Some effective numerical schemes for solving Klein-Gordon-Schrödinger system in 2D based on IEQ method

In the paper, we study some energy-preserving schemes to solve Klein-Gordon-Schrödinger system with periodic boundary condition. However, most of energy-preserving schemes of the system face how to efficiently solve a large nonlinear system at each time step, and the accuracy is often lower. Therefore, the linear implicit and high-order energy-preserving schemes of the Klein-Gordon-Schrödinger system are very valuable research work. First, we change original Klein-Gordon-Schrödinger system into an equivalent form based on the IEQ method, which transforms energy conservation law of the original system into quadratic invariants. Second, the linear implicit Crank-Nicolson scheme is presented for the modified Klein-Gordon-Schrödinger system to arrive at semi-discrete scheme, which only requires to solve decoupled linear system at each time step, and which can preserve energy conservation law of the modified system. In addition, we also show high-order energy-preserving scheme for the modified system by symplectic Runge-Kutta in time, which can preserve the discrete mass and energy conservation laws. Third, the Fourier spectral method is applied to the resulted semi-discrete systems with periodic boundary condition, and the efficient iterative algorithms of the fully-discrete systems are given. Moreover, we show some theoretical analysis such as uniqueness and convergence for fully-discrete linear implicit Crank-Nicolson scheme. Finally, the numerical experiments of some Klein-Gordon-Schrödinger systems are given to verify the correctness of theoretical results.