A split-step finite element method for the space-fractional Schrödinger equation in two dimensions

Area of Science:

• Computational Mathematics
• Numerical Analysis
• Quantum Mechanics

Background:

• The nonlinear Schrödinger equation (NLS) is fundamental in describing wave phenomena in various physical systems.
• Fractional derivatives extend classical models to capture non-local behaviors, crucial for complex dynamics.
• Efficient numerical methods are needed for solving high-dimensional fractional partial differential equations like the space-fractional NLS.

Purpose of the Study:

• To develop and analyze a novel split-step finite element method (FEM) for the 2D nonlinear Schrödinger equation (NLS) with Riesz fractional derivatives.
• To ensure the proposed numerical scheme conserves fundamental physical quantities like mass and energy.
• To reduce the computational cost associated with solving fractional NLS models.

Main Methods:

• Spatial discretization of the fractional NLS using a finite element scheme to obtain a semi-discrete variational formulation.
• Development of a fully discrete split-step FEM, avoiding iterative computations at each time step.
• Mathematical derivation and proof of mass conservation properties and error estimates for the fully discrete scheme.

Main Results:

• The proposed semi-discrete FEM scheme rigorously maintains mass and energy conservation laws.
• The fully discrete split-step FEM significantly reduces computational expense by eliminating time-layer iterations.
• Numerical simulations demonstrate the scheme's effectiveness and capability in capturing the dynamics of wave solutions.

Conclusions:

• The developed split-step FEM is an effective and computationally efficient approach for the 2D Riesz fractional NLS.
• The scheme's ability to conserve mass and energy makes it suitable for long-term simulations of wave phenomena.
• This method provides a robust tool for studying complex wave dynamics in fractional differential equations.