发表机构
University of Ioannina; Institute of Applied and Computational Mathematics, FORTH; The Hong Kong Polytechnic University（约阿尼纳大学; 应用与计算数学研究所，FORTH; 香港理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种时空有限元方法，用于非线性薛定谔方程，能在离散层面保持质量、能量和动量守恒，并通过数值算例验证其高阶收敛性和长期模拟中的误差优势。
AI 中文摘要
本文针对非线性薛定谔方程解的模拟，提出了一种新颖的公式化方法及相应的时空有限元方法。所提算法的一个主要优势在于其内在能力，即在离散层面上保持质量、能量和动量的守恒性。这一点对于由全离散隐式格式确定的数值解得到了证明。基于一个等价公式，本文提出了一种有效的迭代方案来求解非线性系统，该方案建议对解采用牛顿迭代，而对非线性系统中的拉格朗日乘子则不进行迭代。本文提供了大量的数值算例，以展示所提算法在一维马孤子和双孤子以及由非线性薛定谔方程控制的二维孤子模拟中，具有高阶收敛性以及在保持质量、能量和动量方面的有效性。数值结果表明，与不守恒这些量的方法相比，本文设计的质量、能量和动量守恒方法在长时间模拟中还能显著减少数值解的误差。
英文摘要
This paper introduces a novel formulation and an associated space-time finite element method for simulating solutions to the nonlinear Schrödinger equation. A major advantage of the proposed algorithm is its intrinsic ability to preserve the conservation of mass, energy, and momentum at the discrete level. This is proved for the numerical solutions determined by the fully discrete implicit scheme. An effective iterative scheme is proposed for solving the nonlinear system based on an equivalent formulation which suggests using Newton's iteration for the solution and no iteration for the Lagrange multipliers in the nonlinear system. Extensive numerical examples are provided to demonstrate the high-order convergence and effectiveness of the proposed algorithm in conserving mass, energy, and momentum in the simulation of one-dimensional Ma-solitons and bi-solitons, as well as of two-dimensional solitons governed by the nonlinear Schrödinger equation. The numerical results show that the mass-, energy- and momentum-conserving method designed in this paper also significantly reduces the errors of the numerical solutions in long-time simulations compared with methods which do not conserve these quantities.
