Energy-conserving and time-stepping-varying ESAV-Hermite-Galerkin spectral scheme for nonlocal Klein-Gordon-Schrödinger system with fractional Laplacian in unbounded domains

Abstract

The aim of this paper is to construct a linearized and energy-conserving numerical scheme for nonlocal-in-space Klein-Gordon-Schrödinger system in multi-dimensional unbounded domains Rd (d=1,2, and 3), where the nonlocal property of the system is described by the fractional Laplacian. Firstly, we derive the nonlocal energy conversation law of the system. Then, the Hermite-Galerkin spectral method with scaling factor is employed for the spatial approximation. In the form of the exponential scalar auxiliary variable (ESAV) approach, the Crank-Nicolson scheme with adaptive time-stepping is used for the temporal discretization. To adjust the time-stepping, we propose two kinds of time adaptive strategies depending on the time evolution of the considered system. To avoid solving nonlinear algebra system, the nonlinear terms are explicitly treated by the extrapolation technique. The main advantages of proposed numerical scheme are in three aspects: The first one is that the original nonlocal problem is directly solved in the unbounded domains to avoid the errors and singularities introduced by the domain truncation. The second one is that a linearly explicit and energy-conserving scheme is established with constant coefficients. The third one is the high efficiency without sacrificing accuracy when used in conjunction with the adaptive time-stepping strategies. Numerical experiments are given to demonstrate the accuracy, efficiency, and robustness of the proposed algorithm. As the applications of the scheme, we present numerical simulations of the interactions of 2D/3D vector solitons, which can provide a deeper understanding of nonlinear nonlocal processes in the system.