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<p>This paper considers the general linear method for the time discretization of <i>d</i>-dimensional fractional diffusion equation with spectral fractional Laplacian. Combining with the spectral Galerkin method founded on Fourier-like basis functions in space, a class of high order numerical method with matching accuracy in time and in space is constructed. The main contribution of this paper is to prove that the proposed method is stable and convergent with order <i>p</i> in time, when the general linear method is (<i>k</i>,&#xa0;<i>l</i>)-algebraically stable and has general stage order <i>p</i>. It improves the previous results which required the general linear method to be algebraically stable, i.e. (1,&#xa0;0)-algebraically stable. The optimal spatial error estimate with convergence order depends on initial value and <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3116_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">\(f(\varvec{x},t)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mrow> <mi mathvariant="bold-italic">x</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is also derived. Numerical experiments verify and complement our theoretical results.</p>

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Error estimates of general linear and spectral Galerkin methods for the fractional diffusion equation with spectral fractional Laplacian

  • 期刊论文

  • 发表日期：

    2025年2月3日

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  • Yanming Zhang,
  • Yu Li,
  • Yuexin Yu,
  • Wansheng Wang

摘要

This paper considers the general linear method for the time discretization of d-dimensional fractional diffusion equation with spectral fractional Laplacian. Combining with the spectral Galerkin method founded on Fourier-like basis functions in space, a class of high order numerical method with matching accuracy in time and in space is constructed. The main contribution of this paper is to prove that the proposed method is stable and convergent with order p in time, when the general linear method is (k, l)-algebraically stable and has general stage order p. It improves the previous results which required the general linear method to be algebraically stable, i.e. (1, 0)-algebraically stable. The optimal spatial error estimate with convergence order depends on initial value and \(f(\varvec{x},t)\) f ( x , t ) is also derived. Numerical experiments verify and complement our theoretical results.