Unconditional convergence of a linearly implicit energy-conserving scheme for the 2D NLSW equation

A novel linear modified energy-conserving scheme for the Two-Dimensional (2D) fractional Non Linear Schrödinger Wave (NLSW) equation, recently proposed by Hu et al. (2023), demonstrates high efficiency in long-time computations. However, the convergence analysis provided is conditional, as the time-step size depends on the dimension of the approximation space. This paper aims to establish a linearly implicit energy-conserving scheme with unconditional convergence that is as efficient as the aforementioned scheme for the 2D NLSW equation. Theoretically, discrete modified energy conservation and unique solvability are rigorously proved. Furthermore, an optimal discrete $L^2$ convergence analysis is presented without any restrictions on the time-step size, where the “lifting” technique is utilized to address the most significant difficulty, i.e., the discrete $L^{\infty }$ estimate of numerical solutions. On the numerical side, a fast implementation algorithm is briefly outlined, and several numerical experiments are conducted to validate the theoretical analysis and demonstrate the high efficiency of our scheme.