论文标题
GROSS-PITAEVSKII方程式的时间不变的超级会员
Superconvergence of time invariants for the Gross-Pitaevskii equation
论文作者
论文摘要
本文考虑了时间依赖性的毛taevskii方程的数值处理。为了尽可能准确地节省方程式的时间不变，我们提出了一个曲柄 - 尼科尔森型时间离散化，该时间与空间中合适的广义有限元离散化结合在一起。空间离散化基于局部正交分解（LOD）的技术，并允许以$ \ Mathcal {o}（h^6）$的准确性捕获时间不变性。由于时间步进方法的保护特性，保留了这种准确性。此外，我们证明了结果方案近似于$ l^{\ infty}（l^2）$ - 具有订单$ \ MATHCAL {O}（τ^2 + H^4）$的精确解决方案，其中$τ$表示步骤大小。该方法的计算效率在数值实验中证明了具有已知精确解决方案的基准问题。
This paper considers the numerical treatment of the time-dependent Gross-Pitaevskii equation. In order to conserve the time invariants of the equation as accurately as possible, we propose a Crank-Nicolson-type time discretization that is combined with a suitable generalized finite element discretization in space. The space discretization is based on the technique of Localized Orthogonal Decompositions (LOD) and allows to capture the time invariants with an accuracy of order $\mathcal{O}(H^6)$ with respect to the chosen mesh size $H$. This accuracy is preserved due to the conservation properties of the time stepping method. Furthermore, we prove that the resulting scheme approximates the exact solution in the $L^{\infty}(L^2)$-norm with order $\mathcal{O}(τ^2 + H^4)$, where $τ$ denotes the step size. The computational efficiency of the method is demonstrated in numerical experiments for a benchmark problem with known exact solution.
