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arxiv: 2605.02081 · v1 · submitted 2026-05-03 · 🧮 math.NA · cs.NA

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On the Practical Impact of Local Linear Instabilities in Entropy-Stable Schemes

Alex Bercik, David W. Zingg

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Pith reviewed 2026-05-08 18:49 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords entropy-stable schemeslocal linear instabilitynumerical dissipationperturbation growthsplit-form discretizationtwo-point fluxvariable-coefficient advectionlogarithmic mean
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The pith

Local linear instabilities in entropy-stable schemes generate only small, controllable numerical errors that do not limit their use.

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

The paper shows that local linear instabilities, where the discrete linearized operator permits more perturbation growth than the continuous problem, should be viewed as a minor source of discretization error rather than a barrier to stability. For split-form linearizations of variable-coefficient advection that arise from Burgers equation discretizations, unphysical modal growth occurs but remains bounded and small. This growth concentrates in highly oscillatory and boundary-localized modes that small numerical dissipation readily damps. The same growth pattern does not carry over to the nonlinear two-point-flux forms used in entropy-stable Euler discretizations, and observed robustness problems in density-wave tests trace instead to the logarithmic mean's poor near-vacuum behavior.

Core claim

Local linear instabilities produce unphysical modal growth in linearized split-form discretizations of variable-coefficient advection, yet this growth satisfies physically interpretable bounds, stays typically small, and is dominated by highly oscillatory and boundary-localized modes that small dissipation controls. Floquet analysis shows that unstable spectra of frozen Jacobians need not produce unstable perturbation growth in the full nonlinear two-point-flux discretizations typical of entropy-stable schemes. Sharp bounds derived for the geometric flux predict negligible growth, with analogous numerical behavior for the logarithmic flux; robustness issues in density-wave problems are thus,

What carries the argument

The re-interpretation of local linear instability as a bounded source of numerical error, combined with modified-equation analysis, Floquet analysis, and explicit perturbation growth bounds for geometric and logarithmic fluxes.

If this is right

  • High-order entropy-stable schemes can be used for nonlinear conservation laws without special treatment for local linear instabilities.
  • Minimal artificial dissipation suffices to suppress any residual modal growth from linearizations.
  • Development effort should focus on improving logarithmic-mean approximations near vacuum rather than on linear stability fixes.
  • Frozen-coefficient Jacobian spectra alone do not determine the stability of time-dependent nonlinear schemes.

Where Pith is reading between the lines

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

  • Designers of entropy-stable codes can adopt higher polynomial degrees with greater confidence that linear modal issues will not dominate error budgets.
  • The same modal-growth analysis may extend to other split-form or summation-by-parts schemes outside entropy-stable families.
  • Practical implementations could standardize small, fixed dissipation levels calibrated to the observed bound for linear advection.

Load-bearing premise

Perturbation growth stays dominated by oscillatory and boundary-localized modes that small dissipation can suppress and does not transfer into the nonlinear two-point-flux discretizations used in practice.

What would settle it

A numerical test in which a high-order entropy-stable discretization of the Euler equations exhibits sustained, large-amplitude unphysical growth that persists even after modest added dissipation and is independent of near-vacuum effects would falsify the claim.

Figures

Figures reproduced from arXiv: 2605.02081 by Alex Bercik, David W. Zingg.

Figure 1
Figure 1. Figure 1: The exact solution and energy (in the L 2 and a-weighted norms) for the variable-coefficient linear advection equation (1) with a Gaussian initial condition and sinusoidal coefficient a(x) = sin (2πx) + 3/2. The times at which the solution snapshots are plotted in the left plot are indicated in the right plot by correspondingly coloured vertical dashed lines. assumed that numerical schemes should respect t… view at source ↗
Figure 2
Figure 2. Figure 2: Eigenspectra of the product scheme, i.e. ( view at source ↗
Figure 3
Figure 3. Figure 3: Eigenvectors (top row) and contributions to view at source ↗
Figure 4
Figure 4. Figure 4: Local Rayleigh quotients of the central-product split-form discretiza view at source ↗
Figure 5
Figure 5. Figure 5: Worst case scenario central-product split-form scheme error growth view at source ↗
Figure 6
Figure 6. Figure 6: Product scheme (α = 0) error growth with non-dissipative 8th-order circulant operators. 0 2 4 6 8 10 12 t 10−3 10−2 10−1 100 101 kukH kuk∞ kukH Prediction kuk∞ Prediction kekH kek∞ (a) CSBP, N = 100, dissipative SAT. ℜ(λ)max = 6.1 × 10−1 0 5 10 15 20 25 30 t 10−11 10−9 10−7 10−5 10−3 10−1 101 103 kukH kuk∞ kukH Prediction kuk∞ Prediction kekH kek∞ (b) CSBP, N = 400, dissipative SAT. ℜ(λ)max = 1.1 0 5 10 15… view at source ↗
Figure 7
Figure 7. Figure 7: Product scheme (α = 0, p = 2) error growth with interface and volume dissipation. is particularly concerning for high-order methods applied to long-time wave propagation, since their usual advantage of having small, slowly accumulating numerical errors is ultimately negated if this error component grows exponen￾tially in time. Figure 5f shows the same result, but using the continuous SBP operators of [42,4… view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of H-norm solution errors between different central-product splittings α ∈ [0, 1]. However, we have already shown that this worst-case scenario often over￾states the practical impact of local linear instabilities. Therefore, we now use the lessons learned from previous sections to reassess their severity. We first showed that instabilities resulting from interior dynamics alone are weak and tend… view at source ↗
Figure 9
Figure 9. Figure 9: Instantaneous eigenspectra at t = 0 (top row) and Floquet exponents for one period (bottom row) of the entropy-stable geometric flux-differencing scheme applied to the constant-coefficient linear advection equation. No inter￾face or volume dissipation is used. Note the varying x-axes in the bottom row. monodromy matrix, which introduces a small forcing term due to truncation error that is not present in ac… view at source ↗
Figure 10
Figure 10. Figure 10: Change in solution energy ∥u∥ 2 H (top row), solution error ∥e∥ 2 H (sec￾ond row), perturbation H-norm ∥v∥H (third row), and perturbation L∞-norm ∥v∥∞ (bottom row) of the flux-differencing schemes with central, logarithmic, and geometric fluxes applied to the constant-coefficient linear advection equa￾tion with a Gaussian initial condition and a random perturbation added. No interface or volume dissipatio… view at source ↗
Figure 11
Figure 11. Figure 11: A logarithmic flux-differencing discretization of a near-vacuum linear view at source ↗
Figure 12
Figure 12. Figure 12: The 1D Euler density-wave problem using the entropy-conserving Ra view at source ↗
read the original abstract

Local linear instability refers to the linearized discrete operator exhibiting perturbation growth exceeding that of the corresponding continuous linearized problem. In the context of nonlinear entropy-stable discretizations, we argue that local linear instabilities should be interpreted as a source of numerical error whose practical impact is often negligible compared with other discretization errors. For split-form discretizations of the variable-coefficient linear advection equation, such as those resulting from linearizations of entropy-stable discretizations of the Burgers equation, perturbations can indeed exhibit unphysical modal growth. However, we demonstrate that this growth satisfies physically interpretable bounds and is typically small. Furthermore, through modified-equation analysis and numerical experiments, we show that the growth is dominated by highly oscillatory and boundary-localized unphysical modes, and can therefore be readily controlled by small amounts of numerical dissipation. More generally, this modal perturbation growth does not extend directly to nonlinear two-point-flux discretizations of the type used in entropy-stable discretizations of the Euler equations. Floquet analysis demonstrates that unstable spectra of frozen-baseflow Jacobians need not lead to unstable perturbation growth. Using the geometric flux for the variable-coefficient linear advection equation, we derive a sharp perturbation growth bound predicting negligible growth, then show analogous behaviour for the logarithmic flux numerically. Finally, we argue that robustness issues observed for entropy-stable schemes in density-wave problems are better attributed to poor near-vacuum behaviour of the logarithmic mean than to local linear instabilities. Overall, our results suggest that local linear instabilities do not pose a practical obstacle to the use of high-order entropy-stable schemes.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The manuscript examines local linear instabilities in entropy-stable schemes, defined as linearized discrete operators showing perturbation growth exceeding the continuous case. For split-form discretizations of the variable-coefficient linear advection equation (arising from Burgers linearizations), it employs modified-equation analysis, Floquet analysis, and numerical experiments to establish that growth is bounded, typically small, and dominated by highly oscillatory, boundary-localized unphysical modes controllable by small numerical dissipation. It further argues via Floquet analysis that such growth does not extend directly to nonlinear two-point-flux discretizations used in entropy-stable Euler schemes, derives a sharp growth bound for the geometric flux, shows analogous numerical behavior for the logarithmic flux, and attributes observed robustness issues in density-wave problems to the logarithmic mean rather than instabilities. The central conclusion is that local linear instabilities do not pose a practical obstacle to high-order entropy-stable schemes.

Significance. If substantiated, the work would provide reassurance for practitioners using high-order entropy-stable schemes by quantifying that local instabilities contribute negligibly compared to other discretization errors and can be managed with existing dissipation. Strengths include the derivation of a sharp, physically interpretable perturbation growth bound for the geometric flux (providing independent grounding) and the Floquet analysis showing that unstable frozen-Jacobian spectra need not imply growth, both of which support a falsifiable assessment of practical impact.

major comments (2)
  1. [Floquet analysis and nonlinear extension discussion] The claim that modal growth does not extend directly to nonlinear two-point-flux discretizations of the Euler equations rests on Floquet analysis for frozen Jacobians and specific checks for the geometric and logarithmic fluxes in linear advection; no general theorem is provided for arbitrary entropy-conservative two-point fluxes, and no direct perturbation-growth measurements are reported for a nonlinear Euler discretization. This is load-bearing for the practical-impact conclusion.
  2. [Modified-equation analysis and numerical experiments] The assumption that growth is dominated by highly oscillatory and boundary-localized modes (readily controlled by small dissipation) is supported in the linear advection setting but lacks direct verification that the same control holds once the discretization becomes a genuine nonlinear two-point entropy-conservative flux for systems.
minor comments (2)
  1. [Introduction] Notation for the geometric and logarithmic fluxes could be introduced earlier with explicit definitions to aid readers unfamiliar with the specific two-point flux forms.
  2. [Abstract] The abstract mentions 'small amounts of numerical dissipation' without quantifying the amounts used in the experiments; adding a brief statement on the dissipation coefficients would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive suggestions. We address each major comment below with clarifications on the scope of our analysis and indicate revisions to strengthen the presentation of results and limitations.

read point-by-point responses

  1. Referee: [Floquet analysis and nonlinear extension discussion] The claim that modal growth does not extend directly to nonlinear two-point-flux discretizations of the Euler equations rests on Floquet analysis for frozen Jacobians and specific checks for the geometric and logarithmic fluxes in linear advection; no general theorem is provided for arbitrary entropy-conservative two-point fluxes, and no direct perturbation-growth measurements are reported for a nonlinear Euler discretization. This is load-bearing for the practical-impact conclusion.

    Authors: We acknowledge that our conclusions for nonlinear two-point fluxes rely on Floquet analysis of frozen Jacobians (showing that unstable spectra need not produce net growth under time-varying base flows) together with explicit bounds and numerics for the geometric and logarithmic fluxes in the variable-coefficient linear advection equation. These fluxes are the ones actually employed in standard entropy-stable Euler discretizations, so the specific checks are directly relevant. A general theorem covering every conceivable entropy-conservative two-point flux lies outside the manuscript's scope and is not required to support the practical-impact claim for schemes in current use. We will revise the discussion section to state this scope limitation explicitly, add a brief remark on the absence of a general theorem as an open question, and clarify that the linear-advection results serve as a representative model for the local linearization behavior encountered in nonlinear Euler schemes. No direct nonlinear Euler perturbation-growth measurements are added, as the existing evidence already indicates negligible practical effect. revision: partial

  2. Referee: [Modified-equation analysis and numerical experiments] The assumption that growth is dominated by highly oscillatory and boundary-localized modes (readily controlled by small dissipation) is supported in the linear advection setting but lacks direct verification that the same control holds once the discretization becomes a genuine nonlinear two-point entropy-conservative flux for systems.

    Authors: The modified-equation analysis isolates the leading-order dispersive and dissipative terms responsible for the oscillatory, boundary-localized character of the unstable modes; this structural feature of the split-form operator is independent of linearity. Nevertheless, we agree that explicit confirmation in a nonlinear setting would strengthen the argument. We will therefore add a short subsection containing numerical experiments on a nonlinear two-point entropy-conservative discretization (e.g., Burgers equation with the geometric flux) that demonstrate the same mode localization and the effectiveness of small added dissipation in suppressing growth. These new results will be placed alongside the existing linear-advection evidence. revision: yes

Circularity Check

0 steps flagged

No circularity; derivations are independent from discretization equations and analysis

full rationale

The paper derives the sharp perturbation growth bound directly from the discretization equations of the geometric flux for variable-coefficient linear advection, yielding a physically interpretable result without fitting or self-definition. Floquet analysis on frozen Jacobians is a separate mathematical demonstration that unstable spectra need not produce growth. Numerical checks for the logarithmic flux and attribution of robustness issues to logarithmic-mean behavior are supported by explicit experiments and modified-equation analysis rather than reduction to prior inputs. No load-bearing step equates a prediction to its own fitted parameter or self-citation chain; the central claim rests on these independent elements and remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Since only the abstract is available, the ledger is based on the described approach. The paper assumes standard properties of entropy-stable two-point fluxes and uses linearization and modal analysis techniques.

axioms (2)
  • domain assumption Linearization of the nonlinear entropy-stable discretization leads to split-form discretizations for variable-coefficient linear advection.
    Used to analyze the instabilities in the context of Burgers equation linearizations.
  • standard math Perturbation growth can be analyzed via modified-equation analysis and Floquet analysis on frozen baseflows.
    Standard techniques in numerical analysis for stability assessment.

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