Some new numerical schemes for delay sub-diffusion equations via the idea of symmetry

Abstract

In this work, we propose several novel numerical schemes that synergistically integrates the $L1$ method and symmetric fractional-order reduction (SFOR) method, specifically designed for fractional sub-diffusion equations with time delay. Crucially, we rigorously establish an optimal convergence order of $2 - \alpha/2$ in temporal direction, which improves upon the classical rate of $2-\alpha$ under identical smoothness conditions. To further enhance computational efficiency, we develop a weighted alternating direction implicit (ADI) scheme for the two-dimensional case and analyze its coupling with compact spatial discretization---thereby achieving high-order accuracy in space. Moreover, we introduce a new framework for analyzing the error of $L1$-type discretization on graded temporal meshes, which relaxes the mesh grading requirement and still guarantees optimal convergence rate at continuous time points. Some numerical experiments comprehensively validate both the theoretical accuracy and computational efficiency of the resulting schemes.