An averaged L1-type compact difference method for time-fractional mobile/immobile diffusion equations with weakly singular solutions

This paper is concerned with a new numerical method for time-fractional mobile/immobile diffusion equations with weakly singular solutions. We propose an averaged L1-type compact difference method, which improves the temporal convergence order of the traditional L1-type method and is also superior to the WSGD-type and L2−1_σ-type methods in terms of regularity requirements. Taking into account the weak singularity of the solution at the initial time, we prove that the proposed method is unconditionally convergent with the convergence order O(τ²|lnτ|+h⁴), where τ and h are the sizes of the time and spatial steps, respectively. Numerical results confirm the theoretical convergence result.