Mathematical analysis and symmetric fractional-order reduction method for diffusion-wave equations

Caixia Ou; Dakang Cen; Seakweng Vong

arxiv: 2604.05361 · v1 · submitted 2026-04-07 · 🧮 math.NA · cs.NA

Mathematical analysis and symmetric fractional-order reduction method for diffusion-wave equations

Dakang Cen , Caixia Ou , Seakweng Vong This is my paper

Pith reviewed 2026-05-10 19:39 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords fractional wave equationsnonuniform temporal meshesL-type schemessymmetric fractional-order reductionstability analysisconvergence ratesdiffusion-wave equationsnumerical methods

0 comments

The pith

New method solves fractional wave equations accurately on nonuniform meshes

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

The paper introduces a symmetric fractional-order reduction method to create numerical schemes for fractional wave equations on nonuniform time meshes. It adapts L1 and L2-1_sigma schemes for diffusion-wave equations by deriving new optimal parameters that fit the mesh structure. This works under assumptions of only lower regularity on the solution, without needing extra smoothness. Readers would care because many physical fractional models have initial-time singularities that make standard uniform-mesh methods inefficient or unstable. The paper includes stability and convergence analysis plus numerical tests to confirm the schemes perform well in practice.

Core claim

We introduce a symmetric fractional-order reduction (SFOR) method to develop numerical algorithms on nonuniform temporal meshes for fractional wave equations under lower regularity assumptions. The L-type methods including L1 and L2-1_sigma schemes are specifically designed for diffusion-wave equations, and we propose novel optimal parameter selections tailored to nonuniform meshes.

What carries the argument

The symmetric fractional-order reduction (SFOR) method, which symmetrically reduces the fractional derivative order to enable stable L-type time discretizations on nonuniform grids under weak solution regularity.

If this is right

The L1 scheme on nonuniform meshes achieves its optimal convergence rate under the stated lower regularity assumptions.
The L2-1_sigma scheme admits specific parameter choices that preserve accuracy and stability on the given mesh.
Stability and error estimates for both schemes hold without requiring additional solution smoothness.
Numerical experiments confirm the efficiency and accuracy of the resulting algorithms for diffusion-wave problems.

Where Pith is reading between the lines

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

The SFOR reduction could extend to other fractional evolution equations that exhibit similar initial singularities.
Graded versions of the nonuniform meshes might further reduce computational cost while retaining the convergence guarantees.
The parameter optimization procedure may apply to spatial discretizations or to systems in higher dimensions.

Load-bearing premise

The solution has only lower regularity near the initial time, and the nonuniform temporal mesh follows a specific structured pattern that supports derivation of optimal parameters and error bounds.

What would settle it

Numerical tests on a low-regularity solution where the observed convergence rate on a structured nonuniform mesh falls below the predicted order, or where stability fails for the chosen parameters, would disprove the central claims.

read the original abstract

In this work, our aim is to introduce a symmetric fractional-order reduction (SFOR) method to develop numerical algorithms on nonuniform temporal meshes for fractional wave equations under lower regularity assumptions. The $L$-type methods--including $L1$ and $L2$-$1_\sigma$ schemes--are specifically designed for diffusion-wave equations, and we propose novel optimal parameter selections tailored to nonuniform meshes. Finally, several numerical experiments are conducted to validate the efficiency and accuracy of the algorithms.

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.

Desk Editor's Note private letter to a colleague

The paper adapts L1 and L2-1_sigma schemes to nonuniform meshes for fractional diffusion-wave equations via a symmetric reduction and gives tailored parameters, but the convergence under low regularity likely needs explicit mesh-ratio bounds to hold for arbitrary steps.

read the letter

The core point is that this paper takes the standard L1 and L2-1_sigma discretizations for diffusion-wave equations and modifies them with a symmetric fractional-order reduction so they work on nonuniform time meshes while assuming only lower regularity on the solution. It also supplies specific choices for the sigma parameter in the L2-1_sigma scheme that are meant to be optimal for those meshes, followed by numerical tests that check accuracy and efficiency. That is the actual contribution: an incremental but concrete extension of existing L-type methods to a setting where time steps can vary. The experiments appear to support practical performance, which is the part that gives the work some immediate value for people who need to run simulations on adaptive or graded meshes. The approach stays within the usual framework of truncation-error analysis and stability estimates, so the citations and basic setup look consistent with prior work in the area. No machine-checked proofs or released code are mentioned, but that is typical for this style of numerical analysis paper. The soft spot is the one the stress test flags. For the L2-1_sigma scheme the optimal sigma cancels the leading term only when the local step ratios are controlled; the low-regularity hypothesis does not supply a uniform bound on those ratios. If the mesh can have arbitrarily large jumps, the error constants grow and the stated convergence orders no longer follow directly. The paper would need to state clearly whether it restricts to meshes with bounded ratios or absorbs the dependence into the grading parameter; without that clarification the theoretical claims are narrower than the abstract suggests. This is the kind of paper that numerical analysts working on time-fractional PDEs would read for implementation ideas and parameter tuning. A reader who already knows the L1/L2-1_sigma literature can extract the mesh-specific adjustments and test them quickly. It is focused enough and has enough concrete validation to deserve a serious referee, even if the theory needs tightening on the mesh assumptions. I would send it to peer review and ask the referees to check the handling of arbitrary step ratios in the error estimates.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces a symmetric fractional-order reduction (SFOR) method for constructing L-type numerical schemes (L1 and L2-1_σ) for diffusion-wave equations on nonuniform temporal meshes. It claims to derive novel optimal parameter selections for these schemes and to establish stability and convergence under lower regularity assumptions on the solution, with numerical experiments provided to support efficiency and accuracy.

Significance. If the convergence analysis under low regularity holds without hidden mesh-ratio restrictions, the work would offer a useful advance in numerical methods for time-fractional PDEs by permitting optimal-order schemes on graded meshes for solutions with limited smoothness.

major comments (2)

[Convergence analysis for L2-1_σ scheme (likely §4, Theorem on error bound)] The skeptic concern lands: the choice of optimal σ in the L2-1_σ scheme to cancel leading truncation terms on nonuniform meshes produces an error constant depending on local ratios r_n = τ_n/τ_{n-1}. The lower-regularity hypothesis supplies no uniform control on these ratios, so the claimed O(τ^{3-α}) rate in the convergence theorem does not follow from the given estimates. This is load-bearing for the central claim.
[Parameter selection and truncation error analysis (likely §3)] The stability and truncation-error estimates for the adapted L-type schemes on arbitrary nonuniform meshes appear to absorb or assume bounded mesh ratios when deriving the optimal parameters; without an explicit mesh-ratio hypothesis or proof that the lower-regularity data controls the ratios, the optimality claim is not fully justified.

minor comments (2)

[Abstract] The abstract is terse and does not state the specific convergence orders achieved or the precise form of the lower-regularity assumption.
[Introduction / Method definition] Notation for the fractional-order reduction and the definition of the symmetric operator could be clarified with an explicit equation reference early in the paper.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and insightful review of our manuscript. The comments on the convergence analysis for the L2-1_σ scheme and the truncation error estimates raise important points about mesh-ratio dependence that we address below. We will revise the manuscript to clarify these aspects and strengthen the presentation of the results under low regularity assumptions.

read point-by-point responses

Referee: [Convergence analysis for L2-1_σ scheme (likely §4, Theorem on error bound)] The skeptic concern lands: the choice of optimal σ in the L2-1_σ scheme to cancel leading truncation terms on nonuniform meshes produces an error constant depending on local ratios r_n = τ_n/τ_{n-1}. The lower-regularity hypothesis supplies no uniform control on these ratios, so the claimed O(τ^{3-α}) rate in the convergence theorem does not follow from the given estimates. This is load-bearing for the central claim.

Authors: We appreciate this observation and have re-examined the proof of the error bound in Section 4. The SFOR method selects σ locally to cancel the leading term, and the resulting local truncation error involves factors depending on r_n. However, under the low-regularity assumption (solution with time derivatives controlled by t^{β-1} for β > 0), the global error accumulation is handled via discrete fractional integral inequalities and summation-by-parts that absorb the local ratio variations into a mesh-independent constant. The O(τ^{3-α}) rate holds without a uniform bound on r_n. To make this rigorous and explicit, we will expand the proof with an additional lemma bounding the ratio-dependent terms and add a remark in the revised manuscript. revision: partial
Referee: [Parameter selection and truncation error analysis (likely §3)] The stability and truncation-error estimates for the adapted L-type schemes on arbitrary nonuniform meshes appear to absorb or assume bounded mesh ratios when deriving the optimal parameters; without an explicit mesh-ratio hypothesis or proof that the lower-regularity data controls the ratios, the optimality claim is not fully justified.

Authors: We agree that the derivation of the optimal σ in Section 3 is local and involves r_n. The truncation error is shown to be O(τ_n^{3-α}) with a constant that may depend on the local ratio, but the low-regularity hypothesis is used to ensure the summed error remains optimal. We will revise Section 3 to include an explicit statement that the analysis holds for arbitrary nonuniform meshes (no hidden ratio restriction) and provide the detailed estimate showing independence of the global constant from sup r_n. If a mild bounded-ratio assumption is ultimately needed for full rigor, we will add it as a standard hypothesis with justification that it is satisfied by typical graded meshes used in practice. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper introduces the SFOR method as a new construction for adapting L1 and L2-1_σ schemes to nonuniform meshes under lower regularity, then derives optimal parameters and performs stability/convergence analysis from truncation-error estimates. No step reduces a claimed prediction or uniqueness result to a fitted quantity by construction, nor does any load-bearing premise rest solely on a self-citation whose content is itself unverified. The parameter selection for σ is obtained by explicit cancellation of leading error terms on the given mesh, which is an independent calculation rather than a renaming or tautological fit. The analysis therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the approach appears to rest on standard fractional calculus approximations and common assumptions about mesh nonuniformity and solution regularity.

pith-pipeline@v0.9.0 · 5374 in / 1194 out tokens · 36342 ms · 2026-05-10T19:39:45.579236+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Lemma 1.1. … ∂^α_t u(t) = ∂^{α/2}_t (∂^{α/2}_t u(t)) − u'(0) ω^{2-α}(t)
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

r = (4−α)/(2−α) … optimal H¹ convergence

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

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H. Liao, W. McLean, and J. Zhang. A second-order scheme with nonuniform time steps for a linear reaction-subdiffusion problem.Communications in Computational Physics, 30:567–601, 2021

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P. Lyu and S. Vong. Second-order and nonuniform time-stepping schemes for time fractional evolution equations with time-space dependent coefficients.Journal of Scientific Computing, 89:49, 202

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P. Lyu and S. Vong. A symmetric fractional-order reduction method for direct nonuniform approximations of semilinear diffusion-wave equations.Journal of Sci- entific Computing, 93:34, 2022

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K. Mustapha and W. McLean. Superconvergence of a discontinuous galerkin method for fractional diffusion and wave equations.SIAM Journal on Numeri- cal Analysis, 51:491–515, 2013

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K. Mustapha and D. Sch¨ otzau. Well-posedness of hp-version discontinuous galerkin methods for fractional diffusion wave equations.IMA Journal of Numerical Anal- ysis, 34(4):1426–1446, 2014

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J. Shen, M. Stynes, and Z. Sun. Two finite difference schemes for multi-dimensional fractional wave equations with weakly singular solutions.Computational Methods in Applied Mathematics, 21(4):913–928, 2021

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Zhihao Sheng, Yang Liu, and Yonghai Li.l 2error estimate of a nonuniform h2n2 difference scheme for the 2d nonlinear time-fractional reaction-diffusion-wave equa- tion.ZAMM-Journal of Applied Mathematics and Mechanics, 105:e70126, 2025

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Stynes, E

M. Stynes, E. O’Riordan, and J. Gracia. Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation.SIAM Journal on Nu- merical Analysis, 55:1057–1079, 2017

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H. Sun, Z. Sun, and G. Gao. Some temporal second order difference schemes for fractional wave equations.Numerical Methods for Partial Differential Equations, 32(3):970–1001, 2016

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Sun and X

Z. Sun and X. Wu. A fully discrete difference scheme for a diffusion-wave system. Applied Numerical Mathematics, 56(2):193–209, 2006

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Wang and S

Z. Wang and S. Vong. Compact difference schemes for the modified anomalous frac- tional sub-diffusion equation and the fractional diffusion-wave equation.Journal of Computational Physics, 277:1–15, 2014. 25

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[1] [1]

Alikhanov

A. Alikhanov. A new difference scheme for the time fractional diffusion equation. Journal of Computational Physics, 280:424–438, 2015

work page 2015

[2] [2]

N. An, G. Zhao, C. Huang, and X. Yu.α-robust h1-norm analysis of a finite element method for the superdiffusion equation with weak singularity solutions. Computers & Mathematics with Applications, 118:159–170, 2022

work page 2022

[3] [3]

Dakang Cen, Caixia Ou, Zhibo Wang, and Seakweng Vong. Time two-grid tech- nique combined with temporal second order difference method for semilinear frac- tional diffusion-wave equations.Discrete and Continuous Dynamical Systems - Series B, 29(7):3137–3162, 2024

work page 2024

[4] [4]

Lisha Chen, Zhibo Wang, and Seakweng Vong. A second-order weighted adi scheme with nonuniform time grids for the two-dimensional time-fractional telegraph equa- tion.Journal of Applied Mathematics and Computing, 70:5777–5794, 2024

work page 2024

[5] [5]

A fast h3n3-2 σ-based compact adi difference method for time frac- tional wave equations.ZAMM-Journal of Applied Mathematics and Mechanics, 104:e202400431, 2024

Ruilian Du. A fast h3n3-2 σ-based compact adi difference method for time frac- tional wave equations.ZAMM-Journal of Applied Mathematics and Mechanics, 104:e202400431, 2024

work page 2024

[6] [6]

B. Jin, R. Lazarov, and Z. Zhou. An analysis of the l1 scheme for the subdiffusion equation with nonsmooth data.IMA Journal of Numerical Analysis, 36(1):197– 221, 2016

work page 2016

[7] [7]

B. Jin, R. Lazarov, and Z. Zhou. Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data.SIAM Journal on Scientific Computing, 38(1):A146–A170, 2016

work page 2016

[8] [8]

N. Kopteva. Error analysis of the l1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions.Mathematics of Com- putation, 88:2135–2155, 2019

work page 2019

[9] [9]

Kubica, K

A. Kubica, K. Ryszewska, and M. Yamamoto.Time-Fractional Differential Equa- tions: A Theoretical Introduction. Springer, 2020

work page 2020

[10] [10]

Kundaliya and Sudhakar Chaudhary

Pari J. Kundaliya and Sudhakar Chaudhary. Symmetric fractional order reduction method withl1 scheme on graded mesh for time fractional nonlocal diffusion- wave equation of kirchhoff type.Computers and Mathematics with Applications, 149:128–134, 2023

work page 2023

[11] [11]

B. Li, T. Wang, and X. Xie. Analysis of a time-stepping discontinuous galerkin method for fractional diffusion-wave equations with nonsmooth data.Journal of Scientific Computing, 82:4, 2020

work page 2020

[12] [12]

H. Liao, D. Li, and Zhang J. Sharp error estimate of a nonuniform l1 formula for time-fractional reaction subdiffusion equations.SIAM Journal on Numerical Analysis, 56:1112–1133, 2018

work page 2018

[13] [13]

H. Liao, W. McLean, and J. Zhang. A second-order scheme with nonuniform time steps for a linear reaction-subdiffusion problem.Communications in Computational Physics, 30:567–601, 2021

work page 2021

[14] [14]

An energy stable and maximum bound preserving scheme with variable time steps for time fractional allen–cahn equation

Hong-lin Liao, Tao Tang, and Tao Zhou. An energy stable and maximum bound preserving scheme with variable time steps for time fractional allen–cahn equation. SIAM Journal on Scientific Computing, 43:A3503–A3526, 2021

work page 2021

[15] [15]

Lin and C

Y. Lin and C. Xu. Finite difference/spectral approximations for the time-fractional diffusion equation.Journal of Computational Physics, 225:1533–1552, 2007

work page 2007

[16] [16]

H. Luo, B. Li, and X. Xie. Convergence analysis of a petrov-galerkin method for fractional wave problems with nonsmooth data.Journal of Scientific Computing, 80:957–992, 2019. 24

work page 2019

[17] [17]

Lyu and S

P. Lyu and S. Vong. Second-order and nonuniform time-stepping schemes for time fractional evolution equations with time-space dependent coefficients.Journal of Scientific Computing, 89:49, 202

work page

[18] [18]

Lyu and S

P. Lyu and S. Vong. A symmetric fractional-order reduction method for direct nonuniform approximations of semilinear diffusion-wave equations.Journal of Sci- entific Computing, 93:34, 2022

work page 2022

[19] [19]

Mainardi.Fractional Calculus and Waves in Linear Viscoelasticity

F. Mainardi.Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, London 2010

work page 2010

[20] [20]

Mainardi and P

F. Mainardi and P. Paradisi. Fractional diffusive waves.Journal of Computational Acoustics, 9(4):1417–1436, 2001

work page 2001

[21] [21]

Mustapha and W

K. Mustapha and W. McLean. Superconvergence of a discontinuous galerkin method for fractional diffusion and wave equations.SIAM Journal on Numeri- cal Analysis, 51:491–515, 2013

work page 2013

[22] [22]

Mustapha and D

K. Mustapha and D. Sch¨ otzau. Well-posedness of hp-version discontinuous galerkin methods for fractional diffusion wave equations.IMA Journal of Numerical Anal- ysis, 34(4):1426–1446, 2014

work page 2014

[23] [23]

Academic Press, 1999

Igor Podlubny.Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, volume 198 ofMathematics in Science and Engineer- ing. Academic Press, 1999

work page 1999

[24] [24]

J. Shen, M. Stynes, and Z. Sun. Two finite difference schemes for multi-dimensional fractional wave equations with weakly singular solutions.Computational Methods in Applied Mathematics, 21(4):913–928, 2021

work page 2021

[25] [25]

Zhihao Sheng, Yang Liu, and Yonghai Li.l 2error estimate of a nonuniform h2n2 difference scheme for the 2d nonlinear time-fractional reaction-diffusion-wave equa- tion.ZAMM-Journal of Applied Mathematics and Mechanics, 105:e70126, 2025

work page 2025

[26] [26]

Stynes, E

M. Stynes, E. O’Riordan, and J. Gracia. Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation.SIAM Journal on Nu- merical Analysis, 55:1057–1079, 2017

work page 2017

[27] [27]

H. Sun, Z. Sun, and G. Gao. Some temporal second order difference schemes for fractional wave equations.Numerical Methods for Partial Differential Equations, 32(3):970–1001, 2016

work page 2016

[28] [28]

Sun and X

Z. Sun and X. Wu. A fully discrete difference scheme for a diffusion-wave system. Applied Numerical Mathematics, 56(2):193–209, 2006

work page 2006

[29] [29]

Wang and S

Z. Wang and S. Vong. Compact difference schemes for the modified anomalous frac- tional sub-diffusion equation and the fractional diffusion-wave equation.Journal of Computational Physics, 277:1–15, 2014. 25

work page 2014