论文标题
立方非线性schrödinger方程的修改分裂方法
A modified splitting method for the cubic nonlinear Schrödinger equation
论文作者
论文摘要
作为一种经典的时间步变方法，众所周知，奇异分裂方法通过失去两个空间衍生物来达到一阶准确度。在本文中，我们为1D立方非线性schrödinger方程提出了一种修改的分裂方法：\ begin {align*} u^{n+1} = \ mathrm {e}^e}^i \ frac \ frac \ frac \ tau2 \ partial_x^partial_x^2} {\ Mathcal n}_τ\ left [\ mathrm {e}^{i \ frac \ tau2 \ partial_x^2} \ big（π_τ+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ mathrm {e}^{ - 2πiλm_0谢el n} _t（ϕ）= \ mathrm {e}^{ - iλt|π_τx|^2} ϕ，$和$ m_0 $是初始数据的质量。适当地选择过滤器$π_τ$和$π^τ$，它严格地表明，它仅通过损失$ \ frac32 $ - 空间衍生物来达到一阶精度。此外，如果$γ\ in（0,1）$，则该新方法呈现$τ^{\ frac {4γ} {4+γ}} $ in $ l^2 $ -NORM的收敛速率，用于$ H^γ$ -DATA;如果$γ\在[1,2] $中，它表示$τ^{\ frac25（1+γ） - } $的收敛速率，$ l^2 $ - norm in $ h^γ$ -DATA。 ％尤其是％，一阶收敛的初始数据的规律性要求$ l^2 $ -norm仅为$ h^{\ frac32+} $。这些结果比标准（过滤）strang分裂方法的预期结果要好。此外，质量是保守的：$ \ frac1 {2π} \ int _ {\ mathbb t} | u^n（x）|^2 \，d x \ equiv m_0，\ quad n = 0,1，\ ldots，l。 $$的关键想法是基于观察到的，即解决方案的低频和高频组件几乎分开（最多达到一些平滑的组件）。然后，通过分别在低频和高频组件下跟踪溶液行为来构建算法。
As a classical time-stepping method, it is well-known that the Strang splitting method reaches the first-order accuracy by losing two spatial derivatives. In this paper, we propose a modified splitting method for the 1D cubic nonlinear Schrödinger equation: \begin{align*} u^{n+1}=\mathrm{e}^{i\frac\tau2\partial_x^2}{\mathcal N}_τ\left[\mathrm{e}^{i\frac\tau2\partial_x^2}\big(Π_τ+\mathrm{e}^{-2πiλM_0τ}Π^τ\big)u^n\right], \end{align*} with ${\mathcal N}_t(ϕ)=\mathrm{e}^{-iλt|Π_τϕ|^2}ϕ,$ and $M_0$ is the mass of the initial data. Suitably choosing the filters $Π_τ$ and $Π^τ$, it is shown rigorously that it reaches the first-order accuracy by only losing $\frac32$-spatial derivatives. Moreover, if $γ\in (0,1)$, the new method presents the convergence rate of $τ^{\frac{4γ}{4+γ}}$ in $L^2$-norm for the $H^γ$-data; if $γ\in [1,2]$, it presents the convergence rate of $τ^{\frac25(1+γ)-}$ in $L^2$-norm for the $H^γ$-data. %In particular, the regularity requirement of the initial data for the first-order convergence in $L^2$-norm is only $H^{\frac32+}$. These results are better than the expected ones for the standard (filtered) Strang splitting methods. Moreover, the mass is conserved: $$\frac1{2π}\int_{\mathbb T} |u^n(x)|^2\,d x\equiv M_0, \quad n=0,1,\ldots, L . $$ The key idea is based on the observation that the low frequency and high frequency components of solutions are almost separated (up to some smooth components). Then the algorithm is constructed by tracking the solution behavior at the low and high frequency components separately.
