论文标题
非线性Schrödinger方程
High-order mass- and energy-conserving SAV-Gauss collocation finite element methods for the nonlinear Schrödinger equation
论文作者
论文摘要
基于标量辅助变量公式的非线性schrödinger方程提出了任意高阶全分散时空有限元方法的家族，该方法由高斯搭配时间离散化和有限元元素空间离散化组成。事实证明，所提出的方法在离散水平上可以很好地保存质量和能量。建立$ l^\ infty（0，t; h^1）$ norm的$ o（h^p+τ^{k+1}）$的错误限制，其中$ h $和$ h $和$τ$分别表示空间和时间网状尺寸，以及$（p，k）$是空间时间时间的$（p，k）$。提供数值实验来验证收敛速率和保护特性的理论结果。数值结果也证明了所提出的方法在保存孤子波的形状方面的有效性。
A family of arbitrarily high-order fully discrete space-time finite element methods are proposed for the nonlinear Schrödinger equation based on the scalar auxiliary variable formulation, which consists of a Gauss collocation temporal discretization and the finite element spatial discretization. The proposed methods are proved to be well-posed and conserving both mass and energy at the discrete level. An error bound of the form $O(h^p+τ^{k+1})$ in the $L^\infty(0,T;H^1)$-norm is established, where $h$ and $τ$ denote the spatial and temporal mesh sizes, respectively, and $(p,k)$ is the degree of the space-time finite elements. Numerical experiments are provided to validate the theoretical results on the convergence rates and conservation properties. The effectiveness of the proposed methods in preserving the shape of a soliton wave is also demonstrated by numerical results.
