An intelligent non-uniform mesh to improve errors of a stable numerical method for time-tempered fractional advection–diffusion equation with weakly singular solution

This paper introduces a finite volume element method for solving the time-tempered fractional advection–diffusion equation with weakly singular solution at initial time \(t=0\). An innovative approach is proposed to construct an intelligent non-uniform temporal mesh, which significantly reduces errors as compared to using a uniform temporal mesh. The error reduction is quantified in terms of percentage improvement of errors. Due to the presence of a large number of integral calculations involving complicated functions, we used parallel computing techniques to accelerate the computation process. The stability of the method is rigorously proven, and numerical examples are provided to demonstrate the effectiveness of the method and validate the theoretical results.