Second-order $H^1$-norm error analysis for time-fractional advection-dispersion equations based on the fast averaged L1 method
In this paper, based on the fast averaged L1 method, we present an error analysis for time-fractional advection-dispersion equations with a weak singularity at the initial time. An integrating-factor transformation is introduced to convert the tempered fractional derivative into the standard Caputo derivative, which is more suitable for discretization using the fast averaged L1 method. A sum-of-exponentials approximation is then incorporated into the averaged L1 method to reduce computational cost and storage while preserving the desired accuracy. By deriving error estimates for the discrete coefficients and the accumulated truncation errors, we establish the stability and $H^1$-norm convergence analysis, with a convergence order higher than those in the published literature. Numerical examples are tested to validate our theoretical results. The effects of the fractional parameters $α$ and $λ$ on the solution are discussed. The memory effect and long-time tail phenomenon, which are known to exist in real systems yet cannot be captured by classical integer-order equations, are again found in the current fractional case.