary data supply exterior traces and point-operator boundary derivatives, while conservative-flux exterior states must belong to the closed admissible set.Source: §2.5 · pp. 8–9
The weak Lax–Friedrichs condition is verified for scalar and Euler invariant domains, but adding a lower bound on Euler specific entropy requires re-verification.For scalar laws the wave-speed bound is controlled by the maximum local characteristic speed; the entropy-strengthened set may require a larger bound and modified boundary correction.Source: §2.5 · pp. 9–10

2.6 Main theorems

Under (A1)–(A5), prescribed boundary data, and CFL condition (2.24), the hybrid forward Euler update preserves admissibility when the point update uses the exact inverse B = Ψ−1.The framework also allows an admissibility-preserving practical point map.Source: §2.6 · p. 10
For a G-admissible reconstruction, the Lax–Friedrichs conservative update preserves cell-average admissibility under an explicit CFL condition, including admissible exterior states at boundary edges.Theorem 2.9 assumes (A1), (A2), (A3), and (A5).Source: §2.6 · pp. 10–11
SSP Runge–Kutta stages preserve the corresponding hybrid admissible set when they combine forward Euler steps with admissible substep sizes; this includes classical third-order SSP Runge–Kutta.Source: §2.6 · p. 11
For compressible Euler equations, applying cell-average admissibility within the open interior requires finite wave-speed bounds and avoidance of vacuum, with numerical density and pressure cutoffs as one practical safeguard.The stated cutoffs are ρ ≥ ερ > 0 and p ≥ εp > 0.Source: §2.6 · p. 11

3 Point-value update in transformed variables: operator construction and mapped admissibility

This section constructs a multidimensional non-conservative point update in transformed variables, using geometry-aware directional splitting and an admissibility-preserving map to connect the update with admissible point states and flux inputs, while establishing smooth-solution consistency. Together with the conservative CFL guarantee for admissible cell averages, these componentwise properties establish admissibility of the hybrid update.

3.1 Transformed non-conservative formulation

For smooth solutions in the admissible interior, the transformed variable w = Ψ(u) satisfies the stated non-conservative system.The formulation is established under property (AΨ3).Source: §3.1 · p. 11

3.2 Multidimensional upwind splitting in transformed variables

The multidimensional upwind construction builds on a directional decomposition and is presented for two-dimensional convex polygonal cells at third order, with k = 2.The decomposition provides the basis for geometry-consistent upwind splitting.Source: §3.1 · p. 11, §3.2 · p. 11

3.2.1 Local geometry at a mesh skeleton point

At each skeleton point, local edge directions use orthonormal normal–tangent bases and nonnegative angular weights to assemble the vertex operator.At an interior edge midpoint, the sole incident edge has weight one.Source: §3.2.1 · p. 12, §3.2.4 · p. 15

3.2.2 Lax–Friedrichs splitting matrices in transformed space

When the splitting parameter bounds the directional Jacobian’s spectral radius, the Lax–Friedrichs split has nonnegative plus and nonpositive minus characteristic speeds.This sign structure follows under local hyperbolicity on G.Source: §3.2.2 · p. 12

3.2.3 Directional derivatives without evaluating the transform outside admissible set

Intrinsic transformed directional derivatives depend only on an admissible point state and its reconstructed gradient, so the transform need not be evaluated at inadmissible intermediate reconstruction values.For a practical forward map, its derivative is evaluated at the current point state.Source: §3.2.3 · p. 13

3.2.4 Definition of the semi-discrete point operator

The explicit point-stage operator is well-defined in the exact smooth setting and maps point states into the prescribed target set in both exact and practical settings.For a practical setting, the inverse map must be globally defined, admissibility-preserving, and have range contained in the target set.Source: §3.2.4 · pp. 15–16
Under smooth-approximation bounds and boundary derivatives consistent to O(h^p), the semi-discrete point operator is consistent with the transformed PDE.The result assumes the exact solution remains in a compact subset of the admissible interior and the relevant Jacobians are uniformly bounded.Source: §3.2.4 · p. 16

3.3 Point-side boundary data and admissible flux inputs

A point stage preserves the target admissible set, and the prepared reconstruction ensures every trace state supplied to the numerical flux belongs to G.These conclusions hold under the stated exact-smooth or practical-map assumptions.Source: §3.3 · p. 17
Every point-side state σ belongs to Gpt for all σ ∈ Σh.Source: §3.3 · p. 18
Theorem 2.9 guarantees that the updated cell average belongs to G for every K ∈ Th under the conservative CFL condition (2.24).Source: §3.3 · p. 18
Combining the componentwise conclusions proves that the updated hybrid state belongs to Gh(G, Gpt).This completes the proof.Source: §3.3 · p. 18

4 Cell-average update: conditional admissibility and a priori limiters

Section 4 shows that admissible boundary traces alone do not guarantee a conservative invariant-domain-preserving cell-average update, and develops a priori limiters to protect reconstruction control states while preserving conservation and accuracy under stated conditions. With admissible reconstruction and the conservative CFL condition, geometric flux splitting and boundary closure establish conditional admissibility, including for coupled hybrid states and SSP stages.

4.1 Continuous physical fluxes and the need for an IDP flux mechanism

No fixed positive CFL number guarantees admissibility for the continuous-flux update when an internal CAD control value lies outside G, despite admissible averages and boundary quadrature values.The proposed remedy enforces admissibility of internal CAD control states before evaluating the conservative update with an IDP numerical flux; cellwise limiting can make interface traces from the two sides distinct.Source: §4.1 · p. 18

4.2 A priori limiters

The a priori limiters extend Zhang–Shu local scaling to enriched Active Flux or PAMPA reconstruction spaces, distinguishing the accuracy analysis from classical Pk-based DG results.The classical results cited for comparison concern standard Pk-based discontinuous Galerkin schemes for scalar conservation laws.Source: §4.2 · p. 19

4.2.1 Classic limiter

The classic limiter preserves the cell average, makes the limited reconstruction G-admissible, and retains accuracy for sufficiently resolved smooth solutions taking values in the interior of G.Its smooth-solution accuracy result assumes the reconstruction error tends to zero as h tends to zero.Source: §4.2.1 · pp. 19–20
Under the stated slack and boundedness conditions, the classic limiter’s control-point values and trace map are locally Lipschitz continuous.The trace-map result applies on sets where the slack condition holds uniformly.Source: §4.2.1 · p. 20

4.2.2 Alternative limiter

The alternative limiter preserves the cell average, ensures G-admissibility, and achieves O(h^(k+1)) accuracy for smooth interior solutions under its stated CAD-weight condition.The accuracy constants are independent of h and K across a shape-regular mesh family when βK = 0 or βK ≥ β0 > 0.Source: §4.2.2 · p. 21
For both limiters, the perturbation is asymptotically negligible on smooth solutions that remain a positive distance from the boundary of G.The reported perturbation order is O(h^(k+1)) under the stated smooth-regime approximation assumption.Source: §4.2.2 · p. 24

Assume further that all states

The alternative limiter is locally Lipschitz continuous on edge control values under the stated slack and boundedness conditions.For shape-regular meshes with βK ≥ β0 > 0, its local Lipschitz constant is independent of mesh size h and cell K.Source: §4.2.2 · p. 24, Assume further that all states · p. 24

4.3 Geometric IDP property and numerical fluxes

Under Assumption (A5), the geometric split states lie in G, making the associated first-order Lax–Friedrichs scheme IDP on G.The result applies to admissible left and right states and any unit normal, with the viscosity parameter satisfying the stated bound.Source: Then the intermediate split states · p. 25, §4.3 · p. 25

Then the intermediate split states

Choosing local interface viscosity ensures geometric IDP and the algebraic half-space condition for both prepared trace states.For another one-dimensional IDP flux with maximal CFL number c0, the conservative CFL upper bound is scaled by c0.Source: Then the intermediate split states · p. 25

4.4 Proof of the conservative cell-average theorem

With a G-admissible prepared reconstruction and the conservative CFL condition, the conservative cell-average update remains in G.The proof uses CAD to decompose the update, flux splitting to form admissible sub-states, GQL inequalities, and the polygon boundary-normal closure identity.Source: §4.4 · pp. 26–28
Combining the admissible point-stage map with the conservative update preserves the hybrid target set, and SSP stages retain admissibility.For classical third-order SSP Runge–Kutta, the SSP coefficient is unity, so the forward Euler conservative CFL condition remains sufficient.Source: §4.4 · p. 28

5 Geometric realizations and CFL conditions

The section constructs third-order cell-average decompositions and associated CFL bounds for triangular, Cartesian, convex quadrilateral, and general convex polygonal meshes.

5 Geometric realizations and CFL conditions

Third-order CAD realizations are provided for triangles, Cartesian cells, convex quadrilaterals, and general convex polygons, together with CFL bounds and limiter parameters.Source: §5 · p. 28

5.1 IDP CFL conditions formulated via CAD weights

The cell-average IDP time-step restriction is formulated using CAD weights and can be recast through a local CFL indicator.Source: §5.1 · p. 28

5.2.1 Optimal CAD (OCAD) boundary weights

For triangular P2 elements, the OCAD boundary weights and interior completion form a feasible CAD with positive coefficients summing to unity.The decomposition holds at the prescribed Gauss–Lobatto edge nodes.Source: §5.2.1 · p. 29

5.2.2 Associated CFL condition for triangular elements

The triangular realization satisfies the cell-average IDP property under its stated triangular CFL condition.The edge wave-speed bound uses the maximum speed on each edge.Source: §5.2.2 · pp. 29–30

5.3 Rectangular (Cartesian) element realization

The rectangular OCAD realizes the cell-average IDP property on the tensor-product Q2 space under an explicit CFL condition.The directional wave-speed bounds are defined from the maximum speeds on opposing edges.Source: §5.3 · p. 30

5.4 Convex quadrilateral element realization

For shape-regular convex quadrilaterals, the proposed third-order decomposition is a feasible CAD and yields a cell-average IDP guarantee under an explicit CFL condition.Source: §5.4 · pp. 31–32

5.5 General convex polygonal element realization with explicit CAD weights

For every convex polygon, the construction gives a feasible third-order CAD with positive unit-sum weights, boundary quadrature values, and one explicitly constructed interior node.Source: §5.5 · p. 34
The polygonal CFL bound is edgewise, coupling each edge's maximum wave speed to its fan-triangle area and edge length.Source: §5.5 · p. 34

6 Numerical experiments

The experiments test accuracy on smooth scalar problems and admissibility on discontinuous scalar and Euler problems. Across the reported scalar tests, the limiter retains designed accuracy in smooth regimes and prevents admissibility violations in discontinuous cases.

6.1 Numerical setup

Computations use third-order SSP Runge–Kutta, local Lax–Friedrichs flux, and a proxy CFL condition that is not the rigorous trace-based sufficient condition in nonsmooth regimes.The paper therefore validates admissibility in those regimes through reported diagnostics.Source: §6.1 · p. 35

Numerical results

Smooth scalar tests retain designed accuracy, while discontinuous scalar and Euler tests report admissible states or positive density and pressure.The section reports cell-average diagnostics for discontinuous scalar cases and final-time and all-stage diagnostics for Euler benchmarks.Source: §6.2.1 · p. 36, §6.2.3 · p. 37, §6.2.2 · p. 37

6.2.1 Smooth linear convection: accuracy-preservation test

For smooth linear convection, the limiter preserves formal design order and has very small activation fractions.Table 2 reports cell-average and point-value L1 errors alongside limiter activation fractions.Source: §6.2.1 · p. 36, Table 2 · p. 36

6.2.2 Discontinuous linear convection: IDP diagnostics

For discontinuous linear convection, limiter activation keeps cell-average diagnostics nonnegative up to machine roundoff on both Cartesian and triangular meshes; without it, violations occur.The test uses a 200 × 200 Cartesian mesh and a triangular mesh of 92,474 cells.Source: §6.2.2 · p. 37, Table 3 · p. 37

6.2.3 Burgers’ equation: smooth nonlinear accuracy test

The smooth nonlinear Burgers test confirms that the a priori limiter remains essentially inactive away from the invariant-domain boundary.Table 4 reports L1 errors, observed orders, and average limiter activation fraction.Source: §6.2.3 · p. 37, Table 4 · p. 38

6.2.4 Burgers’ equation: oblique Riemann problem

For Burgers’ oblique Riemann problem, the limiter keeps cell-average diagnostics strictly nonnegative, whereas violations occur without it on both meshes.Figure 4 shows cell averages, point values, and a diagonal cut for Cartesian and triangular solutions.Source: §6.2.4 · p. 38, Table 5 · p. 38, Figure 4 · p. 39

6.3 Compressible Euler benchmarks and admissibility diagnostics

The Euler figures display logarithmic density and pressure for the jets and density contours for shock diffraction and reflection.The Sedov blast is shown with density contours and a diagonal profile.Source: Figure 5 · p. 41, Figure 6 · p. 42

7 Conclusions

The paper gives an a priori IDP framework for hybrid schemes whose cell averages and point values use different update mechanisms. It combines transformed point-value evolution with locally scaled admissible flux traces, and reports third-order schemes and numerical tests supporting accuracy and admissibility.

7 Conclusions

The framework couples conservative cell-average updates with possibly non-conservative point-value updates through compatible mechanisms under an explicit trace-based CFL condition.The point update uses transformed variables; the cell-average update uses CAD, geometric quasilinearization, local a priori scaling, and admissible trace inputs.Source: §7 · pp. 42–44
Admissible continuous traces alone cannot guarantee a conservative invariant-domain update for any prescribed CFL number when internal reconstruction values remain uncontrolled.The obstruction already occurs for scalar linear advection; the construction supplies generally discontinuous admissible flux inputs to provide local one-sided control.Source: §7 · p. 44
Third-order constructions cover triangular, Cartesian, convex quadrilateral, and general convex polygonal meshes, with local scaling that preserves cell averages and is inactive on admissible CAD data.The scaling has a quantified high-order accuracy cost when activated and acts before flux evaluation rather than as a post-update repair.Source: §7 · p. 44
Numerical tests report designed third-order accuracy for smooth scalar solutions and preservation of scalar bounds and strict density and pressure positivity in nonsmooth scalar and Euler tests.The paper identifies multi-entropy stability under the EPO framework as future work.Source: §7 · p. 44

Statements and Declarations

The authors report no competing interests and state that no external datasets were used.

Statements and Declarations

The authors declare that they have no competing interests.Source: Statements and Declarations · p. 44
No external datasets were used, and numerical configurations and implementation details are specified in the paper.The configurations are in Section 6; limiter and boundary-treatment details are in Appendix D.Source: Statements and Declarations · p. 44

A.1 Constructive admissibility-transform theorem

The theorem constructs a barrier–Legendre transform from the open admissible interior to the full transformed space. Its inverse is globally defined and 1-Lipschitz, and the transform is compatible with smooth solutions of the governing PDE.

A.1 Constructive admissibility-transform theorem

The barrier–Legendre transform is a bijection from the open admissible interior to R^m, with a unique minimizer characterizing each inverse value.The proof establishes convexity and strong convexity of the potential and barrier blow-up at the boundary.Source: §A.1 · pp. 45–46
The transform is a C1 diffeomorphism whose inverse is globally 1-Lipschitz, and it maps smooth PDE solutions to the stated transformed equation.Source: §A.1 · p. 45

A.2 Point-layer transform and admissibility-preserving map pairs

The appendix lists concrete point-layer transform and admissibility-preserving map pairs for scalar conservation laws and compressible Euler equations.The listed choices include a softplus-based global diffeomorphism and an affine map paired with a clipped admissibility-preserving map for scalar intervals.Source: §A.2 · pp. 47–48

B.1 Algebraic verification of the quadrilateral CAD

The quadrilateral CAD verification uses quadratic exactness along edges and coefficient identities to show that its edge-based decomposition matches the point-value decomposition from Section 5.4.

B.1 Algebraic verification of the quadrilateral CAD

Simpson’s rule applies because each target third-order polynomial restricts to a quadratic polynomial on every edge.Source: §B.1 · p. 48
The coefficient identities follow from the explicit definitions of λK,i and βK,i and the geometric relations for opposite edge lengths.The identities are stated for i = 1, . . . , 4 with cyclic indices.Source: §B.1 · p. 48
The substituted CAD ansatz matches the Section 5.4 point-value decomposition up to cyclic relabeling, proving the stated exactness property.The relabeling arises from edge-based numbering of the vertex terms.Source: §B.1 · p. 48

C.1 Proof of the obstruction theorem

This section proves the continuous-flux obstruction and analyzes limiter accuracy and regularity, trace-speed comparisons, and wave-speed assumptions. It also verifies Assumption (A5) for the Euler invariant domain by establishing its directional inequalities when α is at least |v · n| + c_s.

C.1 Proof of the obstruction theorem

No fixed positive CFL number guarantees an admissible conservative cell average when internal reconstruction values are uncontrolled, even if all boundary quadrature values are admissible.The proof constructs an interior admissibility violation and chooses ε arbitrarily small, forcing the sufficient time-step bound to shrink with ε.Source: §C.1 · p. 49

C.2 Accuracy part of the classic limiter lemma

For smooth exact solutions, the classic limiter’s accuracy argument bounds scaling loss using uniform approximation and Lipschitz concave constraints.The proof uses boundedness of the reconstruction and the slack condition on the cell average for sufficiently fine meshes.Source: §C.2 · pp. 49–50

C.3 Lipschitz continuity of the classic limiter

The classic limiter is Lipschitz continuous because concavity bounds perturbations of constraint roots and finite minima preserve Lipschitz continuity.The proof handles inactive, active, and mixed constraint cases, then extends the bound to the limited control values.Source: §C.3 · pp. 50–52

C.4 Lipschitz continuity of the alternative limiter

The alternative limiter’s Lipschitz proof follows the classic limiter argument but additionally controls an auxiliary state when βK > 0.When βK = 0, the auxiliary constraint is absent and that control step is vacuous.Source: §C.4 · p. 52, Step 0: Lipschitz bound for the… · p. 52

Step 0: Lipschitz bound for the auxiliary state u∗

For βK > 0, positivity of quadrature weights and edge quadrature exactness support a Lipschitz bound for the auxiliary state used by the alternative limiter.The proof then incorporates that state into the candidate scaling parameters and bounds the limited edge controls.Source: Step 0: Lipschitz bound for the… · pp. 52–53

C.5 Comparison between trace speeds and proxy CFL surrogates for smooth solutions

For smooth solutions staying compactly inside the invariant domain, trace-based wave speeds and pre-stage cell-average proxies have the same asymptotic scaling.This requires locally Lipschitz wave-speed bounds; the stated comparison applies on sufficiently fine meshes and extends to globalized maxima in triangular and Cartesian settings.Source: §C.5 · p. 54

C.6 Verification of Assumption (A5) for the scalar invariant domain

The scalar-domain proof checks Assumption (A5) for states in the scalar invariant domain and its two interval endpoints.It uses the interval structure and the inward normals at the lower and upper bounds.Source: §C.6 · p. 55

C.7 Verification of Assumption (A5) for the Euler invariant domain

The Euler-domain proof begins verifying Assumption (A5) for interior Euler states, detailing the two-dimensional case and stating that the three-dimensional case follows analogously.The supplied passage identifies the verification setup but does not provide its subsequent result.Source: §C.7 · p. 55
The verification considers any unit direction n and α ≥ ˜α(u, n).The resulting threshold is ˜α(u, n) = |v · n| + c_s.Source: §C.7 · pp. 55–56
For j = 1, the density-related inequality holds strictly because ρ(α − |v · n|) ≥ ρ c_s > 0.This establishes (2.21) for j = 1.Source: §C.7 · p. 55
For j = 2, the corresponding inequality is positive, including when v∗ = v, yielding the same threshold ˜α(u, n) = |v · n| + c_s.The equality case reduces to (α ± v · n) p/(γ − 1) ≥ c_s p/(γ − 1) > 0.Source: §C.7 · pp. 55–56

D Implementation details and boundary treatments

The implementation uses a tightened numerical admissible set for limiter checks, with practical and untightened decisions eventually coinciding under a smooth-interior condition.

D.1 Implementation details for the limiters

The limiter uses a slightly tightened closed numerical admissible set; scalar invariant intervals need no modification.Source: §D.1 · p. 56
For sufficiently fine meshes, practical and untightened limiter checks yield identical outcomes when sampled states converge uniformly to exact states in a compact interior subset.The implementation cutoffs must remain below the subset’s interior margin.Source: §D.1 · p. 56
For Euler simulations, the density cutoff is 10^-13 and the cellwise internal-energy cutoff is max{1, ĒK}10^-13.Proposition D.1 applies to these cutoffs when sampled states remain in a compact interior subset of G.Source: §D.1 · p. 56

D.2 Boundary treatments used in the numerical experiments

The boundary treatments supply exterior states and directional derivatives for the point operator, while the optional OE preprocessing contracts reconstructions toward cell averages without changing them.

D.2 Boundary treatments used in the numerical experiments

Point-operator ghost extensions need not be admissible, unlike the admissible limited interior and boundary-flux states used by the conservative update.Source: §D.2 · p. 57
Periodic, inflow or exact-data, outflow, and reflective treatments are consistent with their boundary conditions and locally Lipschitz under the stated regularity assumptions.The reflective treatment changes only normal momentum while preserving density and total energy.Source: §D.2 · p. 58

D.3 Optional oscillation-eliminating (OE) preprocessing

The optional OE preprocessing contracts the reconstruction toward the cell average, preserving that average and not expanding the admissible target enforced afterward.It is introduced to suppress spurious oscillations near strong shocks.Source: §D.3 · p. 58