Journal of Scientific Computing

, Volume 75, Issue 2, pp 1102–1127

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

• Xue-lei Lin
• Michael K. Ng
• Hai-Wei Sun
Article

## 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 ($$j\ge 3$$) 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 $$\ell ^2$$-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.

## Keywords

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

## Mathematics Subject Classification

26A33 35R11 65M06 65M12

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## Authors and Affiliations

• Xue-lei Lin
• 1
• Michael K. Ng
• 1
• Hai-Wei Sun
• 2
1. 1.Department of MathematicsHong Kong Baptist UniversityKowloon TongHong Kong
2. 2.Department of MathematicsUniversity of MacauMacaoPeople’s Republic of China