低正则势与非线性下分数阶非线性薛定谔方程的指数波积分傅里叶谱方法的最优误差界
Optimal error bounds on an exponential wave integrator Fourier spectral method for fractional nonlinear Schrödinger equations with low regularity potential and nonlinearity
AI总结：
针对低正则势与非线性下的分数阶非线性薛定谔方程，提出指数波积分傅里叶谱方法，证明时间半离散和全离散的最优误差界，并揭示与经典方程不同的演化模式。
摘要：
我们为具有低正则势和/或非线性的空间分数阶非线性薛定谔方程（SFNLSE）建立了指数波积分（EWI）方法的最优误差界。对于时间半离散化，在$L^\infty$-势、$C^1$-非线性和$H^\alpha$-解（其中$1<\alpha\leq 2$为$(-\Delta)^{\frac{\alpha}{2}}$的分数阶指数）的假设下，我们证明了半离散数值解的最优一阶$L^2$-范数误差界$O(\tau)$和一致的$H^\alpha$-范数界，其中$\tau$为时间步长。我们进一步用傅里叶谱方法对EWI进行空间离散化，并在不引入任何CFL型时间步长限制的情况下，获得了$L^2$-范数下的最优误差界$O(\tau+h^m)$，其中$h$为空间步长，$m$为精确解的正则性。此外，在稍强的正则性假设下，我们在$H^{\frac{\alpha}{2}}$-范数（与能量相关的范数）下获得了最优误差界$O(\tau)$和$O(\tau+h^{m-{\frac{\alpha}{2}}})$。我们提供了大量数值示例来验证最优误差界并展示其尖锐性。我们还发现了SFNLSE与经典非线性薛定谔方程之间不同的演化模式。
英文摘要：
We establish optimal error bounds on an exponential wave integrator (EWI) for the space fractional nonlinear Schrödinger equation (SFNLSE) with low regularity potential and/or nonlinearity. For the semi-discretization in time, under the assumption of $L^\infty$-potential, $C^1$-nonlinearity, and $H^α$-solution with $1<α\leq 2$ being the fractional index of $(-Δ)^\fracα{2}$, we prove an optimal first-order $L^2$-norm error bound $O(τ)$ and a uniform $H^α$-norm bound of the semi-discrete numerical solution, where $τ$ is the time step size. We further discretize the EWI in space by the Fourier spectral method and obtain an optimal error bound in $L^{2}$-norm $O(τ+h^{m})$ without introducing any CFL-type time step size restrictions, where $h$ is the spatial step size, $m$ is the regularity of the exact solution. Moreover, under slightly stronger regularity assumptions, we obtain optimal error bounds $O(τ)$ and $O(τ+h^{m-{\fracα{2}}})$ in $H^\fracα{2}$-norm, which is the norm associated to the energy. Extensive numerical examples are provided to validate the optimal error bounds and show their sharpness. We also find distinct evolving patterns between the SFNLSE and the classical nonlinear Schrödinger equation.
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