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<p>This paper is concerned with the exact solution and pointwise error estimate of supergeneralized viscous Burgers’ (SVB) equation under a conservative and energy-preserving difference scheme. The cut-off function method is adopted for constructing the discrete scheme, and we present the theoretical analysis for the suggested scheme. It is proved that, compared with the original continuous model, this scheme preserves conservation of energy similarly when <i>q</i> and <i>p</i> in the model take different positive integer values. In addition, we use the Browder theorem to validate the conservative difference scheme can be solved, and we further prove the uniqueness of the solution. At last, we come to the conclusion that it is convergent in pointwise sense. More importantly, compared with the previous work Zhang (Appl Math Lett 112:106719, 2021) where the uniqueness of the numerical solution is not proved, we introduce a new method to give a detailed proof of the uniqueness of its solution to fill this gap in this paper. We further refine the model with one parameter <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3222_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(p\ge 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> in Zhang (2021). Numerical results demonstrate the validity of the numerical analysis.</p>

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The pointwise error estimate of a new energy-preserving nonlinear difference method for supergeneralized viscous Burgers’ equation

  • 期刊论文

  • 发表日期：

    2025年4月30日

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  • Yang Shi,
  • Xuehua Yang

摘要

This paper is concerned with the exact solution and pointwise error estimate of supergeneralized viscous Burgers’ (SVB) equation under a conservative and energy-preserving difference scheme. The cut-off function method is adopted for constructing the discrete scheme, and we present the theoretical analysis for the suggested scheme. It is proved that, compared with the original continuous model, this scheme preserves conservation of energy similarly when q and p in the model take different positive integer values. In addition, we use the Browder theorem to validate the conservative difference scheme can be solved, and we further prove the uniqueness of the solution. At last, we come to the conclusion that it is convergent in pointwise sense. More importantly, compared with the previous work Zhang (Appl Math Lett 112:106719, 2021) where the uniqueness of the numerical solution is not proved, we introduce a new method to give a detailed proof of the uniqueness of its solution to fill this gap in this paper. We further refine the model with one parameter \(p\ge 1\) p ≥ 1 in Zhang (2021). Numerical results demonstrate the validity of the numerical analysis.