In this work, two Crank-Nicolson schemes without corrections are developed for sub-diffusion equations. First, we propose a Crank-Nicolson scheme without correction for problems with regularity assumptions only on the source term. Second, since the existing Crank-Nicolson schemes have a severe reduction of convergence order for solving sub-diffusion equations with singular source terms in time, we then extend our scheme and propose a new Crank-Nicolson scheme for problems with singular source terms in time. Second-order error estimates for both the two Crank-Nicolson schemes are rigorously established by a Laplace transform technique, which are numerically verified by some numerical examples.

翻译：本文针对子扩散方程，提出了两种无需校正的Crank-Nicolson格式。首先，针对仅对源项具有正则性假设的问题，我们提出了一种无需校正的Crank-Nicolson格式。其次，鉴于现有Crank-Nicolson格式在求解含时间奇异源项的子扩散方程时存在收敛阶严重降低的问题，我们在原有格式基础上进行拓展，提出了一种适用于时间奇异源项问题的新型Crank-Nicolson格式。通过Laplace变换技术，严格建立了两种Crank-Nicolson格式的二阶误差估计，并通过数值算例进行了验证。