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

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 H¹ norm convergence are proved. The convergence order is min\2-{/2, \;1+\} 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 O (^{ min\2-{/2, \;1+\}}+h⁴). Numerical experiments are included to support the theoretical results, and comparisons with the related works are presented to show the effectiveness of our method.