高维非线性Schrödinger方程具有几乎质量守恒的低正则Fourier积分算法

A low-regularied Fourier integrator for the high dimensions nonlinear Schr dinger equation with almost mass conservation

构建了一种Fourier积分器来求解高维三次非线性Schrödinger方程,这种指数型积分器是显式的,且可通过快速Fourier变换实现一阶收敛.通过严格的分析,证明对任意的γ>d/2,该格式对于Hγ+1空间中的任何初始数据都提供了一阶精度,且满足几乎质量守恒定律.即,固定时间T,存在常数C=C(T,‖u‖L∞([0,T];H^(γ+1)))>0,使得‖u^(n)-u(t_(n))‖H_(γ(Td))≤Cτ,|M(u_(n)-M(u 0)|≤Cτ^(3),其中u^(n)为在t n=nτ处的数值解,M为质量泛函.同时,适当增加修正项,质量可以达到任意阶精度.

In this paper,we propose a Fourier integrator for solving the cubic nonlinear Schrodinger equation in high dimension.The scheme is explicit and can be implemented by using the fast Fourier transform to achieve the first-order.By a rigorous analysis,we prove that the scheme provides the first order accuracy for any initial data belonging to Hγ+1,and obeys the almost mass conservation law.That is,up to some fixed time of T,there exists some constant C=C(T,‖u‖_(L∞([0,T]);H^(γ+1)))>0,such that‖u^(n)-u(t_(n))‖Hγ(T_d)≤Cτ,|M(u ^(n))-M(u 0)|≤Cτ^(3),where u n denotes the numerical solution at t _(n)=nτ.At the same time,any order accuracy can be achieved by adding the modified term appropriately.