A second-order scheme with nonuniform time steps for a linear reaction-sudiffusion problem

TL;DR

This paper develops a second-order accurate numerical scheme with nonuniform time steps for linear reaction-subdiffusion equations, ensuring stability and convergence by novel analysis techniques that account for initial singularities.

Contribution

It introduces a new second-order scheme with nonuniform time steps for reaction-subdiffusion equations, utilizing a quadratic interpolation-based Caputo derivative approximation and a novel consistency analysis.

Findings

The scheme is stable and convergent on graded meshes.

Achieves second-order accuracy with proper grading.

Numerical example confirms sharpness of theoretical error estimates.

Abstract

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.