A fast second‐order difference scheme for the space–time fractional equation. Issue 4 (17th January 2019)

• Record Type: Journal Article
• Authors: Xu, Weiyan; Sun, Hong
• Is Part Of: Numerical methods for partial differential equations. Volume 35:Issue 4(2019)
• Page Start: 1326
• Page End: 1342
• Publication Date: 2019-01-17
• Subjects: convergence -- fast method -- finite difference scheme -- space–time fractional equation -- stability; Differential equations, Partial -- Numerical solutions -- Periodicals; 515.353
• DOI: 10.1002/num.22352
• Languages: English
• ISSNs: 0749-159X

Abstract

Abstract : In this paper, a fast second‐order accurate difference scheme is proposed for solving the space–time fractional equation. The temporal Caputo derivative is approximated by ℱ L 2‐1 σ formula which employs the sum‐of‐exponential approximation to the kernel function appeared in Caputo derivative. The second‐order linear spline approximation is applied to the spatial Riemann–Liouville derivative. At each time step, a fast algorithm, the preconditioned conjugate gradient normal residual method with a circulant preconditioner (PCGNR), is used to solve the resulting system that reduces the storage and computational cost significantly. The unique solvability and unconditional convergence of the difference scheme are shown by the discrete energy method. Numerical examples are given to verify numerical accuracy and efficiency of the difference schemes.