Mass-, Energy-, and Momentum-Preserving Spectral Scheme for Klein-Gordon-Schrödinger System on Infinite Domains

Abstract

For the nonlinear conservative system, how to design an efficient scheme to preserve as manyinvariants as possible is a challenging task. The aim of this paper is to construct the finite difference/spectral method for the Klein–Gordon–Schrödinger (KGS) system on infinite domains ( , and 3) to conserve three kinds of the most important invariants, namely, the mass, the energy, and the momentum. Regarding the mass- and momentum-conservation laws as globally physical constraints, we elaborately combine the exponential scalar auxiliary variable (ESAV) approach and Lagrange multiplier approach to construct the ESAV/Lagrange multiplier reformulation of the KGS system to preserve its original energy-conservation law. When solving the ESAV/Lagrange multiplier reformulation, we employ the Hermite–Galerkin spectral method for the spatial approximation and apply the Crank–Nicolson scheme for the temporal discretization. At each time level, we only need to solve linearly algebraic systems with constant coefficients and a set of quadratic algebraic equations which can be efficiently solved by Newton iteration. We establish the conservation properties of the proposed method at the fullydiscrete level, which indicates that our scheme can preserve conserved quantities including mass, energy, and momentums of the KGS system. Numerical experiments are carried out to demonstrate the accuracy and conservation properties of the proposed scheme. As the applications of our scheme, we simulate the nonlinear interactions of vector solitons for KGS system in 2 dimensional/3 dimensional cases.