Analysis and efficient implementation of alternating direction implicit finite volume method for Riesz space‐fractional diffusion equations in two space dimensions. Issue 1 (2nd October 2020)

• Record Type: Journal Article
• Authors: Liu, Huan; Zheng, Xiangcheng; Fu, Hongfei; Wang, Hong
• Is Part Of: Numerical methods for partial differential equations. Volume 37:Issue 1(2021)
• Page Start: 818
• Page End: 835
• Publication Date: 2020-10-02
• Subjects: ADI -- fast solution algorithm -- finite volume method -- space‐fractional diffusion equation -- stability and convergence analysis; Differential equations, Partial -- Numerical solutions -- Periodicals; 515.353
• DOI: 10.1002/num.22554
• Languages: English
• ISSNs: 0749-159X

Abstract

Abstract: In this article, we develop a Crank–Nicolson alternating direction implicit finite volume method for time‐dependent Riesz space‐fractional diffusion equation in two space dimensions. Norm‐based stability and convergence analysis are given to show that the developed method is unconditionally stable and of second‐order accuracy both in space and time. Furthermore, we develop a lossless matrix‐free fast conjugate gradient method for the implementation of the numerical scheme, which only has O ( N ) memory requirement and O ( N log N ) computational complexity per iteration with N being the total number of spatial unknowns. Several numerical experiments are presented to demonstrate the effectiveness and efficiency of the proposed scheme for large‐scale modeling and simulations.