A backward Euler alternating direction implicit difference scheme for the three‐dimensional fractional evolution equation. Issue 3 (18th December 2017)

• Record Type: Journal Article
• Authors: Chen, Hongbin; Xu, Da; Cao, Jiliang; Zhou, Jun
• Is Part Of: Numerical methods for partial differential equations. Volume 34:Issue 3(2018)
• Page Start: 938
• Page End: 958
• Publication Date: 2017-12-18
• Subjects: backward Euler ADI -- convolution quadrature -- difference scheme -- numerical experiments -- three‐dimensional fractional evolution equation -- unconditional stability and convergence; Differential equations, Partial -- Numerical solutions -- Periodicals; 515.353
• DOI: 10.1002/num.22239
• Languages: English
• ISSNs: 0749-159X

Abstract

Abstract : A backward Euler alternating direction implicit (ADI) difference scheme is formulated and analyzed for the three‐dimensional fractional evolution equation. In our method, the Riemann‐Liouville fractional integral term is treated by means of first order convolution quadrature suggested by Lubich. Meanwhile, an ADI technique is adopted to reduce the multidimensional problem to a series of one‐dimensional problems. A fully discrete difference scheme is constructed with space discretization by finite difference method. Two new inner products and corresponding norms are defined to analyze the scheme. The verification of stability and convergence is based on the nonnegative character of the real quadratic form associated with the convolution quadrature. Numerical experiments are reported to demonstrate the efficiency of our scheme.