An efficient fourth-order structure-preserving scheme for the nonlocal Klein-Gordon-Schrödinger system

The paper presents a family of efficient and high-accurate conservative schemes, which is achieved by using a combination of the diagonally implicit Runge-Kutta method and the quadratic auxiliary variable approach for the nonlocal Klein-Gordon Schrödinger system with Riesz fractional derivative. The process involves first expressing the equation as a new system with modified energy, original energy, and mass by introducing a new variable. The symplectic Runge-Kutta method is then applied to approximate the equivalent system and generate a class of semi-discrete systems with high-order accuracy in the temporal direction. The semi-discrete system is then discretized in space using the fourth-order difference method to obtain a fully-discrete scheme that preserves the original energy and mass. A fast solver is provided to implement the proposed methods in practical computations effectively. The accuracy and conservation properties of the new schemes are demonstrated through numerical experiments.