AI 中文总结
本评论指出被评论文提出的“无流量曲线”实为DPY方法的时空积分曲线，其归属存在错误，后续部分论文引用该构造时未正确归属DPY研究，且未质疑后续离散工作
AI 中文摘要
本评论针对E. Abreu、A. Espirito Santo、W. Lambert与J. Perez发表于《Numer. Methods Partial Differential Equations》第39卷（2023年，第2400-2443页）的arXiv:2106.08363v3 [math.NA]论文。该论文的格式基于时空控制体构建，其侧边界被称为“无流量曲线”，满足微分方程dσ/dt = H(u)/u，且被表述为新提出的对象。我们逐方程证明，该对象是Douglas、Pereira和Yeh提出的局部守恒拉格朗日-欧拉方法（DPY方法，发表于《Comput. Geosci.》第4卷，2000年，第1-40页）的时空积分曲线，无流量区域是这些曲线围成的管域，其标量前向追踪形式已见于Mancuso、Pereira和de Souza的研究（发表于《TEMA》第8卷，2007年，第269-276、277-286页）。争议点并非移动控制体这一通用操作，而是移动控制体的曲线选择：仅当通量H(u)为线性时，H'(u)与H(u)/u完全一致，且仅后者能使侧边界质量通量为零。被评论论文称2000年的DPY论文是首个引入带积分曲线围成的积分管的时空局部守恒的研究，而其摘要却称该对象由其作者首次提出。全文文献记录了术语与归属的变化：2025和2026年发表的多篇论文使用相同构造，却引用后续研究而非DPY论文；另有一篇2026年论文归属表述矛盾，称无流量曲线是DPY积分曲线的“扩展”，但该论文自身的零通量定义和比率常微分方程表明二者完全一致，还指出扩散与弥散不会改变定义向量场，其扩展的是模型、格式与分析，而非连续曲线；后续的离散相关工作未受质疑。
英文摘要
This Comment concerns arXiv:2106.08363v3 [math.NA] by E. Abreu, A. Espirito Santo, W. Lambert and J. Perez, published in Numer. Methods Partial Differential Equations 39 (2023) 2400-2443. That article builds its scheme on space-time control volumes whose lateral boundaries, called "no-flow curves", solve dsigma/dt = H(u)/u and are presented as new. We show, equation by equation, that this object is the space-time integral curve of the locally conservative Eulerian-Lagrangian method of Douglas, Pereira and Yeh [Comput. Geosci. 4 (2000) 1-40], that the no-flow region is the tube those curves bound, and that its scalar forward-tracked form appears in Mancuso, Pereira and de Souza [TEMA 8 (2007) 269-276, 277-286]. At issue is not moving a control volume, which is generic, but which curve moves it: H'(u) and H(u)/u coincide identically only for a linear flux, and only the latter makes lateral mass flux vanish. The commented article itself calls the 2000 paper the first to introduce space-time local conservation, with an integral tube bounded by integral curves, while its abstract calls the same object introduced by the authors. A full-text corpus documents the changing terminology and attribution. Papers published in 2025 and 2026 use the same construction while citing later work but not DPY. Another 2026 paper gives mixed attribution: it calls the no-flow curve an "extension" of the DPY integral curve, although its own zero-flux definition and ratio ODE show identity, and it states that diffusion and dispersion do not modify the defining vector field. It extends the model, scheme and analysis, not the continuous curve. The later discrete contributions are not challenged.
Comments18 pages, 1 figure, 2 tables. Comment on arXiv:2106.08363v3 [math.NA], published as Numer. Methods Partial Differential Equations 39 (2023) 2400-2443, doi:10.1002/num.22972. All quotations and all page, section and equation references are to the published version
