  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.
