  A reaction-diffusion problem with a Caputo time derivative is considered. An
integral discretization scheme on a graded mesh along with a decomposition of
the exact solution is proposed. The truncation error estimate of the
discretization scheme is derived by using the remainder formula of the linear
interpolation and some inequality estimate techniques. It is proved that the
scheme is second-order convergent by applying a difference analogue of
Gronwall's inequality, which exhibits an enhancement in the convergence rate
compared with the L1 schemes. Numerical experiments are presented to support
the theoretical result.
