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<p>In this paper, we focus on the high-order fitted scheme for the nonlinear time fractional Gray-Scott model with initial singularity. The regularity of the solution is derived by Euler’s beta function. Subsequently, we enhance the regularity of the solution through decomposition techniques. The fitted difference scheme, based on the non-uniform <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2552_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(L2\)</EquationSource> </InlineEquation>-<InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2552_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(1_\sigma\)</EquationSource> </InlineEquation> method, is then constructed with <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2552_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="168" /> </InlineMediaObject> <EquationSource Format="TEX">\(O(N^{-\min\{2r\alpha,2\}}+M^{-2})\)</EquationSource> </InlineEquation> convergence rate. Here, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2552_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(N\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2552_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(M\)</EquationSource> </InlineEquation> are the number of grids, <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2552_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">\(\alpha\in(0,1)\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2552_Article_IEq7.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">\(r\geq0\)</EquationSource> </InlineEquation> are fractional order and grid grading parameter, respectively. Furthermore, the stability and convergence of the resulting system are rigorously proven by the energy method. Eventually, numerical experiments validate the theoretical analyses, providing strong evidence for the validity of the entire study.</p>

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A fitted scheme for the nonlinear time fractional Gray-Scott model with nonsmooth solutions

  • 期刊论文

  • 发表日期：

    2025年6月22日

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  • Xueyun Deng,
  • Caixia Ou,
  • Zhibo Wang,
  • Seakweng Vong

摘要

In this paper, we focus on the high-order fitted scheme for the nonlinear time fractional Gray-Scott model with initial singularity. The regularity of the solution is derived by Euler’s beta function. Subsequently, we enhance the regularity of the solution through decomposition techniques. The fitted difference scheme, based on the non-uniform \(L2\) - \(1_\sigma\) method, is then constructed with \(O(N^{-\min\{2r\alpha,2\}}+M^{-2})\) convergence rate. Here, \(N\) and \(M\) are the number of grids, \(\alpha\in(0,1)\) and \(r\geq0\) are fractional order and grid grading parameter, respectively. Furthermore, the stability and convergence of the resulting system are rigorously proven by the energy method. Eventually, numerical experiments validate the theoretical analyses, providing strong evidence for the validity of the entire study.