A second–order approximation scheme for Caputo–Hadamard derivative and its application in fractional Allen–Cahn equation

In light of the advantages of the Caputo–Hadamard fractional derivative in characterizing ultra-slow diffusion phenomena, this paper proposes a second-order approximation scheme to approximate it. Then, for the Allen–Cahn equation with the Caputo–Hadamard fractional derivative in time, a numerical algorithm is designed. This algorithm employs the proposed second-order formula for time discretization. Considering the potential anisotropic behavior of the solution in space, the anisotropic nonconforming quasi-Wilson finite element method is utilized for spatial approximation. The error in the $ L^2 $-norm and the superclose error in the $ H^1 $-norm of this algorithm are analyzed. The global superconvergence in the $ H^1 $-norm is demonstrated through interpolation postprocessing techniques. Numerical examples are given to verify the theoretical results and further investigate the influence of different time derivatives on the dynamic behavior of the solution.