Temporal second-order H1-norm convergence of the integral-averaged L1 method for time-fractional mobile–immobile diffusion equations

This paper studies the integral-averaged L1 (IAL1) method of the Caputo time-fractional derivative of order α∈(0,1). Under a certain singular regularity condition, Zheng and Wang (2024) proved that on uniform time meshes, the IAL1 method for time-fractional mobile–immobile (MIM) diffusion equations has a temporal convergence order of min{3−2α,2}, which means that when α>1/2, the IAL1 method cannot achieve temporal second-order convergence on uniform time meshes. Furthermore, the numerical experiments by Zheng and Wang (2024) indicate that on a suitably graded time mesh, the IAL1 method can achieve temporal second-order convergence even when α>1/2, but the corresponding convergence analysis remains unavailable. To fill this gap, under relatively weak regularity conditions, we give an H¹-norm error estimate for the IAL1 method of time-fractional MIM diffusion equations on graded time meshes. For certain cases of initially weakly singular solutions, we prove that the method can achieve temporal second-order convergence under a weaker mesh grading constraint for all α∈(0,1). Our proof relies on the positive definiteness of the IAL1 derivative operator established on general time meshes. Numerical results confirm the theoretical results.