Two linearly implicit energy preserving exponential scalar auxiliary variable approaches for multi-dimensional fractional nonlinear Schrödinger equations

Abstract

In this paper, we propose two linearly implicit energy preserving schemes with constant coefficient matrix for multi-dimensional fractional nonlinear Schrödinger equations. By introducing an exponential auxiliary variable for the nonlinear energy, we first reformulate the original system into an equivalent system, which admits mass and energy conservation laws. Further, by virtue of the Lawson transformation, we rewrite the modified system into another equivalent system, which possesses both the mass and energy conservation laws. As for the two kinds of modified systems, we construct two classes of linearly implicit energy preserving schemes by combining the implicit midpoint method and the extrapolation strategy. Moreover, the proposed schemes enjoy the second-order accuracy in time and spectral accuracy in space, respectively. Numerical results are reported to demonstrate that the proposed numerical schemes have high efficiency for energy preservation in long-time computations.