Gaussian wave packet transform based numerical scheme for the semi-classical Schrodinger equation with random inputs

JOURNAL OF COMPUTATIONAL PHYSICS

卷:401

Article

Jin, Shi^1,2  Liu, Liu³  Russo, Giovanni⁴  Zhou, Zhennan⁵ 

[1] Shanghai Jiao Tong Univ, Sch Math Sci, Inst Nat Sci, MOE LSC, Shanghai, Peoples R China

[2] Shanghai Jiao Tong Univ, SHL MAC, Shanghai, Peoples R China

[3] Univ Texas Austin, ICES, Austin, TX 78712 USA

[4] Univ Catania, Dept Math & Comp Sci, Via A Doria 6, I-95125 Catania, Italy

[5] Peking Univ, Beijing Int Ctr Math Res, Beijing, Peoples R China

关键词: Schrodinger equation;    Uncertainty quantification;    Gaussian wave packet;    Stochastic collocation;   

DOI  :  10.1016/j.jcp.2019.109015

来源: Elsevier

【 摘 要 】

In this work, we study the semi-classical limit of the Schrodinger equation with random inputs, and show that the semi-classical Schrodinger equation produces O(epsilon) oscillations in the random variable space. With the Gaussian wave packet transform, the original Schrodinger equation is mapped to an ordinary differential equation (ODE) system for the wave packet parameters coupled with a partial differential equation (PDE) for the quantity win rescaled variables. Further, we show that the w equation does not produce epsilon dependent oscillations, and thus it is more amenable for numerical simulations. We propose multi-level sampling strategy in implementing the Gaussian wave packet transform, where in the most costly part, i.e. simulating the w equation, it is sufficient to use epsilon independent samples. We also provide extensive numerical tests as well as meaningful numerical experiments to justify the properties of the numerical algorithm, and hopefully shed light on possible future directions. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free