A Second-Order Scheme with Nonuniform Time Steps for a Linear Reaction-Subdiffusion Problem

It is reasonable to assume that a discrete convolution structure dominates the local truncation error of any numerical Caputo formula because the fractional time derivative and its discrete approximation have the same convolutional form. We suggest an error convolution structure (ECS) analysis for a class of interpolation-type approximations to the Caputo fractional derivative. Our assumptions permit the use of adaptive time steps, such as is appropriate for accurately resolving the initial singularity of the solution and also certain complex behavior away from the initial time. The ECS analysis of numerical approximations has two advantages: (i) to localize (and simplify) the analysis of the approximation error of a discrete convolution formula on general nonuniform time grids; and (ii) to reveal the error distribution information in the long-time integration via the global consistency error. The core result in this paper is an ECS bound and a global consistency analysis of the nonuniform Alikhanov approximation, which is constructed at an offset point by using linear and quadratic polynomial interpolation. Using this result, we derive a sharp $L^2$-norm error estimate of a second-order Crank-Nicolson-like scheme for linear reaction-subdiffusion problems. An example is presented to show the sharpness of our analysis.