A Linearly-Implicit Energy-Preserving Algorithm for the Two-Dimensional Space-Fractional Nonlinear Schrödinger Equation Based on the SAV Approach (Journal of Computational Mathematics, 2023)
The main objective of this paper is to present an efficient structure-preserving scheme,which is based on the idea of the scalar auxiliary variable approach, for solving the two-dimensional space-fractional nonlinear Schrödinger equation. First, we reformulate theequation as an canonical Hamiltonian system, and obtain a new equivalent system viaintroducing a scalar variable. Then, we construct a semi-discrete energy-preserving schemeby using the Fourier pseudo-spectral method to discretize the equivalent system in spacedirection. After that, applying the Crank-Nicolson method on the temporal directiongives a linearly-implicit scheme in the fully-discrete version. As expected, the proposedscheme can preserve the energy exactly and more efficient in the sense that only decoupledequations with constant coefficients need to be solved at each time step. Finally, numericalexperiments are provided to demonstrate the efficiency and conservation of the scheme.