Kernkonzepte
This paper proposes a novel fast fractional block-centered finite difference method to efficiently and accurately solve two-sided space-fractional diffusion equations on general nonuniform grids, addressing the computational bottleneck of traditional methods by employing a sum-of-exponentials approximation technique for fast matrix-vector multiplications.
Zusammenfassung
Bibliographic Information: Kong, M., & Fu, H. (2024). A fast fractional block-centered finite difference method for two-sided space-fractional diffusion equations on general nonuniform grids. Fract. Calc. Appl. Anal.
Research Objective: This paper aims to develop an efficient and accurate numerical method for solving two-sided variable-coefficient space-fractional diffusion equations (SFDEs) with fractional Neumann boundary conditions on general nonuniform grids.
Methodology: The authors propose a fractional Crank-Nicolson block-centered finite difference (CN-BCFD) method, introducing an auxiliary fractional flux variable and utilizing a sum-of-exponentials (SOE) approximation technique to efficiently evaluate the Riemann-Liouville fractional integrals. This approach enables fast matrix-vector multiplications within a Krylov subspace iterative solver (BiCGSTAB), significantly reducing computational complexity.
Key Findings: The proposed fast fractional CN-BCFD method achieves second-order accuracy in both space and time on general nonuniform grids. The SOE approximation allows for efficient evaluation of the fractional integrals, reducing the computational cost of matrix-vector multiplications to O(MNexp) per iteration, where Nexp is significantly smaller than the number of spatial unknowns (M).
Main Conclusions: The fast fractional CN-BCFD method offers a computationally efficient and accurate approach for solving SFDEs on nonuniform grids, overcoming the limitations of traditional methods that struggle with the dense coefficient matrices arising from discretization.
Significance: This research provides a valuable tool for modeling and simulating anomalous diffusion phenomena, particularly in scenarios where nonuniform grids are necessary to capture solution behavior near boundaries.
Limitations and Future Research: The paper primarily focuses on one-dimensional SFDEs. Further research could explore extending the method to higher dimensions and investigating its applicability to other types of fractional differential equations. Additionally, rigorous stability and convergence analysis of the method on general nonuniform grids remains an open question.