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 H-1 norm convergence are proved. The convergence order is 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 O(tau(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.