A class of arbitrarily high-order energy-preserving method for nonlinear Klein–Gordon–Schrödinger equations

In this paper, we develop a class of arbitrarily high-order energy-preserving time integrators for the nonlinear Klein–Gordon–Schrödinger equations. We employ Fourier pseudo-spectral method for spatial discretization, resulting in a semi-discrete system. Subsequently, we employ the Petrov-Galerkin method in time to obtain a fully-discrete system. We rigorously demonstrate that the proposed scheme preserves the original energy of the target system. Furthermore, we have proved that the mass of the system is also approximately preserved. To assess the accuracy of our method, we provide a simple estimate of the local error, revealing that the proposed approach achieves a temporal order of 2s. Additionally, we extend the proposed methods to the damped Klein–Gordon–Schrödinger equations, and the corresponding fully-discrete scheme preserves the original energy dissipation law of the system. We present numerical examples to verify the accuracy and robustness of the proposed scheme.