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<p>In this article, a family of explicit one-step nonlinear numerical schemes is developed and then combined with standard fourth-order compact finite differences to solve a well-known third-order partial differential equation, namely, the Korteweg–de Vries (KdV) equation. The developed numerical scheme turns out to be <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2919_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathcal {A}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation>-stable. The stability of the first-order differential systems arising from the semi-discretization of the KdV equation is also analyzed. Various numerical experiments have been performed by considering test problems related to the KdV equation to evaluate the performance of the proposed scheme. The observed numerical convergence rates align well with the theoretical predictions. Furthermore, the proposed scheme yields more accurate results than several existing methods reported in the literature. The numerical experiments demonstrate that this combined approach offers a robust and effective alternative for solving problems of the type considered.</p>

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A Family of \({\mathcal {A}}\)-Stable Nonlinear Time Integrators Coupled with Compact Finite Differences for Solving Third-Order Korteweg–de Vries Equation

  • 期刊论文

  • 发表日期：

    2025年7月28日

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  • Akansha Mehta,
  • Gurjinder Singh,
  • Higinio Ramos

摘要

In this article, a family of explicit one-step nonlinear numerical schemes is developed and then combined with standard fourth-order compact finite differences to solve a well-known third-order partial differential equation, namely, the Korteweg–de Vries (KdV) equation. The developed numerical scheme turns out to be \({\mathcal {A}}\) A -stable. The stability of the first-order differential systems arising from the semi-discretization of the KdV equation is also analyzed. Various numerical experiments have been performed by considering test problems related to the KdV equation to evaluate the performance of the proposed scheme. The observed numerical convergence rates align well with the theoretical predictions. Furthermore, the proposed scheme yields more accurate results than several existing methods reported in the literature. The numerical experiments demonstrate that this combined approach offers a robust and effective alternative for solving problems of the type considered.