Mass-, and Energy Preserving Schemes with Arbitrarily High Order for the Klein–Gordon–Schrödinger Equations

We present a class of arbitrarily high-order conservative schemes for the Klein–Gordon Schrödinger equations. These schemes combine the symplectic Runge–Kutta method with the quadratic auxiliary variable approach. We first introduce an auxiliary variable that satisfies a quadratic equation to reformulate the original system into an equivalent one. This reformulated system possesses two strong quadratic invariants: energy and mass. Next, we discretize the reformulated system using symplectic Runge–Kutta methods, yielding a class of semi-discrete systems with arbitrarily high-order accuracy in time. The semi-discrete systems naturally preserve the discrete contour part of the strong invariants and the relationship of the quadratic equation. By eliminating the intermediate variable, we obtain the original energy conservation law. Then, the Fourier pseudo-spectral method is employed to obtain the fully discrete scheme that preserves the original energy and mass. We provide a fast solver to implement the proposed methods effectively. Numerical experiments demonstrate the expected accuracy and conservation of proposed schemes.