An Explicit Finite Difference Scheme for Solving the Space Fractional Nonlinear Schrödinger Equation

Abstract

This paper uses the finite difference method to numerically solve the space fractional nonlinear Schrodinger equation. First, we give some properties of the fractional Laplacian $Δ_h^\alpha.$ Then we construct a numerical scheme which satisfies the mass conservation law without proof and the scheme's order is $o(\tau^2+h^2)$ in the discrete $L^\infty$ norm. Moreover, The scheme conserves the mass conservation and is unconditionally stable about the initial values. Finally, this article gives a numerical example to verify the relevant properties of the scheme.