High-order Lagrange multiplier method for the coupled Klein-Gordon-Schrödinger system

Abstract

In this work, a novel class of high-order energy-preserving algorithms are developed for simulating the coupled Klein-Gordon-Schrödinger equations. We introduce a Lagrange multiplier approach and derive a new system which warrants exactly the original total energy, instead of the modified quadratic energy in the previous energy quadratization approach. The Lagrange system is then reformulated equivalently into an optimization problem subject to PDE constraints. We discretize the optimization system by applying the Gauss collocation method together with the prediction-correction technique in time and the sine pseudo-spectral method in space, which leads to a family of fully discrete high-order energy-preserving schemes. In the numerical experiments, we compare this method with other previous methods and verify the high accuracy as well as its ability to maintain the original energy at each time layer. The feasibility and validity of new method are also witnessed for the KGS model in the nonrelativistic limit regime. Additionally, enlightened by this seminal idea, another type of high-order numerical algorithms conserving two invariants of mass and energy are also presented and relevant experimental results are listed to verify the power of the suggested method.