Crank–Nicolson Schemes for Sub-Diffusion Equations with Nonsingular and Singular Source Terms in Time

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 verified by some numerical examples.