A Fourier Spectral Collocation Method for Space Fractional Klein‐Gordon‐Schrödinger Equations

ABSTRACT In the article, we study an efficient numerical scheme to solve a class of space fractional Klein‐Gordon‐Schrödinger equations with periodic boundary condition. First, we propose finite difference scheme to discrete time derivative, and the space fractional derivative is approximated by Fourier spectral collocation method. Second, we prove that the arising numerical scheme preserves discrete mass and energy conservation laws, respectively. Moreover, the existence, uniqueness, and convergence of the obtained numerical scheme are discussed, and it is shown that the scheme is of the accuracy . However, the resulting numerical scheme faces how to efficiently solve a large fully implicit system at each time step, and it takes too much time in the numerical simulation. Then, we introduce split‐step Fourier spectral collocation method to improve calculation speed. Finally, the numerical experiments including one‐dimensional and two‐dimensional fractional Klein‐Gordon‐Schrödinger systems are given to verify the correctness of theoretical results.