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<p>When numerically simulating problems involving moving interfaces using the generalized finite element method, there arises a significant challenge due to the variation of the approximation spaces. Therefore, the time-stepping scheme in stable generalized finite element method (SGFEM) has to be treated carefully. The complexity and precision of algorithms are important considerations when addressing problems with moving interfaces. This paper aims at proposing a robust, efficient, and high-precision numerical scheme for solving one-dimensional parabolic problems with moving interfaces. The discrete scheme is derived by using the Crank-Nicolson scheme or central difference scheme in temporal direction, while employing the modified SGFEM in spatial direction, in which an exponential enrichment function is utilized. The proposed methods reduce the complexity of the spatial discretization and decreases the number of degrees of freedom as compared to the corrected extended finite element method (XFEM), and it has a second-order convergence rate in both the temporal and spatial directions with respect to the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2471_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(\boldsymbol{L^2}\)</EquationSource> </InlineEquation> norm. Finally, extensive numerical experiments are carried out to verify the optimal <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2471_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(\boldsymbol{L^2}\)</EquationSource> </InlineEquation> convergence for the proposed algorithms. The numerical results also show that the proposed algorithms are free from the ratio of the time step to the spatial mesh parameter, and possess the capability to capture the weak discontinuities of moving interface problems.</p>

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Numerical simulation of one-dimensional parabolic problems with moving interfaces by using two second order time-stepping schemes in SGFEM

  • 期刊论文

  • 发表日期：

    2025年4月28日

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  • Pengfei Zhu,
  • Ming Xu,
  • Kai Liu

摘要

When numerically simulating problems involving moving interfaces using the generalized finite element method, there arises a significant challenge due to the variation of the approximation spaces. Therefore, the time-stepping scheme in stable generalized finite element method (SGFEM) has to be treated carefully. The complexity and precision of algorithms are important considerations when addressing problems with moving interfaces. This paper aims at proposing a robust, efficient, and high-precision numerical scheme for solving one-dimensional parabolic problems with moving interfaces. The discrete scheme is derived by using the Crank-Nicolson scheme or central difference scheme in temporal direction, while employing the modified SGFEM in spatial direction, in which an exponential enrichment function is utilized. The proposed methods reduce the complexity of the spatial discretization and decreases the number of degrees of freedom as compared to the corrected extended finite element method (XFEM), and it has a second-order convergence rate in both the temporal and spatial directions with respect to the \(\boldsymbol{L^2}\) norm. Finally, extensive numerical experiments are carried out to verify the optimal \(\boldsymbol{L^2}\) convergence for the proposed algorithms. The numerical results also show that the proposed algorithms are free from the ratio of the time step to the spatial mesh parameter, and possess the capability to capture the weak discontinuities of moving interface problems.