Comments on the paper ”A conservative linear difference scheme for the 2D regularized long-wave equation”, by Xiaofeng Wang, Weizhong Dai and Shuangbing Guo [Applied Mathematics and Computation, 342 (2019) 55-70]
•The convergence analysis and the techniques used in Lemma 3.3 by X. Wang et al. in [1] are based essentially on Sobolev’s inequality in one dimension and they cannot be extended to two dimensions.•All results based on Lemma 3.3 are false, in particular Theorem 3.3, Theorem 4.1, Theorem 5.1 and Theo...
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| Vydáno v: | Applied mathematics and computation Ročník 410; s. 126455 |
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| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Elsevier Inc
01.12.2021
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| Témata: | |
| ISSN: | 0096-3003, 1873-5649 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | •The convergence analysis and the techniques used in Lemma 3.3 by X. Wang et al. in [1] are based essentially on Sobolev’s inequality in one dimension and they cannot be extended to two dimensions.•All results based on Lemma 3.3 are false, in particular Theorem 3.3, Theorem 4.1, Theorem 5.1 and Theorem 5.2.•Regarding the conservation of mass and energy, the initial terms Q0 and E0 depends on U0 and on U1 and the authors consider only Eq. (4) without taking into account Eq. (7). Therfore, there is an error in Theorem 3.1 and Theorem 3.2.•In Theorem 4.1, the authors did not show that u1 is uniquely determined by Eq. (7).
In the above article, a second-order convergence in the maximum norm of the two-dimensional regularized long-wave equation is proved. However, due to a wrong inequality in Lemma 3.3 used in X. Wang et al.[1], there are some fundamental errors in this paper, in particular the proof of Theorem 3.3 and the convergence Theorem are no longer correct. The present brief paper contains clarifying comments. |
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| ISSN: | 0096-3003 1873-5649 |
| DOI: | 10.1016/j.amc.2021.126455 |