三次非线性薛定谔方程的低正则性全局误差估计
Low regularity full error estimates for the cubic nonlinear Schrödinger equation
AI总结：
本文针对周期边界三次非线性薛定谔方程，提出结合伪谱法与滤波李分裂的数值格式，证明其对低正则初值的收敛阶并通过数值算例验证。
摘要：
针对带周期边界条件的三次非线性薛定谔方程的数值求解问题，本文研究了空间方向采用伪谱法、时间方向结合滤波李分裂格式的数值方案。证明了该格式即使对正则性极低的初值也具备收敛性，特别地，对属于$H^s(\mathbb T^2)$（其中$s>0$）的初值，证明了其在$L^2$空间下的收敛阶为$\mathcal O(τ^{s/2}+N^{-s})$，其中$τ$表示时间步长，$N$为所采用的傅里叶模态数。该结果的证明在离散Bourgain空间的抽象框架下完成，最终收敛结果在$L^2$空间下给出，所述收敛特性通过若干数值算例进行了验证。
英文摘要：
For the numerical solution of the cubic nonlinear Schrödinger equation with periodic boundary conditions, a pseudospectral method in space combined with a filtered Lie splitting scheme in time is considered. This scheme is shown to converge even for initial data with very low regularity. In particular, for data in $H^s(\mathbb T^2)$, where $s>0$, convergence of order $\mathcal O(τ^{s/2}+N^{-s})$ is proved in $L^2$. Here $τ$ denotes the time step size and $N$ the number of Fourier modes considered. The proof of this result is carried out in an abstract framework of discrete Bourgain spaces, the final convergence result, however, is given in $L^2$. The stated convergence behavior is illustrated by several numerical examples.
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