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Fast permutation preconditioning for fractional diffusion equations.

Sheng-Feng Wang1, Ting-Zhu Huang1, Xian-Ming Gu1

  • 1School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731 Sichuan People's Republic of China.

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|August 2, 2016
PubMed
Summary
This summary is machine-generated.

This study introduces a stable numerical method for fractional diffusion equations, significantly reducing computation time using fast matrix-vector products and novel iterative solvers. The findings demonstrate the effectiveness of these advanced preconditioners for complex scientific computing tasks.

Keywords:
BiCGT methodBiCRT methodFast Fourier transformsFractional diffusion equationsShifted Grünwald formulaToeplitz matrix

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Area of Science:

  • Numerical analysis
  • Computational mathematics
  • Partial differential equations

Background:

  • Fractional diffusion equations model anomalous diffusion processes.
  • Efficient numerical methods are crucial for solving these complex equations.
  • Existing methods may face computational challenges with large datasets.

Purpose of the Study:

  • To develop and analyze an unconditionally stable implicit finite difference scheme for fractional diffusion equations.
  • To reduce the computational complexity of discretizing these equations.
  • To propose efficient preconditioned iterative methods for solving the resulting linear systems.

Main Methods:

  • Discretization using an implicit finite difference scheme with the shifted Grünwald formula.
  • Leveraging the Toeplitz structure of the coefficient matrix for fast matrix-vector products (reducing complexity from O(N^2) to O(N log N)).
  • Developing and applying two preconditioned iterative methods: bi-conjugate gradient for Toeplitz matrix (BCGToep) and bi-conjugate residual for Toeplitz matrix (BCRToep).

Main Results:

  • The proposed finite difference scheme is unconditionally stable.
  • Computational complexity is significantly reduced due to the fast Toeplitz matrix-vector product.
  • Numerical experiments confirm the effectiveness of the proposed preconditioners (BCGToep and BCRToep) in solving the discretized systems.

Conclusions:

  • The developed numerical scheme provides an efficient and stable approach for fractional diffusion equations.
  • The proposed preconditioned iterative methods offer substantial computational advantages.
  • This work contributes to the advancement of numerical methods for fractional partial differential equations.