半经典薛定谔方程在向量势中的高斯波包变换
The Gaussian wave packets transform for the semi-classical Schrödinger equation with vector potentials
AI总结：
本文利用高斯波包变换将电磁场中的半经典薛定谔方程转换为非振荡的w方程，提出两种数值方法解决w方程，采用半拉格朗日方法处理对流项，傅里叶谱方法处理动能项，方法具有无条件稳定性、谱精度和二阶时间精度。
摘要：
本文通过高斯波包变换将电磁场中的半经典薛定谔方程转换为非振荡的w方程，从而减少计算成本。我们提出两种数值方法求解w方程，其中哈密顿量被分为动能、势能和对流部分，或动能和势能-对流部分。对流或势能-对流部分采用半拉格朗日方法处理，动能部分采用傅里叶谱方法求解。数值方法被证明是无条件稳定的，空间上具有谱精度，时间上具有二阶精度，并且原则上可以扩展到更高阶的时间方案。提供了一维和多维的数值测试以验证所提方法的性质。
英文摘要：
In this paper, we reformulate the semi-classical Schrödinger equation in the presence of electromagnetic field by the Gaussian wave packets transform. With this approach, the highly oscillatory Schrödinger equation is equivalently transformed into another Schrödinger type wave equation, the $w$ equation, which is essentially not oscillatory and thus requires much less computational effort. We propose two numerical methods to solve the $w$ equation, where the Hamiltonian is either divided into the kinetic, the potential and the convection part, or into the kinetic and the potential-convection part. The convection, or the potential-convection part is treated by a semi-Lagrangian method, while the kinetic part is solved by the Fourier spectral method. The numerical methods are proved to be unconditionally stable, spectrally accurate in space and second order accurate in time, and in principle they can be extended to higher order schemes in time. Various one dimensional and multidimensional numerical tests are provided to justify the properties of the proposed methods.
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