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<p>This work presents a numerical method for solving a three-dimensional time-fractional reaction-diffusion equation (TFRDE). The solution to this problem has a weak singularity near the initial time. The fractional time derivative is discretized using the <i>L</i>1 method on a nonuniform time grid, while the spatial derivatives are approximated by a fourth-order compact finite difference (CFD) scheme. The resulting fully discrete formulation is computationally expensive, therefore, an alternating direction implicit (ADI) technique is introduced to improve efficiency. The stability and convergence of the proposed scheme are rigorously analyzed. Two numerical experiments are conducted to verify the accuracy and computational efficiency of the proposed method. Theoretical analysis demonstrates that the proposed scheme attains a temporal convergence rate of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\min \{ 2 - \gamma ,\, r\gamma ,\, 1 + \gamma \}\)</EquationSource> </InlineEquation> and fourth-order spatial accuracy. Numerical findings validate the theoretical convergence rates. To demonstrate the advantage of the proposed method, the numerical results obtained by the proposed method are compared with the result reported in Xiao et al., (Commun. Anal. Mech. 16(1):53–70, 2024).</p>

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A fourth-order compact ADI scheme for solving a three-dimensional time-fractional reaction-diffusion equation

  • 期刊论文

  • 发表日期：

    2025年12月20日

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  • Pradip Roul,
  • Vivek Pathak

摘要

This work presents a numerical method for solving a three-dimensional time-fractional reaction-diffusion equation (TFRDE). The solution to this problem has a weak singularity near the initial time. The fractional time derivative is discretized using the L1 method on a nonuniform time grid, while the spatial derivatives are approximated by a fourth-order compact finite difference (CFD) scheme. The resulting fully discrete formulation is computationally expensive, therefore, an alternating direction implicit (ADI) technique is introduced to improve efficiency. The stability and convergence of the proposed scheme are rigorously analyzed. Two numerical experiments are conducted to verify the accuracy and computational efficiency of the proposed method. Theoretical analysis demonstrates that the proposed scheme attains a temporal convergence rate of \(\min \{ 2 - \gamma ,\, r\gamma ,\, 1 + \gamma \}\) and fourth-order spatial accuracy. Numerical findings validate the theoretical convergence rates. To demonstrate the advantage of the proposed method, the numerical results obtained by the proposed method are compared with the result reported in Xiao et al., (Commun. Anal. Mech. 16(1):53–70, 2024).