Sharp Error Estimate of the Nonuniform L1 Formula for Linear Reaction-Subdiffusion Equations

Abstract

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.