An efffcient numerical scheme for two-dimensional nonlinear time fractional Schrödinger equation
Jun Ma, Tao Sun, Hu Chen
TL;DR
This work addresses the two-dimensional nonlinear time-fractional Schrödinger equation with Caputo derivatives, focusing on initial singularities. It proposes a linearized fully discrete scheme that blends the L1 time discretization for $D_t^\alpha$, a backward-type linearization of the nonlinear term, and a five-point finite-difference spatial discretization. The authors establish unconditional stability and a pointwise-in-time convergence result, attaining the local error bound $\|e^n\| \le C(\tau t_n^{\alpha-1} + h^2)$ without any grid-ratio restrictions, aided by a discrete fractional Grönwall framework and truncation-error analysis. Numerical experiments corroborate the theory, demonstrating correct local/global convergence rates for smooth and non-smooth initial data and confirming the method’s robustness across various grid ratios.
Abstract
In this paper, a linearized fully discrete scheme is proposed to solve the two-dimensional nonlinear time fractional Schrödinger equation with weakly singular solutions, which is constructed by using L1 scheme for Caputo fractional derivative, backward formula for the approximation of nonlinear term and five-point difference scheme in space. We rigorously prove the unconditional stability and pointwise-in-time convergence of the fully discrete scheme, which does not require any restriction on the grid ratio. Numerical results are presented to verify the accuracy of the theoretical analysis.