An improved finite integration method for nonlocal nonlinear Schrödinger equations

In this paper, we study the nonlocal nonlinear Schrödinger equation (NNLSE), which is an important extension of the nonlinear Schrödinger equation (NLSE). Since the nonlinearity of NNLSE involves a convolution, numerical approximation of its solution is very expensive. To achieve an efficient numerical method for solving the NNLSE, according to the property of convolution, we firstly present a partial differential equation (PDE) method to transfer the NNLSE from an integro-differential equation to an equivalent or approximate system of PDEs, and the numerical solution of the NNLSE can then be obtained by solving these PDEs. Based on the recently developed finite integration method (FIM), we derive an improved method (IFIM) in this paper. The novelty of this IFIM is that the quadrature method is only used once instead of multiple times for multi-layer integral, so it consumes less running time and provides higher accuracy. In addition, we adapt a second-order operator-splitting method (OS2) at each time-step to ensure a convergent solution for long-time integration. Several numerical experiments are given to verify the efficiency and accuracy of the proposed IFIM for solving the NNLSE.