This paper introduces the energy-preserving scheme for the nonlinear fractional Klein–Gordon Schrödinger equation, which uses the scalar auxiliary variable approach. By a scalar variable, the system is transformed into a new equivalent system. Then applying the extrapolated Crank–Nicolson method on the temporal direction and Fourier pseudospectral method on space direction, we give a linear implicit energy-preserving scheme. Moreover, it proved that at each discrete time the scheme preserves the corresponding discrete mass and energy. The unique solvability and convergence of the numerical solution are also investigated. In particular, it shows the method has the second-order accuracy in time and the spectral accuracy in space. Finally, it gives the algorithm implementation. Several numerical examples illustrate the efficiency and accuracy of the numerical scheme.

本文介绍了非线性分数阶 Klein-Gordon Schrödinger 方程的能量保持方案，它使用标量辅助变量方法。通过一个标量变量，系统被转化为一个新的等价系统。然后在时间方向上应用外推Crank-Nicolson方法和在空间方向上应用傅立叶伪谱方法，我们给出了线性隐式能量保持方案。此外，它证明了在每个离散时间，该方案都保留了相应的离散质量和能量。还研究了数值解的独特可解性和收敛性。特别是，它表明该方法在时间上具有二阶精度，在空间上具有光谱精度。最后给出了算法实现。