The impact of Richardson extrapolation on the asymptotic global error estimation of uniform \(\text {L2-1}_\sigma \) scheme for subdiffusion equation with multiple delays

A numerical algorithm with convergence order \(O(\varDelta ^{3-\alpha }+h^{4})\) $O({\Delta }^{3-\alpha }+h^4)$ is developed for solving subdiffusion equations with multiple time delays, where \(\varDelta \) $\Delta$ and h denote the temporal and spatial step sizes, respectively. The proposed approach is primarily designed to investigate and demonstrate the improvement produced by applying Richardson extrapolation to the uniform L2- \(1_{\sigma }\) $1_{\sigma }$ scheme. Starting from the basic L2- \(1_{\sigma }\) $1_{\sigma }$ method, which conventionally provides second-order temporal convergence, we first derive the asymptotic expansion of its global error. Based on this expansion, a Richardson extrapolation technique in the time direction is constructed to enhance the temporal accuracy. Auxiliary lemmas and theorems are established and proved to characterize the asymptotic behavior of the global error of the uniform L2- \(1_{\sigma }\) $1_{\sigma }$ scheme and to justify the improved convergence properties of the extrapolated method. Numerical experiments are finally presented to confirm the theoretical analysis and to verify the effectiveness of Richardson extrapolation in improving the accuracy of the uniform L2- \(1_{\sigma }\) $1_{\sigma }$ method.