  This article concerns second-order time discretization of subdiffusion
equations with time-dependent diffusion coefficients. High-order
differentiability and regularity estimates are established for subdiffusion
equations with time-dependent coefficients. Using these regularity results and
a perturbation argument of freezing the diffusion coefficient, we prove that
the convolution quadrature generated by the second-order backward
differentiation formula, with proper correction at the first time step, can
achieve second-order convergence for both nonsmooth initial data and
incompatible source term. Numerical experiments are consistent with the
theoretical results.
