A discrete Grönwall inequality for L2-type difference schemes with application to multi-term time-fractional nonlinear Sobolev-type convection-diffusion equations with delay

This paper presents a comprehensive numerical analysis of a linearized high-order L2-compact difference scheme. The approach is developed for solving multi-term time-fractional nonlinear Sobolev-type convection-diffusion equations (STCDEs) involving a constant time delay. The multi-term time-fractional derivatives are defined in the Caputo sense with orders ${\beta }_k,{\alpha }_k\in (0,1)$. Temporal discretization employs an L2-type formula to approximate the Caputo time-fractional derivatives, combined with a third-order extrapolation for the nonlinear term. Spatial discretization utilizes a high-order compact difference operator to achieve enhanced accuracy. A novel technique is proposed for analyzing high-order L2-type difference schemes for nonlinear time-fractional differential equations. The approach involves reformulating the L2 formula to enable the application of the discrete fractional Grönwall inequality, which was originally developed for the L1 formula. This strategy is employed to analyze the stability and convergence of the convection-diffusion problem. The results demonstrate that the L2-compact scheme achieves temporal accuracy of order $(3-\max \{{\beta }_k,{\alpha }_k\})$ and fourth-order spatial accuracy. The analysis is extended to a high-order linearized L2-compact difference scheme for two-dimensional multi-term time-fractional nonlinear delayed STCDEs. Several numerical examples are presented to demonstrate the performance and applicability of the proposed schemes, and to validate the corresponding theoretical results.