A Single-Step Correction Scheme of Crank-Nicolson Convolution Quadrature for the Subdiffusion Equation

We develop a new correction scheme for time discretization of the subdiffusion equation based on the fractional Crank-Nicolson convolution quadrature. Due to the weak singularity of solution near time t = 0, a single-step initial correction of the scheme is proposed with rigorous analysis to render the time discretization of second-order accuracy. Optimal error estimates of the numerical schemes are proved for L-2 initial data based on the integral representations of solutions and resolvent estimates of elliptic operator, with regularity assumptions only on the source term. Numerical examples are presented to demonstrate the performance of the proposed method and the consistency with the theoretical analysis.