Stability and Convergence Analysis of Finite Difference Schemes for Time-Dependent Space-Fractional Diffusion Equations with Variable Diffusion Coefficients

Xue lei Lin, Kwok Po NG*, Hai Wei Sun

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

9 Citations (Scopus)

Abstract

In this paper, we study and analyze Crank–Nicolson temporal discretization with high-order spatial difference schemes for time-dependent Riesz space-fractional diffusion equations with variable diffusion coefficients. To the best of our knowledge, there is no stability and convergence analysis for temporally 2nd-order or spatially jth-order (j≥ 3) difference schemes for such equations with variable coefficients. We prove under mild assumptions on diffusion coefficients and spatial discretization schemes that the resulting discretized systems are unconditionally stable and convergent with respect to discrete ℓ2-norm. We further show that several spatial difference schemes with jth-order (j= 1 , 2 , 3 , 4) truncation error satisfy the assumptions required in our analysis. As a result, we obtain a series of temporally 2nd-order and spatially jth-order (j= 1 , 2 , 3 , 4) unconditionally stable difference schemes for solving time-dependent Riesz space-fractional diffusion equations with variable coefficients. Numerical results are presented to illustrate our theoretical results.

Original languageEnglish
Pages (from-to)1102-1127
Number of pages26
JournalJournal of Scientific Computing
Volume75
Issue number2
DOIs
Publication statusPublished - 1 May 2018

Scopus Subject Areas

  • Software
  • Theoretical Computer Science
  • Numerical Analysis
  • Engineering(all)
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics

User-Defined Keywords

  • Convergence
  • High-order finite difference schemes
  • Stability
  • Time-dependent space-fractional diffusion equation
  • Variable diffusion coefficients

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