Mass- and energy-conserving Gauss collocation methods for the nonlinear Schrödinger equation with a wave operator

A fully discrete finite element method with a Gauss collocation in time is proposed for solving the nonlinear Schrödinger equation with a wave operator in the d-dimensional torus, d ∈ { 1 , 2 , 3 } . Based on Gauss collocation method in time and the scalar auxiliary variable technique, the proposed method preserves both mass and energy conservations at the discrete level. Existence and uniqueness of the numerical solutions to the nonlinear algebraic system, as well as convergence to the exact solution with order O (hp+ τ^k+1) in the L^∞(0 , T; H¹) norm, are proved by using Schaefer’s fixed point theorem without requiring any grid-ratio conditions, where (p, k) is the degree of the space-time finite elements. The Newton iterative method is applied for solving the nonlinear algebraic system. The numerical results show that the proposed method preserves discrete mass and energy conservations up to machine precision, and requires only a few Newton iterations to achieve the desired accuracy, with optimal-order convergence in the L^∞(0 , T; H¹) norm.

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