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<p>In this paper, we propose and analyze a second-order accurate numerical scheme for the Swift–Hohenberg equation, without relying on any global Lipschitz assumptions. After introducing a linear stabilization technique, we successfully derive the maximum norm of numerical solutions at every stage to obtain the lower bound of the stabilization parameter <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_2839_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\kappa \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>κ</mi> </math></EquationSource> </InlineEquation>. A rigorous original energy stability theory is then established. Furthermore, we present an optimal <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_2839_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">\(\ell ^{\infty }(0,T;\ell ^{2})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>ℓ</mi> <mi>∞</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo>;</mo> <msup> <mi>ℓ</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> convergence estimate and provide several numerical simulations to demonstrate the dynamics.</p>

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On the Convergence and Energy Stability Analysis for a Second-Order Scheme of the Swift–Hohenberg Equation

  • 期刊论文

  • 发表日期：

    2025年4月16日

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  • Jingwei Sun,
  • Haifeng Wang,
  • Hong Zhang,
  • Xu Qian,
  • Songhe Song

摘要

In this paper, we propose and analyze a second-order accurate numerical scheme for the Swift–Hohenberg equation, without relying on any global Lipschitz assumptions. After introducing a linear stabilization technique, we successfully derive the maximum norm of numerical solutions at every stage to obtain the lower bound of the stabilization parameter \(\kappa \) κ . A rigorous original energy stability theory is then established. Furthermore, we present an optimal \(\ell ^{\infty }(0,T;\ell ^{2})\) ℓ ∞ ( 0 , T ; ℓ 2 ) convergence estimate and provide several numerical simulations to demonstrate the dynamics.