A coupled numerical method for time-fractional convection-diffusion equations based on compressed Legendre polynomials

In this paper, we develop a novel coupled numerical method for time-fractional convection-diffusion equations. First, in the temporal dimension, a derivative-based linear interpolation is utilized to approximate the time partial derivative. Then, we construct a high-order approximate scheme for the temporal fractional derivative, which transforms the original partial differential equation into a semi-discrete ordinary differential system involving only spatial variables. In the spatial dimension, a standard orthogonal basis is constructed by using compressed Legendre polynomials. Combining the concept of $\epsilon$-approximate solutions, we derive the numerical solution of the semi-discrete scheme. Convergence analysis and numerical experiments demonstrate that the proposed algorithm achieves second-order convergence in time, and the spatial convergence accuracy improves as the Legendre polynomial degree and compression level increase.