Improved High-Order Difference Scheme for the Conservation of Mass and Energy in the Two-Dimensional Spatial Fractional Schrödinger Equation

In this paper, our primary objective is to develop a robust and efficient higher-order structure-preserving algorithm for the numerical solution of the two-dimensional nonlinear spatial fractional Schrödinger equation. This equation, which incorporates fractional derivatives, poses significant challenges due to its non-local nature and nonlinearity, making it essential to design numerical methods that not only achieve high accuracy but also preserve the intrinsic physical and mathematical properties of the system. To address these challenges, we employ the scalar auxiliary variable (SAV) method, a powerful technique known for its ability to maintain energy stability and simplify the treatment of nonlinear terms. Combined with the composite Simpson’s formula for numerical integration, which ensures high precision in approximating integrals, and a fourth-order numerical differential formula for discretizing the Riesz derivative, we construct a highly effective finite difference scheme. This scheme is designed to balance computational efficiency with numerical accuracy, making it suitable for long-time simulations. Furthermore, we rigorously analyze the conserving properties of the numerical solution, including mass and energy conservation, which are critical for ensuring the physical relevance and stability of the results.