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<p>The three-step backward difference formula (BDF) method is considered for the two-dimensional incompressible Navier–Stokes equations with <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_2837_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">\(H^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> initial data. Utilizing the energy technique with two multipliers, third-order convergence in time of a semi-implicit three-step BDF method is established on modified graded meshes, which are designed to address the initial singularity. Numerical examples are performed to illustrate the theoretical analysis. To the best of our knowledge, this is the first topic on third-order convergence analysis for the Navier-Stokes equations with <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_2837_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">\(H^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> initial data.</p>

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Third-Order Convergence of Three-Step BDF Method for Navier–Stokes Equations with \(H^1\) Initial Data

  • 期刊论文

  • 发表日期：

    2025年3月10日

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  • Fan Yu,
  • Minghua Chen

摘要

The three-step backward difference formula (BDF) method is considered for the two-dimensional incompressible Navier–Stokes equations with \(H^1\) H 1 initial data. Utilizing the energy technique with two multipliers, third-order convergence in time of a semi-implicit three-step BDF method is established on modified graded meshes, which are designed to address the initial singularity. Numerical examples are performed to illustrate the theoretical analysis. To the best of our knowledge, this is the first topic on third-order convergence analysis for the Navier-Stokes equations with \(H^1\) H 1 initial data.