二维非线性薛定谔方程的线性隐式保能积分因子方法及收敛性分析
Linearly implicit energy-preserving integrating factor methods for the 2D nonlinear Schrödinger equation with wave operator and convergence analysis
摘要：
本文将标量辅助变量法与积分因子方法相结合，发展了一类求解二维非线性波算子薛定谔方程的线性保能积分因子方法.首先提出了一个二阶格式，并严格证明了该格式是能量守恒的。通过使用能量方法，我们分析了其在$H^1$范数下的最优收敛性，而不对网格比做任何限制，其中提出了一种新的技术和一种改进的归纳论证，以克服数值解的先验$L^\infty$估计不可用所带来的困难.基于积分因子龙格-库塔法，我们将所提出的格式推广到任意高阶，该格式也是线性的和保守的。数值实验验证了理论分析的正确性，并证明了所提方法的优越性。
英文摘要：
In this paper, we develop a novel class of linear energy-preserving integrating factor methods for the 2D nonlinear Schrödinger equation with wave operator (NLSW), combining the scalar auxiliary variable approach and the integrating factor methods. A second-order scheme is first proposed, which is rigorously proved to be energy-preserving. By using the energy methods, we analyze its optimal convergence in the $H^1$ norm without any restrictions on the grid ratio, where a novel technique and an improved induction argument are proposed to overcome the difficulty posed by the unavailability of a priori $L^\infty$ estimates of numerical solutions. Based on the integrating factor Runge-Kutta methods, we extend the proposed scheme to arbitrarily high order, which is also linear and conservative. Numerical experiments are presented to confirm the theoretical analysis and demonstrate the advantages of the proposed methods.
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