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<p>Numerically solving parabolic equations with quasiperiodic coefficients is a significant challenge due to the space-filling quasiperiodic solutions that lack translational symmetry and decay. In this work, we propose a highly accurate numerical framework for solving time-dependent quasiperiodic parabolic equations by combining the projection method (PM) for spatial discretization with <i>k</i>-th order backward differentiation formulas (BDF-<i>k</i>) for temporal discretization, named as the PM-BDF method. As a concrete example, we present the PM-BDF2 method, and provide the convergence analysis. Our theoretical analysis and numerical results demonstrate that this approach achieves spectral accuracy in space and second-order accuracy in time. The framework can be easily extended to PM-BDF<i>k</i> schemes (<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_2945_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(k&gt;2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>&gt;</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>), preserving spectral convergence in space while achieving <i>k</i>-th order temporal accuracy.</p>

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Convergence Analysis of PM-BDF Method for Quasiperiodic Parabolic Equations

  • 期刊论文

  • 发表日期：

    2025年5月27日

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  • Kai Jiang,
  • Meng Li,
  • Juan Zhang,
  • Lei Zhang

摘要

Numerically solving parabolic equations with quasiperiodic coefficients is a significant challenge due to the space-filling quasiperiodic solutions that lack translational symmetry and decay. In this work, we propose a highly accurate numerical framework for solving time-dependent quasiperiodic parabolic equations by combining the projection method (PM) for spatial discretization with k-th order backward differentiation formulas (BDF-k) for temporal discretization, named as the PM-BDF method. As a concrete example, we present the PM-BDF2 method, and provide the convergence analysis. Our theoretical analysis and numerical results demonstrate that this approach achieves spectral accuracy in space and second-order accuracy in time. The framework can be easily extended to PM-BDFk schemes ( \(k>2\) k > 2 ), preserving spectral convergence in space while achieving k-th order temporal accuracy.