On banded M-splitting iteration methods for solving discretized spatial fractional diffusion equations

• Published in: BIT Vol. 59; no. 1; pp. 1 - 33
• Main Authors: Bai, Zhong-Zhi, Lu, Kang-Ya
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 04.03.2019; Springer Nature B.V
• Subjects: Computational Mathematics and Numerical Analysis; Convergence; Diffusion; Finite difference method; Iterative methods; Linear systems; Mathematics; Mathematics and Statistics; Numeric Computing; Splitting; Time dependence; 65Z05; 65F10; Convergence property; 65N06; Matrix splitting; Iteration method; Spatial fractional diffusion equation; 65M22; 65F08; 65N22
• ISSN: 0006-3835; 1572-9125
• DOI: 10.1007/s10543-018-0727-8

For solving time-dependent one-dimensional spatial-fractional diffusion equations of variable coefficients, we establish a banded M -splitting iteration method applicable to compute approximate solutions for the corresponding discrete linear systems resulting from certain finite difference schemes at every temporal level, and demonstrate its asymptotic convergence without imposing any extra condition. Also, we provide a multistep variant for the banded M -splitting iteration method, and prove that the computed solutions of the discrete linear systems by employing this iteration method converge to the exact solutions of the spatial fractional diffusion equations. Numerical experiments show the accuracy and efficiency of the multistep banded M -splitting iteration method.