Stability and convergence of the L1 formula on nonuniform time grids are studied for solving linear reaction-subdiffusion equations with the Caputo derivative. A discrete fractional Grönwall inequality is developed for the nonuniform L1 formula by introducing a discrete convolution kernel of Riemann--Liouville fractional integral. To simplify the consistency analysis of the nonuniform L1 formula, we bound the local truncation error in a discrete convolution form and consider a global convolution error involving the discrete Riemann--Liouville integral kernel. With the help of discrete fractional Grönwall inequality and global consistency error analysis, a sharp error estimate reflecting the regularity of solution is obtained for a simple L1 scheme. Numerical examples are provided to verify the sharpness of the error analysis.

We consider a class of numerical approximations to the Caputo fractional derivative. Our assumptions permit the use of nonuniform time steps, such as is appropriate for accurately resolving the behavior of a solution whose temporal derivatives are singular at $t=0$. The main result is a type of fractional Grönwall inequality and we illustrate its use by outlining some stability and convergence estimates of schemes for fractional reaction-subdiffusion problems. This approach extends earlier work that used the familiar L1 approximation to the Caputo fractional derivative, and will facilitate the analysis of higher order and linearized fast schemes.

The variable two-step backward differentiation formula (BDF2) is revisited via a new theoretical framework using the positive semi-definiteness of BDF2 convolution kernels and a class of orthogonal convolution kernels. We prove that, if the adjacent time-step ratios $r_k\coloneq \tau _k/\tau _{k-1}\le (3+\sqrt {17})/2\approx 3.561$, the adaptive BDF2 time-stepping scheme for linear reaction-diffusion equations is unconditionally stable and (maybe, first-order) convergent in the $L^2$ norm. The second-order temporal convergence can be recovered if almost all of time-step ratios $r_k\le 1+\sqrt {2}$ or some high-order starting scheme is used. Specially, for linear dissipative diffusion problems, the stable BDF2 method preserves both the energy dissipation law (in the $H^1$ seminorm) and the $L^2$ norm monotonicity at the discrete levels. An example is included to support our analysis.