  Stability and convergence of a time-weighted discrete scheme with nonuniform
time steps are established for linear reaction-subdiffusion equations. The
Caupto derivative is approximated at an offset point by using linear and
quadratic polynomial interpolation. Our analysis relies on two tools: a
discrete fractional Gr\"{o}nwall inequality and the global consistency
analysis. The new consistency analysis makes use of an interpolation error
formula for quadratic polynomials, which leads to a convolution-type bound for
the local truncation error. To exploit these two tools, some theoretical
properties of the discrete kernels in the numerical Caputo formula are crucial
and we investigate them intensively in the nonuniform setting. Taking the
initial singularity of the solution into account, we obtain a sharp error
estimate on nonuniform time meshes. The fully discrete scheme generates a
second-order accurate solution on the graded mesh provided a proper grading
parameter is employed. An example is presented to show the sharpness of our
analysis.
