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Numerical methods for time-fractional subdiffusion equations: Convolution quadrature with block generalized Adams

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This study presents a novel numerical method for time-fractional subdiffusion equations. The findings demonstrate high-order convergence for this important class of partial differential equations.

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Area of Science:

  • Numerical Analysis
  • Partial Differential Equations
  • Mathematical Physics

Background:

  • Time-fractional subdiffusion equations model anomalous diffusion processes.
  • Accurate numerical solutions are crucial for understanding these complex phenomena.
  • Existing methods may face challenges with temporal discretization accuracy.

Purpose of the Study:

  • To develop and analyze a high-order numerical scheme for time-fractional subdiffusion equations.
  • To ensure both stability and convergence of the proposed temporal and spatial approximations.
  • To provide a robust method for solving these equations efficiently.

Main Methods:

  • Temporal approximation using convolution quadrature via the block generalized Adams method.
  • Incorporation of a correction term for improved temporal accuracy.
  • Spatial approximation employing the spectral collocation method.

Main Results:

  • Derivation of the convergence bound for the semi-discrete scheme in time.
  • Analysis of the stability of the convolution quadrature.
  • Demonstration of high-order convergence using both theoretical and numerical evidence, even with uniform temporal grids.

Conclusions:

  • The proposed numerical scheme effectively solves time-fractional subdiffusion equations with high accuracy.
  • The method exhibits excellent convergence properties and stability.
  • This work offers a valuable tool for researchers studying anomalous diffusion.