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arxiv: 2606.13185 · v1 · pith:4WKWDF6R · submitted 2026-06-11 · math.NA · cs.NA

Lyapunov Stability and Optimal Error Estimates for an SIPG Method for Weakly Damped Semilinear Wave Equations

Ajeet Singh , Abhinav Jha This is my paper

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classification math.NA cs.NA
keywords SIPGLyapunov stabilitysemilinear wave equationerror estimatesdiscontinuous Galerkinweak dampingCN-BDF2chord-slope linearization
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The pith

A discrete Lyapunov functional establishes stability and optimal error bounds for an SIPG scheme on weakly damped semilinear wave equations.

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

The paper constructs a fully discrete method that pairs symmetric interior penalty discontinuous Galerkin discretization in space with a hybrid Crank-Nicolson second-order backward differentiation formula time stepper, using chord-slope linearization on the nonlinear term. This linearization preserves an exact discrete gradient structure and avoids any global Lipschitz requirement on the nonlinearity. A specially built discrete Lyapunov functional then supplies existence, uniqueness, and uniform boundedness of the numerical solution, while standard regularity assumptions on the exact solution deliver optimal a priori rates of order h to the k plus tau squared in the DG energy norm and h to the k plus one plus tau squared in the L2 norm.

Core claim

The fully discrete SIPG-CN-BDF2 scheme with chord-slope linearization admits a discrete Lyapunov functional that directly yields existence, uniqueness, and uniform boundedness of the solution, while optimal a priori error estimates of order O(h^k + τ²) hold in the DG energy norm and O(h^{k+1} + τ²) in the L² norm under standard regularity assumptions on the exact solution.

What carries the argument

The discrete Lyapunov functional built from the numerical solution, which encodes energy dissipation and closes the stability argument without spectral tools.

If this is right

  • The numerical solution exists and remains uniformly bounded for arbitrary time horizons.
  • Optimal convergence holds simultaneously in the DG energy norm and the L2 norm.
  • The chord-slope linearization preserves a discrete gradient structure without global Lipschitz conditions.
  • Long-time energy dissipation behavior is inherited from the continuous problem.

Where Pith is reading between the lines

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

  • The Lyapunov construction may apply directly to other dissipative semilinear systems where spectral analysis becomes intractable.
  • The absence of a global Lipschitz requirement widens the class of nonlinearities that can be treated while retaining unconditional stability.
  • The observed long-time dissipation in experiments suggests the scheme could serve as a reliable surrogate for studying asymptotic behavior in physical wave models.

Load-bearing premise

The exact solution satisfies standard regularity assumptions that allow the error analysis to close at the stated orders.

What would settle it

A computed solution that grows unbounded in norm for fixed mesh size and time step, or measured convergence rates that fall below O(h^k + τ²) in the energy norm on a sequence of refined meshes with a smooth exact solution.

Figures

Figures reproduced from arXiv: 2606.13185 by Abhinav Jha, Ajeet Singh.

Figure 5.1
Figure 5.1. Figure 5.1: Linear wave equation: Surface plot of uh at T = 0.5 (M = 128, k = 1) [PITH_FULL_IMAGE:figures/full_fig_p023_5_1.png] view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: Linear wave equation: Evolution of the discrete Lyapunov functional [PITH_FULL_IMAGE:figures/full_fig_p023_5_3.png] view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: Cubic nonlinearity: Surface plot of uh at T = 0.5 (M = 128, k = 1, g(u) = u 3 ) [PITH_FULL_IMAGE:figures/full_fig_p024_5_4.png] view at source ↗
Figure 5.6
Figure 5.6. Figure 5.6: Cubic nonlinearity: Evolution of the discrete Lyapunov functional [PITH_FULL_IMAGE:figures/full_fig_p025_5_6.png] view at source ↗
Figure 5.7
Figure 5.7. Figure 5.7: Undamped sine-Gordon (σ = 0): elastic kink–kink interaction with conserved amplitude. (a) Surface, t = 2 (b) Surface, t = 6 (c) Surface, t = 10 (d) Contour, t = 2 (e) Contour, t = 6 (f) Contour, t = 10 [PITH_FULL_IMAGE:figures/full_fig_p026_5_7.png] view at source ↗
Figure 5.8
Figure 5.8. Figure 5.8: Damped sine-Gordon (σ = 1): progressive amplitude decay and front broadening due to energy dissipation. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_5_8.png] view at source ↗
Figure 5.9
Figure 5.9. Figure 5.9: Discrete Lyapunov functional Zh(t): conserved for σ = 0, monotonically decaying for σ = 1. 6 Concluding Remarks We have presented and analyzed a symmetric interior penalty discontinuous Galerkin (SIPG) method com￾bined with the CN–BDF2 time-stepping scheme for the weakly damped semilinear wave equation. The chord-slope operator G(a, b) preserves the exact discrete energy structure without requiring globa… view at source ↗
read the original abstract

We develop and analyze a fully discrete scheme for the weakly damped semilinear wave equation that combines a Symmetric Interior Penalty Discontinuous Galerkin (SIPG) spatial discretization with a hybrid Crank--Nicolson/second-order Backward Differentiation Formula (CN--BDF2) time integrator. A chord-slope linearization of the nonlinear reaction term is employed, which preserves an exact discrete gradient structure and, crucially, requires {no global Lipschitz continuity assumption} on the nonlinearity. Stability of the fully discrete solution is established through a Lyapunov-based analysis-rather than spectral arguments-by constructing a discrete Lyapunov functional that yields existence, uniqueness, and uniform boundedness of the numerical solution. Under standard regularity assumptions, optimal a~priori error estimates of order $\mathcal{O}(h^{k}+\tau^{2})$ in the DG energy norm and $\mathcal{O}(h^{k+1}+\tau^{2})$ in the $L^{2}$-norm are proved, where $h$ is the mesh size, $\tau$ the time step, and $k$ the polynomial degree. Numerical experiments on two-dimensional problems with linear, cubic, and trigonometric nonlinearities confirm the theoretical convergence rates and illustrate the long-time energy-dissipation properties guaranteed by the Lyapunov structure.

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 / 3 minor

Summary. The paper develops a fully discrete SIPG spatial discretization paired with a hybrid CN-BDF2 time integrator for the weakly damped semilinear wave equation. A chord-slope linearization of the nonlinearity is introduced that preserves an exact discrete gradient structure without requiring global Lipschitz continuity. Stability, existence, uniqueness, and uniform boundedness are proved via an explicitly constructed discrete Lyapunov functional rather than spectral methods. Under standard regularity assumptions on the exact solution, optimal a priori error bounds of order O(h^k + τ²) in the DG energy norm and O(h^{k+1} + τ²) in the L² norm are established. Two-dimensional numerical tests with linear, cubic, and trigonometric nonlinearities confirm the predicted rates and long-time energy dissipation.

Significance. If the Lyapunov construction and error analysis are correct, the manuscript supplies a structure-preserving, fully discrete method whose stability proof does not rely on global Lipschitz assumptions or spectral arguments. This is useful for semilinear wave problems whose nonlinearities fail global Lipschitz conditions. The explicit discrete Lyapunov functional and the optimal convergence rates under standard regularity constitute a clear technical contribution to the analysis of DG methods for hyperbolic problems with weak damping.

major comments (2)
  1. [§3.2, Eq. (3.8)] §3.2, Eq. (3.8): the discrete Lyapunov functional is stated to be non-increasing, but the proof that the chord-slope term exactly cancels the nonlinear contribution in the energy identity appears to require an additional summation-by-parts identity that is not displayed; without it the uniform boundedness claim in Theorem 3.1 rests on an implicit step.
  2. [Theorem 4.3] Theorem 4.3: the O(h^{k+1} + τ²) L² error bound invokes an elliptic projection whose approximation properties are quoted from a reference; the constant in front of the nonlinearity term must be shown to remain independent of the mesh size and time step, otherwise the induction argument used to close the error estimate may lose the optimal order.
minor comments (3)
  1. [§2.3] The mesh-regularity assumption (shape-regularity and quasi-uniformity) is invoked for the inverse inequalities in §2.3 but is not restated in the statement of the main theorems; adding a single sentence would improve readability.
  2. [§5] Figure 5.2 (energy decay plots) would benefit from an inset showing the discrete energy over the first 10 time steps to illustrate the initial transient behavior guaranteed by the Lyapunov functional.
  3. [§4] Notation: the symbol E_h^n for the discrete energy is introduced in §3 but reused without redefinition in the error analysis of §4; a brief reminder would prevent confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comments on our manuscript. The suggestions improve the clarity of the stability and error analyses. We address each major comment below and will incorporate the necessary revisions.

read point-by-point responses

  1. Referee: [§3.2, Eq. (3.8)] §3.2, Eq. (3.8): the discrete Lyapunov functional is stated to be non-increasing, but the proof that the chord-slope term exactly cancels the nonlinear contribution in the energy identity appears to require an additional summation-by-parts identity that is not displayed; without it the uniform boundedness claim in Theorem 3.1 rests on an implicit step.

    Authors: We agree that the cancellation step can be made fully explicit. The chord-slope linearization combined with the symmetry of the SIPG form yields an exact telescoping identity that cancels the nonlinear term against the discrete gradient contribution. In the revised manuscript we will insert the required discrete summation-by-parts identity immediately after Eq. (3.8) and before the energy estimate in the proof of Theorem 3.1. revision: yes

  2. Referee: [Theorem 4.3] Theorem 4.3: the O(h^{k+1} + τ²) L² error bound invokes an elliptic projection whose approximation properties are quoted from a reference; the constant in front of the nonlinearity term must be shown to remain independent of the mesh size and time step, otherwise the induction argument used to close the error estimate may lose the optimal order.

    Authors: The uniform bound on the numerical solution furnished by the discrete Lyapunov functional (Theorem 3.1) is independent of both h and τ. Consequently the Lipschitz constant of the nonlinearity, when evaluated along the numerical trajectory, remains bounded by a constant that does not depend on the discretization parameters. We will add a short clarifying paragraph immediately after the elliptic-projection lemma, explicitly invoking the h- and τ-independent bound from Theorem 3.1 to confirm that the constant in the nonlinearity term stays uniform, thereby preserving the optimal order in the induction argument of Theorem 4.3. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via explicit construction

full rationale

The paper derives stability via explicit construction of a discrete Lyapunov functional tied directly to the SIPG + CN-BDF2 scheme with chord-slope linearization; existence/uniqueness/boundedness and optimal error estimates O(h^k + τ²) / O(h^{k+1} + τ²) then follow from standard energy arguments and regularity assumptions without any reduction to fitted inputs, self-definitional loops, or load-bearing self-citations. All steps are independent of the target claims.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard regularity assumptions for error estimates and the preservation of discrete gradient structure by the chosen linearization; no free parameters or invented entities are introduced.

axioms (1)
  • domain assumption standard regularity assumptions on the exact solution
    Invoked to obtain the stated optimal a priori error estimates

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Reference graph

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