Error Estimates of Crank-Nicolson-Type Difference Schemes for the Subdiffusion Equation

  • Authors:

  • Ya-nan Zhang;Zhi-zhong Sun;Hong-wei Wu

  • Affiliations:

  • zynseu@hotmail.com and hwwu@seu.edu.cn;zzsun@seu.edu.cn;-

  • Venue:
  • SIAM Journal on Numerical Analysis
  • Year:
  • 2011

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Abstract

A Crank-Nicolson-type difference scheme is proposed for solving the subdiffusion equation with fractional derivative, and the truncation error is analyzed in detail. At each temporal level, only a tridiagonal linear system needs to be solved and the Thomas algorithm may be used. The solvability, unconditional stability, and $H^1$ norm convergence are proved. The convergence order is ${\rm min}\{2-{\gamma}/{2},\;1+\gamma\}$ in the temporal direction and two in the spatial direction. By the Sobolev embedding inequality, we obtain the maximum norm error estimate. A spatial compact scheme based on the Crank-Nicolson-type difference scheme is also presented, and similar results are given. The convergence order is $\mathcal{O}(\tau^{{\rm min}\{2-{\gamma}/{2},\;1+\gamma\}}+h^4)$. Numerical experiments are included to support the theoretical results, and comparisons with the related works are presented to show the effectiveness of our method.