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<p>This paper develops an efficient numerical scheme for solving variable-order fractional diffusion equations with time-dependent. We propose a new numerical treatment for handling situations where three types of derivatives (first-order derivatives, Riemann-Liouville fractional derivatives, and Caputo variable-order fractional derivatives) coexist. Spatial discretization is carried out using the local discontinuous Galerkin method, while temporal discretization is implemented via the L1 formula of variable-order fractional derivatives. An analysis of the stability and convergence of the proposed method is provided. Notably, the proposed scheme achieves a convergence rate of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(O(h^{k+1}+{\Delta t}^{2-\alpha(t_n)}+{\Delta t}^{\alpha(t_n)/2}h^{k+1/2})\)</EquationSource> </InlineEquation>, where the temporal convergence is jointly governed by both the temporal discretization nodes <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({t_n}\)</EquationSource> </InlineEquation> and the variable-order function <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\alpha(t)\)</EquationSource> </InlineEquation>. In the numerical experiments, we performed numerical approximations for solutions with different levels of smoothness. Extensive experimental results confirm the theoretical analysis. The scheme exhibits distinct convergence rates under different variable-order functions <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\alpha(t)\)</EquationSource> </InlineEquation> and final time <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(T\)</EquationSource> </InlineEquation>.</p>

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Analysis of local discontinuous Galerkin method for variable-order fractional diffusion equations with time-dependent

  • 期刊论文

  • 发表日期：

    2025年12月16日

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  • Jun Cao,
  • Xingzhou Jiang,
  • Xiongbo Zheng

摘要

This paper develops an efficient numerical scheme for solving variable-order fractional diffusion equations with time-dependent. We propose a new numerical treatment for handling situations where three types of derivatives (first-order derivatives, Riemann-Liouville fractional derivatives, and Caputo variable-order fractional derivatives) coexist. Spatial discretization is carried out using the local discontinuous Galerkin method, while temporal discretization is implemented via the L1 formula of variable-order fractional derivatives. An analysis of the stability and convergence of the proposed method is provided. Notably, the proposed scheme achieves a convergence rate of \(O(h^{k+1}+{\Delta t}^{2-\alpha(t_n)}+{\Delta t}^{\alpha(t_n)/2}h^{k+1/2})\) , where the temporal convergence is jointly governed by both the temporal discretization nodes \({t_n}\) and the variable-order function \(\alpha(t)\) . In the numerical experiments, we performed numerical approximations for solutions with different levels of smoothness. Extensive experimental results confirm the theoretical analysis. The scheme exhibits distinct convergence rates under different variable-order functions \(\alpha(t)\) and final time \(T\) .