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A High-Order Geometric-Compatible DGSEM with Subcell Limiting for the GPR System

Jielin Yang

arXiv:2610.34755v1math.NAcs.NA

TL;DR

GPR discretization must handle shocks, nonconservative products, stiff sources, and state constraints, including the determinant–density relation on fixed Eulerian meshes. The paper constructs compatible FV and high-order DGSEM schemes and blends them on subcells. In a five-wave nonlinear-elastic test, the method resolves the waves and plateaus without visible Gibbs oscillations, with a reported normalized mass drift of about 9.4 × 10^-13.

  • Problem

    GPR combines shocks, nonconservative products, material-dependent stiff sources, and state constraints, while geometric compatibility targets semidiscrete preservation of ρ = ρ0 det A on fixed Eulerian meshes.

    Source: §1 · p. 1

  • Method

    The paper constructs a compatible FV scheme and its SBP-based high-order DGSEM extension, then blends them with a global subcell limiter and modal/face-jump indicator.

    Source: §1 · p. 2, §5.1 · p. 23

  • Results

    In the five-wave nonlinear-elastic test, the method resolves five waves and intervening plateaus without visible Gibbs oscillations, with normalized fully discrete mass drift about 9.4 × 10^-13.

    Source: §7.2.1 · p. 31

  • Takeaways & Limitations

    The global subcell blend retains the hidden density law while transitioning from accurate GC-DG in smooth elements toward more dissipative GC-FV near under-resolved discontinuities.

    Source: §1 · p. 2, §5.1 · p. 23

    A local subcell extension requires a new interfacewise indicator and is left for future investigation; the computations use the global elementwise limiter.

    Source: §5.3 · p. 27

Abstract

from arXiv · show

We construct an equation-of-state-independent geometric-compatible finite-volume (GC-FV) method and a high-order nodal discontinuous Galerkin spectral-element extension (GC-DG) for a reduced Godunov-Peshkov-Romenski system on fixed Eulerian meshes. Density is recovered from $\rho=\rho_0\det(\mathbf{A})$, so mass conservation is a hidden companion law of the distortion equation. Starting from a straight-path Dal Maso-LeFloch-Murat fluctuation pair, minimum-change cofactor projections produce a single-valued density flux and exact induced local Lax-Friedrichs dissipation without jump quotients. The summation-by-parts property lifts the geometric two-state identity to high order and yields an equivalent subcell representation of GC-DG. We then blend GC-DG with the compatible GC-FV operator on the same Legendre-Gauss-Lobatto subcells. Because both operators induce the same boundary density flux, the global subcell blend retains the exact semidiscrete mass balance while suppressing shock-generated oscillations. A complementary modal/face-jump indicator supplies the elementwise blending coefficient. Numerical experiments verify high-order convergence, the expected temporal scaling of the fully discrete mass drift, and robust shock capturing for fluid, nonlinear-elastic, thermal-wave, and elastoplastic problems, including genuinely multidimensional configurations.

1. Introduction

The GPR system couples nonconservative products, shocks, stiff sources, and state constraints, while fixed-grid preservation of its determinant–density law has received little attention. This work develops compatible FV and DGSEM schemes and a subcell limiter that blends them while retaining the hidden density law.

1. Introduction

  • The determinant–density companion law is a distinct structural objective; thermodynamic compatibility, curl-type involutions, and physical admissibility are separate properties.Satisfying one structural objective does not automatically secure the others.Source: §1 · p. 1
  • The reduced fixed-grid GPR strategy recovers density from det A and treats its conservation as a hidden law, but the prior construction was first order and used jump-quotient corrections.Its Rusanov-type dissipation was interpreted through vanishing-viscosity regularization, complicating direct high-order extension.Source: §1 · pp. 1–2
  • A combined element-modal and scale-normalized face-jump indicator supplies the elementwise blending coefficient, addressing unresolved modes and jumps across element faces.The two channels provide complementary information.Source: §1 · p. 2
  • The paper develops a two-state companion identity, extends it through SBP to high-order DGSEM, and blends the resulting GC-DG operator with compatible GC-FV.The global subcell blend retains the hidden density law because both operators satisfy it.Source: §1 · p. 2
  • The method is assessed for smooth convergence, general constitutive laws, nonlinear-elastic and thermal-wave Riemann problems, and genuinely multidimensional configurations.The GPR system also encompasses inviscid and viscous fluids and elastic, viscoelastic, and plastic solids.Source: §1 · pp. 1, 3

2. The GPR model and constitutive scope

The paper uses an energy-primary GPR model with density, momentum, energy, distortion, and thermal impulse, and restricts distortion relaxation so it does not alter the determinant law. Its planar computations use two spatial coordinates while retaining the three-dimensional constitutive law.

2.1. Complete energy-primary GPR system

  • The complete energy-primary GPR state includes density, momentum, total energy, distortion, and thermal impulse, with conjugate thermodynamic variables defined by energy gradients.The dual variables include velocity and temperature as well as forces associated with density, distortion, and thermal impulse.Source: §2.1 · p. 3

2.2. Constitutive assumptions

  • The geometric construction requires the isochoric identity w : S_A = 0, while otherwise allowing smooth admissible GPR closures without requiring a particular caloric EOS or entropy convexity.The construction is not restricted to the displayed closures; a closure's dissipation coefficient still comes from its acoustic tensor or full normal Jacobian.Source: §2.2 · pp. 3–5

2.3. Fixed planar 2D–3C convention

  • The planar convention uses two spatial coordinates but retains three-component constitutive vectors and 3 × 3 distortion tensors; A33 evolves while the four cross-block entries remain zero.Only J1 and J2 evolve, while determinants and cofactors remain three-dimensional.Source: §2.3 · p. 5

3. First-order geometric-compatible finite-volume discretization

The GC-FV method removes the independent mass equation and constructs a compatible distortion update whose cofactor contraction induces a conservative determinant, and hence density, equation. It derives the update from finite-jump identities and a projected straight-path fluctuation pair.

3. First-order geometric-compatible finite-volume discretization

  • GC-FV evolves momentum, energy, distortion, and thermal impulse, recovers density as ρ = ρ0 det A, and adds projected LF stabilization for admissible EOS satisfying w : S_A = 0.Its conservative momentum and energy fluxes are paired with one-sided distortion fluctuations.Source: §3 · p. 5

3.1.1. Boscheri reduction and hidden companion laws

  • Removing the independent mass equation makes the determinant relation a hidden conservation law, since cofactor contraction of the distortion equation yields ∂t g + ∇·(gv) = 0.With ρ = ρ0g, this also gives the conservative density equation.Source: §3.1.1 · p. 6

3.1.2. Necessary algebraic formulas

  • Finite-jump determinant identities replace the differential chain rule and provide the algebra needed to evaluate the straight-path DLM jump without determinant jump quotients.The construction uses exact two-state determinant algebra.Source: §3.1.2 · p. 6, §3.1.3 · p. 7

3.1.4. Companion projection of the baseline fluctuations

  • The distortion baseline uses an exchange-symmetric allocation of the generalized jump along a selected straight path, transporting the distortion jump and representing velocity-gradient coupling.This baseline is path-conservative; the later companion projection need not preserve its fluctuation sum.Source: §3.1.4 · pp. 7–8
  • A Tadmor-type two-state identity constrains the one-sided fluctuations to induce one single-valued determinant flux, and the chosen arithmetic target is symmetric and EOS independent.The identity is the discrete chain rule for the hidden law, rather than the DLM path-sum condition.Source: §3.1.4 · p. 9
  • The minimum-change projection modifies the straight-path baseline to meet the prescribed density-flux constraint, yielding a unique companion pair without jump quotients for positive endpoint determinants.The two endpoint reconstructions agree on the target flux.Source: §3.1.4 · pp. 9–10

It also has the fixed-normal exchange symmetry

The projected companion pair has fixed-normal exchange symmetry and a smooth equal-state extension. Projected LF dissipation corrects the raw matrix jump increments so that their cofactor contractions reproduce the exact density jump.

It also has the fixed-normal exchange symmetry

  • The companion pair is unique and has endpoint reconstruction equality, the summed contraction identity, fixed-normal exchange symmetry, and a smooth equal-state extension on compact admissible sets.These properties supply the algebraic content needed from the nondissipative GC-FV core.Source: It also has the fixed-normal exchange… · p. 10

3.2. Dissipative fluxes and fluctuations for GC-FV

  • Raw LF distortion increments miss the exact density jump at finite states because det A is nonlinear, so the method projects them to enforce density LF dissipation.The cofactor-parallel correction leaves determinant-neutral planar modes with raw LF damping while supplying the determinant-active mode's exact density dissipation.Source: §3.2 · p. 11

Then its endpoint increments satisfy

The projected LF increments preserve exact density dissipation by modifying the distortion increment, and the resulting GC-FV update induces a conservative determinant law. The section also specifies wave-speed estimates and the scope of their use.

Then its endpoint increments satisfy

  • The cofactor projection preserves the raw LF increment in determinant-tangent directions and yields the exact scalar LF increment in the induced density law, at the cost of incomplete distortion-equation LF strength.The authors report no experimental instability attributable to this tradeoff.Source: Then its endpoint increments satisfy · pp. 11–12
  • The conservative momentum and total-energy blocks use ordinary LF increments, while the thermal-impulse block needs no projection because it does not enter the determinant constraint.The thermal-impulse increments are added to the straight-path baseline.Source: Then its endpoint increments satisfy · p. 12

3.3.1. GC-FV update

  • For admissible states, constant ρ0, and an isochoric source, the complete GC-FV update induces a conservative determinant-density law while momentum and total energy remain conservative.The proof uses the closed-cell identity and the single-valued central–LF flux in the conservative block.Source: §3.3.1 · pp. 12–13
  • Compared with Boscheri et al., GC-FV uses bounded endpoint projections without jump quotients and gives exact induced density LF dissipation, but claims only geometric compatibility.Boscheri et al. retain the selected path sum and enforce a thermodynamic companion law.Source: §3.3.1 · p. 13
  • The two GC-FV path-sum defects are O(∥∆U∥^3) and O(ℓf ∥∆U∥^2) for smooth pairs, but may remain finite across shocks.The smooth-pair estimates are not a complete convergence-order analysis; no reported experiment showed instability attributable to path-sum inconsistency.Source: §3.3.1 · p. 13

3.3.2. Wave-speed estimate

  • The mechanical wave-speed estimate uses the acoustic tensor without a full mechanical-Jacobian eigendecomposition, with closed-form evaluations for the two EOS classes considered.At finite deformation, the Barton–Romenski result remains directionally anisotropic rather than reducing to one scalar sound speed.Source: §3.3.2 · pp. 13–14
  • For J-active ideal or stiffened-gas cases, the calibrated estimate adds thermal-wave protection; κJ = 10 is a reproducible working value, not a universal spectral bound.For a general caloric EOS, the thermal contribution must be bounded from the full normal quasilinear matrix.Source: §3.3.2 · p. 14

4. High-order nodal DGSEM extension

This section develops the high-order geometric-compatible DGSEM by lifting the GC-FV two-state pair through SBP flux differencing and deriving a nodal determinant companion law. It presents an equivalent LGL-subcell representation, expresses GC-FV on the same subcells, and establishes exact global semidiscrete mass balance for the compatible operators.

4.1. Nodal space and SBP operators

  • The nodal space uses tensor-product LGL nodes and diagonal-norm SBP operators satisfying Q + Q^T = BSBP and Q1 = 0.These identities underpin the high-order flux-differencing construction.Source: §4.1 · pp. 14–15

4.2. The two-dimensional high-order GC-DG scheme

  • GC-DG applies the SBP lift to the nondissipative GC-FV pair and retains the complete dissipative GC-FV operator at element faces through SATs; LF dissipation is not used in the volume.The volume uses the nondissipative pair because SBP differentiation-matrix entries have both signs.Source: §4.2 · pp. 15–16
  • Under the stated mesh, state, density, and source assumptions, GC-DG induces at every LGL node a density equation using the prescribed symmetric volume flux and complete GC-FV face flux.The assumptions are an affine conforming Cartesian mesh, admissible nodal states, constant rho0, and an isochoric source.Source: §4.2 · p. 16
  • The conservative momentum and total-energy blocks retain standard DGSEM balances; equal-state limits and DSBP1 = 0 imply constant preservation without sources.The companion projection is smooth and nonsingular at equal states, with a correction that vanishes quadratically with the state jump.Source: §4.2 · p. 16

4.3. Equivalent subcell form

  • The equivalent subcell rewrite exposes a high-order flux or directed fluctuation at each LGL subcell interface for constructing the limiter, without defining a new spatial discretization.The rewrite uses a conservative flux-differencing construction and a nonconservative construction, with a projection for distortion.Source: §4 · p. 14, §4.3 · p. 17

4.3.1. Subcell form of the conservative terms

  • The conservative blocks of GC-DG, including the auxiliary density equation, are exactly represented by high-order subcell fluxes with GC-FV boundary fluxes.The equivalence is established by subtracting successive prefix sums, including at boundary nodes.Source: §4.3.1 · p. 18

4.3.2. Subcell form of the nonconservative terms

  • The nonconservative rewrite gives an exact subcell form for the thermal-impulse block, but the raw straight-path distortion form does not retain the determinant companion law.The distortion fluctuations therefore require modification, unlike the J equation.Source: §4.3.2 · p. 19

4.3.3. Geometric-compatible subcell form for the distortion

  • The algebraically equivalent subcell form projects raw distortion fluctuations by minimum change so their cofactor contractions match the subcell density flux, restoring geometric compatibility.The resulting distortion subcell equation is identical at every nodal point to the distortion block of GC-DG.Source: §4.3.3 · pp. 19–20
  • The complete subcell scheme is identical to the high-order DGSEM at every nodal point, and its distortion equation induces the nodal density equation.The induced equation is equivalent at each node to the high-order density equation.Source: §4.3.3 · p. 21

4.4. The GC-FV scheme in low-order subcell form

  • GC-FV is written on LGL nodes as algebraic subcells, with no added grid or polynomial reconstruction and shared boundary quantities from the same complete GC-FV face operator.Its low- and high-order subcell quantities coincide at element boundaries with the corresponding GC-FV face-operator blocks.Source: §4.4 · p. 21

4.5. Global semidiscrete mass balance

  • Single-valued density fluxes make internal contributions cancel, yielding exact global semidiscrete mass balance; periodic boundaries imply conservation, while finite Runge–Kutta updates need not preserve it exactly.The GC-FV, GC-DG, and low-order GC-FV subcell operators share this global structure.Source: §4.5 · p. 22

5. Geometric-compatible global subcell limiter

The limiter blends compatible high- and low-order spatial residuals rather than postprocessing the distortion polynomial. Complementary modal and face-jump indicators set the blend, while local interfacewise blending is left for future work.

5. Geometric-compatible global subcell limiter

  • Generic postprocessing can preserve the distortion average without preserving the derived cell mass, because the average of det A differs from det of the average.Such limiting generally breaks the fully discrete companion structure, so its added mass drift is not controlled by the temporal truncation estimate.Source: §5 · p. 23

5.1. Elementwise high–low blending

  • One elementwise coefficient blends high- and low-order subcell residuals, recovering GC-DG as αK approaches zero and GC-FV as it approaches one.The coefficient is recomputed at each Runge–Kutta stage.Source: §5.1 · p. 23, §5 · p. 23
  • Blending GC-DG and GC-FV residuals preserves the nodal determinant companion law and exact semidiscrete global mass balance.The common induced density flux makes the density residual undergo the same elementwise limiting action.Source: §5.1 · pp. 23–24

5.2.1. Elementwise modal indicator

  • The modal sensor groups ρp, A, and J, using shared norms for vector and tensor components while omitting a separate modal velocity group.The added A and J groups target elastic and thermal waves; the modal energy compares high-order bands with total energy.Source: §5.2.1 · p. 24

5.2.2. Face-jump indicator

  • Face sensing combines interior state and highest-derivative jumps, with highest-shell weighting to suppress activation when corresponding high-order content is negligible.The state-jump channel detects jumps of the state, while the highest-derivative channel detects mismatches in the highest resolved derivative.Source: §5.2.2 · p. 26

5.2.3. Final indicator

  • For N≥3, the final indicator combines modal and face-jump channels to balance oscillation control with fine-structure resolution.For N=1 the modal and highest-derivative face channels are disabled; for N=2 the highest quadratic modal band is retained but the penultimate band is omitted. The face-sensor threshold is the only user-tuned limiter parameter and is problem-dependent.Source: §5.2.3 · pp. 26–27

5.2. Complementary modal and face-jump indicators

  • Modal sensing can erase fine structures, while face-jump sensing can leave oscillations near strong isolated shocks.The complementary channels are combined for N≥3 to balance these behaviors.Source: §5.2.3 · p. 26

5.3. Local subcell blending as a lower-dissipation extension

  • Local interfacewise blending retains the exact semidiscrete mass balance and adds no limiter-induced mass drift when evaluated at every Runge–Kutta stage.Its indicator design and validation are outside the work's scope; all computations use the global elementwise limiter.Source: §5.3 · p. 27

6. Time integration

The spatial schemes have exact semidiscrete mass balance, but Runge–Kutta updates do not preserve the nonlinear determinant chain rule exactly. For SSP–RK3, the accumulated drift scales as O(∆t^3) under smoothness assumptions.

6. Time integration

  • Runge–Kutta integration does not preserve the exact semidiscrete companion balance because a finite update of A fails to satisfy the nonlinear determinant chain rule.Mass drift is the discrepancy between the fully discrete mass change and the boundary flux used by that Runge–Kutta step.Source: §6 · pp. 27–28
  • For SSP–RK3, the one-step discrepancy is O(∆t^4) and accumulated mass drift is O(∆t^3).This estimate assumes a sufficiently smooth semidiscrete trajectory with uniformly bounded time derivatives.Source: §6 · p. 28
  • The smooth-trajectory drift estimate does not cover the initial transient from nodally sampled discontinuous data, which creates O(1) jumps between neighboring nodes.Any larger early drift is a fully discrete determinant-chain-rule defect, not an additional defect of the spatial subcell blend.Source: §6 · p. 28

7. Numerical experiments

The numerical experiments assess GC-DG accuracy, mass behavior, wave resolution, shock capturing, and multidimensional symmetry, including regimes with and without stiff relaxation sources. The cylindrical elastoplastic shock test reports close agreement with a radial path-conservative reference at the elastic and plastic fronts, circular fields without visible grid imprinting, and a small fully discrete mass drift.

7.1. Translating isentropic vortex

  • Vortex tests demonstrate high-order convergence toward N + 1 (less clearly for even N) and third-order mass-drift scaling, with observed orders 2.998, 3.000, and 3.004.The temporal study reports a factor-of-eight drift decrease for each halving of the time step.Source: Table 1 · p. 29, §7.1 · pp. 29–30, Table 2 · p. 30

7.2. One-dimensional Riemann problems

  • The planar Riemann tests use thin two-dimensional strips to exercise the multidimensional implementation; zero far-field normal velocities and no boundary-reaching waves make their accumulated boundary mass contribution vanish.Both tests report boundary-compensated mass drift using the discrete flux and SSP-RK3 stage update.Source: §7.2 · p. 30

7.2.2. Seven-wave homogeneous GPR Riemann problem

  • GC-DG separates the seven homogeneous-GPR waves and closely reproduces intermediate plateaus; blending localizes near fronts, with normalized fully discrete mass drift 6.6 × 10^-9.The test activates both elastic shear and thermal waves.Source: §7.2.2 · p. 32, Figure 3 · p. 32, Figure 4 · p. 33

7.3. Circular explosion

  • Fluid and solid circular explosions reproduce principal wave structures and remain radially symmetric, with boundary-compensated mass drifts 1.00 × 10^-7 and 1.25 × 10^-7, respectively.The degree-2 fluid limiter activates across more of the rarefaction-to-shock fan than the degree-4 solid limiter; most later drift accumulation is small after the initial rise.Source: §7.3.1 · p. 33, Figure 5 · p. 34, §7.3.2 · pp. 34–35

7.4. Elastic solid rotor

  • The elastic solid rotor reproduces wave locations and nested structures, retains its rotational and spiral-annular patterns, and has periodic-boundary mass drift 1.97 × 10^-10.After t = 0.01, the drift changes by only 8.16 × 10^-11.Source: §7.4 · p. 36

7.5. Cylindrical elastoplastic shock

  • The test demonstrates robustness for a general solid energy with active nonlinear relaxation, beyond the ideal- and stiffened-gas specializations used in preceding tests.The relaxation law is softened for explicit SSP-RK3 while preserving the copper energy and yield-stress scale of.Source: §7.5 · pp. 37, 39
  • GC-DG closely follows the radial WENO-Z comparison solution through the principal elastic and plastic fronts and reproduces the narrow stress structure near the initial interface.The comparison uses a characteristic fifth-order path-conservative WENO-Z discretization on 6000 radial cells.Source: §7.5 · p. 37, Figure 12 · p. 39
  • The Cartesian-mesh density and velocity-magnitude maps remain circular without visible grid imprinting, while blending is confined to physical wave regions.The final boundary-compensated fully discrete mass drift is 6.02 × 10^-11 and changes by only 1.77 × 10^-13 after t = 10^-7 s.Source: §7.5 · pp. 37, 39, Figure 13 · p. 39

8. Concluding remarks

The paper develops equation-of-state-independent GC-FV and high-order GC-DG schemes for the reduced GPR system on fixed Eulerian meshes. Their compatible subcell blend preserves the hidden semidiscrete mass law while supporting high-order accuracy and shock capturing.

8. Concluding remarks

  • The GC-FV construction and its high-order DGSEM extension are equation-of-state-independent geometric-compatible schemes for the reduced GPR system on fixed Eulerian meshes.The distortion equation induces the conservative determinant law, which maintains ρ = ρ0 det A semidiscretely when initially satisfied.Source: §8 · pp. 39–40
  • A minimum-change projection of the path-conservative baseline enforces a geometric two-state identity, which SBP lifts to high order.Rewriting GC-DG and GC-FV on the same LGL subcells permits a global subcell blend controlled by modal and face-jump indicators.Source: §8 · p. 40
  • The global subcell limiter preserves the hidden mass law semidiscretely while supplying dissipation near discontinuities.Numerical experiments confirm the expected temporal scaling of fully discrete mass drift.Source: §8 · p. 40
  • Experiments report high-order accuracy and robust shock capturing for fluid, nonlinear-elastic, thermal-wave, and elastoplastic problems.Future work includes less dissipative local blending, positivity-preserving stabilization, curvilinear and moving meshes, and simultaneous geometric and thermodynamic compatibility.Source: §8 · p. 40

Appendix A. Acoustic-tensor reduction for the mechanical wave speed

The appendix reduces the mechanical wave-speed calculation to a directional acoustic-tensor problem for frozen admissible states. It identifies how the shear-energy closure enters the directional mechanical response and relates the resulting eigenvalues to wave speeds.

Appendix A. Acoustic-tensor reduction for the mechanical wave speed

  • A density-dependent shear speed contributes to both isotropic pressure and shear traction.The appendix obtains the relevant shear-energy derivatives and applies product and chain rules to the stress and shear-energy terms.Source: Appendix A · p. 40
  • For an admissible frozen state, an isentropic plane perturbation in direction n is reduced to directional variations after eliminating distortion through the linearized kinematic equation.The analysis treats two mechanical closures with constant logarithmic density exponent ζb: zero for the constant-shear gas closure and βB for the Barton–Romenski closure.Source: Appendix A · p. 40
  • The linearized normal momentum flux yields an acoustic-tensor action built from direct distortion variations and the density variation of ρb2.For the constant-shear gas closure b2 = c2 s, ζb = 0; for the Barton–Romenski closure ζb = βB.Source: Appendix A · p. 40
  • The two eigenvalues of Wn are the squared in-plane mechanical wave speeds relative to vn, and the larger root yields equation (89).This is an exact reduction of the frozen homogeneous mechanical Jacobian.Source: Appendix A · p. 40

Appendix B. Algebraic properties and proofs for GC-FV and GC-DG

The appendix establishes the minimum-change Frobenius projection used for distortion constraints and applies it to the GC-FV and projected LF corrections. It also distinguishes admissible endpoint denominators from a degenerate jump-based correction.

Appendix B.1. Minimum-change Frobenius projection

  • The one-constraint Frobenius projection gives a unique feasible matrix by changing the baseline only in the direction of the constraint matrix.The proof uses orthogonality to show that every other feasible matrix has greater distance from the baseline.Source: Appendix B.1 · p. 41
  • Applying the projection independently at the endpoints gives the GC-FV pair and projected LF increments.Admissibility makes their denominators endpoint quantities rather than vanishing jump norms.Source: Appendix B.1 · p. 41
  • A zero-sum correction to the cubic defect instead produces a denominator proportional to the squared state jump, which can degenerate.This contrasts with the endpoint denominators used by the projected construction.Source: Appendix B.1 · p. 41

Appendix B.2. Exchange and face-orientation identities

The appendix treats two distinct exchange symmetries: fixed-normal exchange for the nondissipative volume pair and simultaneous state exchange and normal reversal for the element-face identity. It verifies that the complete dissipative face pair retains the latter orientation property.

Appendix B.2. Exchange and face-orientation identities

  • The fixed-normal exchange property supports the volume pair, while simultaneous state exchange and normal reversal support the element-face identity.The appendix states these as distinct orientation properties and proves them separately.Source: Appendix B.2 · p. 41
  • Exchanging states while keeping n fixed reverses the state jumps but leaves arithmetic averages unchanged in the nondissipative volume-pair argument.The resulting identities establish the fixed-normal exchange property.Source: The complete dissipative face pair… · pp. 41–42, Appendix B.2 · p. 41

The complete dissipative face pair (80)–(81) satisfies the orientation identity

  • The complete dissipative face pair satisfies the orientation identity after the LF blocks are added to the nondissipative identity.The proof accounts for the LF coefficient's invariance under simultaneous state exchange and normal reversal, including the distortion correction.Source: The complete dissipative face pair… · p. 42

Appendix B.3. Smooth-state size of the projection and path-sum defects

The appendix quantifies projection and path-sum defects for neighboring states sampled from smooth solutions, while restricting those asymptotic estimates to the smooth regime.

Appendix B.3. Smooth-state size of the projection and path-sum defects

  • The estimates are local as states coalesce and do not provide a bound for finite jumps at shocks.Neither defect order should be interpreted as a uniform shock estimate.Source: Appendix B.3 · p. 42
  • With δ = ||ΔU|| = O(h), determinant and velocity jumps are O(δ), while aΔA = O(δ^2) and dΔA = O(δ^3) on compact admissible sets.These smooth-neighbor scalings are used to estimate the departures from the straight-path fluctuation sum.Source: Appendix B.3 · p. 42
  • Endpoint modifications vanish quadratically at equal states, and the leading quadratic contributions to the path-sum defect cancel.The appendix analyzes these orders as neighboring states approach one another.Source: Appendix B.3 · p. 42
  • For smooth neighboring states, nondissipative and dissipative path-sum defects vanish to cubic and quadratic order, respectively.At discontinuities, δ is not small, so these orders are not uniform shock estimates.Source: Appendix B.3 · p. 42

Appendix B.4. Smooth equal-state regularity of the companion projection

The companion projection extends smoothly to equal states on compact admissible sets bounded away from g = 0, preserving the straight-path pair’s diagonal value and first derivative.

Appendix B.4. Smooth equal-state regularity of the companion projection

  • When g > 0 on compact admissible sets, the endpoint projector is smooth and bounded, and its corrections vanish as O(τ^2) independently of direction V.This gives a smooth equal-state extension of the projected pair.Source: Appendix B.4 · pp. 42–43
  • The projected pair preserves the baseline’s first diagonal linearization and adds only an O(τ^2) perturbation as states coalesce.Its correction therefore vanishes at the equal-state diagonal without changing the first diagonal derivative.Source: Appendix B.4 · p. 43
  • The projection has no jump denominator and is locally consistent with the high-order SBP construction without reducing spatial accuracy.The regularity argument is limited to the local coalescence of the two states.Source: Appendix B.4 · pp. 42–43
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