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Error estimates of Crank-Nicolson-type difference schemes for the subdiffusion equation. (English) Zbl 1251.65132
The authors propose a Crank-Nicolson-type difference scheme for solving the subdiffusion equation with fractional derivative. Only a tridiational linear system needs to be solved at each temporal level. The solvability, unconditional stability, and ${H}^{1}$ norm convergence are proved. The convergence order is $min\left\{2-\gamma /2,1+\gamma \right\}$ in the temporal direction and two in the spatial direction. Using the Sobolev embedding inequality, the maximum norm error estimate is obtained. A spatial compact scheme based on the Crank-Nicolson-type difference scheme with the convergence order of $O\left({\tau }^{min\left\{2-\gamma /2,1+\gamma \right\}}+{h}^{4}\right)$ is also presented. Numerical examples support the theoretical analysis. Comparisons with the related existing works are presented to show the effectiveness of the present method. This is an interesting paper.
##### MSC:
 65M15 Error bounds (IVP of PDE) 65M06 Finite difference methods (IVP of PDE) 65M12 Stability and convergence of numerical methods (IVP of PDE) 35R11 Fractional partial differential equations 35K05 Heat equation