Adaptive time-stepping Hermite spectral scheme for nonlinear Schrödinger equation with wave operator: Conservation of mass, energy, and momentum

Abstract

The aim of this paper is to establish an efficient numerical scheme for nonlinear Schrödinger equation with wave operator (NLSW) on unbounded domains to simultaneously conserve the first three kinds of invariants, namely the mass, the energy, and the momentum conservation laws. Regarding the mass and momentum conservation laws as the globally physical constraints, we elaborately combine the exponential scalar auxiliary variable (ESAV) method with Lagrange multiplier approach to build up the algorithm-friendly reformulation which links between the invariants and existing numerical methods. We employ the Crank-Nicolson and Hermite-Galerkin spectral methods for temporal discretization and spatial approximation, respectively. Additionally, we design a new adaptive time-stepping strategy based on the variation of the solution to improve the efficiency of our scheme. At each time level, we only need to solve a linear system plus a set of quadratic algebraic equations which can be efficiently solved by Newton's method. To enhance the applicability of the proposed scheme, we extend our methodology to N-coupled NLSW system where the mass, the energy, and the momentum are simultaneously conserved at the fully-discrete level. Numerical experiments are provided to show the convergence rates, the efficiency, and the conservation properties of the proposed scheme. In addition, the nonlinear dynamics of 2D/3D solitons are simulated to deepen the understanding of NLSW model.