A divide-and-conquer fast finite difference method for space–time fractional partial differential equation

Hongfei Fu, Kwok Po NG, Hong Wang*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

23 Citations (Scopus)

Abstract

Fractional partial differential equations (FPDEs) provide better modeling capabilities for challenging phenomena with long-range time memory and spatial interaction than integer-order PDEs do. A conventional numerical discretization of space–time FPDEs requires O(N2+MN) memory and O(MN3+M2N) computational work, where N is the number of spatial freedoms per time step and M is the number of time steps. We develop a fast finite difference method (FDM) for space–time FPDE: (i) We utilize the Toeplitz-like structure of the coefficient matrix to develop a matrix-free preconditioned fast Krylov subspace iterative solver to invert the coefficient matrix at each time step. (ii) We utilize a divide-and-conquer strategy, a recursive direct solver, to handle the temporal coupling of the numerical scheme. The fast method has an optimal memory requirement of O(MN) and an approximately linear computational complexity of O(NM(logN+log2M)), without resorting to any lossy compression. Numerical experiments show the utility of the method.

Original languageEnglish
Pages (from-to)1233-1242
Number of pages10
JournalComputers and Mathematics with Applications
Volume73
Issue number6
DOIs
Publication statusPublished - 15 Mar 2017

Scopus Subject Areas

  • Modelling and Simulation
  • Computational Theory and Mathematics
  • Computational Mathematics

User-Defined Keywords

  • Anomalous diffusion
  • Divide-and-conquer method
  • Finite difference method
  • Krylov subspace iterative solver
  • Space–time fractional partial differential equation

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