Fully-discrete energy-preserving scheme for the space-fractional Klein–Gordon equation via Lagrange multiplier type scalar auxiliary variable approach

A family of effective fully-discrete energy-preserving schemes for the space-fractional Klein–Gordon equation is developed in this paper. First, the recently developed Lagrange multiplier type scalar auxiliary variable approach is employed to obtain a new equivalent system from the original space-fractional Klein–Gordon system. Then, a family of special second-order implicit, explicit and implicit approximations to respectively discretize the linear parts, nonlinear parts and time-derivative parts are obtained in the above equivalent system to establish a family of semi-discrete (continuous in space) energy-preserving schemes. Furthermore, the Fourier pseudo-spectral method is used to discretize the space for extending to the fully-discrete case and rigorous theoretical proofs guarantee its conservation of original energy. Especially, the well-known implicit–explicit Crank–Nicolson type scheme is only one of the above-mentioned schemes. It is inspiring that the main computational efforts of this method in each time step are only to solve two linear, decoupled differential equations with constant coefficients different from non-homogeneous terms, which thus can be effectively solved. Finally, numerical experiments are carried out to verify the theoretical results of the accuracy, efficiency and conservation of original energy.