基于虚时间演化与谱方法的一类基态Wigner函数计算方法

Wigner Ground State Calculation Based on Imaginary Time Propagation Method and Spectral Method

基于虚时间演化法,实现基态Wigner函数的数值计算。考虑到Wigner函数是一个高维相空间函数,通过引入合适的谱方法以降低问题的维度,并处理所得控制方程中的全局算子。首先,从Schr9dinger波函数的虚时间演化方程出发,推导得到适用于Wigner函数的控制方程。然后,根据方程中卷积项的不同表达,基于简化Grad矩方法和傅里叶伪谱方法,分别设计两类p空间上的离散方法,并阐述其中的求解细节。此外,基于所提出的数值框架,探讨基于密度泛函理论的基态计算。其中,针对简化Grad矩方法下的密度泛函计算,提出一类密度高阶导数的重构方式。最后,数值算例展示所提方法的有效性,以及方法在多体问题中的应用前景。

Based on imaginary time propagation method,we realize the Wigner ground state calculation.Since Wigner function is a high dimensional phase space function,appropriate spectral methods are introduced to reduce the dimensionality and to handle the global operators.Firstly,with the aid of the governing equation in Schrödinger imaginary time propagation method,governing equation of Wigner version is derived.Then,according to the expression of the convolution term in the equation,discretization methods of momentum space are designed with simplified Grad moment method and Fourier pseudo⁃spectral method,respectively.Corresponding numerical details are discussed.In particular,we consider density functional theory calculation in the framework of our methods.A reconstruction method of high order derivative of density is proposed for simplified Grad moment method.Several numerical experiments demonstrate validity of our methods and the potential in many⁃body problems.