Arbitrarily high-order explicit energy-conserving methods for the generalized nonlinear fractional Schrödinger wave equations

A novel category of explicit conservative numerical methods with arbitrarily high-order is introduced for solving the nonlinear fractional Schrödinger wave equations in one and two dimensions. The proposed method is based on the scalar auxiliary variable approach. The equations studied is first transformed into an equivalent system by introducing a scalar auxiliary variable, and the energy is then reformulated as a sum of three quadratic terms. Applying the explicit relaxation Runge–Kutta methods in temporal and the Fourier pseudo-spectral discretization in spatial, the resulting time–space full discrete scheme is proved to preserve the reformulated energy in the discrete level to machine accuracy. The proposed methods improve the numerical stability during long-term computations, as demonstrated through numerical experiments. Also this idea can be easily extended to other similar equations, such as the nonlinear fractional wave equation and the fractional Klein–Gordon–Schrödinger equation.