Graded Temporal Meshes for Weakly Singular Time-Fractional Diffusion Problems: Error Recovery in the L1 Scheme

Abstract

Weak initial singularities are one of the principal reasons why high-order time discretisations for fractional diffusion equations fail to achieve their formal accuracy on uniform temporal meshes. This paper examines graded meshes as a direct and transparent remedy. The Caputo diffusion problem is reduced to its first sine eigenmode and the exact solution  is used so that the temporal singularity is explicit. A nonuniform L1 formula is applied on meshes . The maximum-in-time temporal error is measured for several grading parameters. For α=0.5, the uniform mesh shows a slowly improving observed order that reaches 0.39 at N=1280, whereas r=2 gives about first-order behaviour and r=3 reaches an observed order of 1.46, close to the theoretical ceiling . For α=0.8, moving from r=1 to the theoretically motivated r=1.5 improves the observed order from 0.75 to 1.09 at N=1280, with continued convergence towards the expected 1.2 limit. The findings show that mesh grading is not merely an efficiency device; it is an accuracy-recovery mechanism that redistributes temporal resolution towards the initial layer. The paper also explains the rate law , the role of first-step error, and the practical trade-offs between aggressive grading, conditioning and fast history evaluation.