High-order numerical solution for solving multi-dimensional Schrödinger-Poisson equation

Abstract

This paper explores the numerical solution of the Schrödinger-Poisson equation in one, two, and three dimensions, which has significant applications in quantum mechanics, cosmology, Bose-Einstein condensates, and nonlinear optics. To address the nonlinear aspects of the problem, we employ the split-step method, which decomposes the equation into linear and nonlinear components. The linear part is discretized using the compact finite difference (CFD) method, while the nonlinear component is solved exactly. For temporal discretization, we utilize the Crank-Nicolson method across all dimensions, achieving a second-order convergence rate of $O\left({\tau }^2\right)$. Spatial discretization is carried out using a CFD scheme, ensuring a fourth-order convergence rate of $O\left(h^4\right)$. In the case of two- and three-dimensional Schrödinger equations, the alternating direction implicit (ADI) method is applied. We establish that the proposed numerical schemes are convergent, unconditionally stable, and maintain the conservation of mass and energy at the discrete level. Numerical experiments in one, two, and three dimensions validate the effectiveness of our approach. Specifically, we compare the split-step CFD scheme with alternative methods, and for higher-dimensional cases, we evaluate the ADI-split-step CFD scheme against the standard split-step CFD method. The results demonstrate that the proposed methods significantly reduce computational time while maintaining accuracy.