A novel accelerated tempered algorithm with nonuniform time-stepping compact ADI scheme for 2D tempered-fractional nonlinear Schrödinger equations with weak initial singularity

This study numerically examines nonlinear fractional Schrödinger equations with the Caputo tempered fractional derivatives (TFD). We present a novel efficient algorithm for enhanced simulation of Caputo TFD. We develop a fast tempered ${}^{\lambda }\mathrm{FL}1$ scheme with parameter λ to significantly reduce computational and storage requirements, crucial for large-scale problems. This algorithm relies on sum of exponents(SOE) technique. The spatial discretization is achieved through the use of the compact finite difference method. Introducing some small perturbation terms yields fully discrete alternating direction implicit (ADI) schemes. By implementing an adaptive time-stepping strategy, we effectively manage long-time simulations while alleviating the inherent initial singularity through the use of a graded mesh in the temporal domain. A crucial Grönwall-type inequality is derived to rigorously analyze the convergence and stability of the proposed ${}^{\lambda }\mathrm{FL}1$-ADI-CD scheme. Numerical findings are consistent with the anticipated theoretical outcomes, improving precision while substantially lowering computational demands and memory requirements in contrast to classical schemes. This efficiency is further evidenced by a substantial reduction in CPU time. The robustness and reliability of the proposed numerical method are thoroughly validated through extensive numerical experiments. This appears to be first accelerated ${}^{\lambda }\mathrm{FL}1$-ADI-CD time-stepping approach for nonlinear tempered time fractional Schrödinger equation(NL-TFSE).