Abstract
This paper develops two efficient numerical methods for solving two-dimensional advection–diffusion equations involving the Caputo–Hadamard fractional derivative. The Caputo-Hadamard fractional derivative is discretized using the L1-2 scheme, which achieves a temporal accuracy of O ( τ 3 − α ) with 0 < α < 1 . To increase spatial accuracy, both methods utilize sixth–order combined compact difference (CCD) formulations with a three–point stencil. The alternating direction implicit (ADI) technique serves to factor the two-dimensional problem into a series of one-dimensional subproblems, which facilitates efficient implementation. Unconditional stability of both schemes is established via Fourier analysis. Convergence of the proposed methods is proved, a contribution absent from the existing literature on combined compact ADI methods. Numerical experiments confirm the agreement between theoretical and computed convergence orders, as well as the high accuracy and computational efficiency of the proposed schemes across a range of test problems.