Frozen Gaussian approximation based domain decomposition methods for the linear Schrödinger equation beyond the semi-classical regime

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• 作者：E. Lorin^a ; ^b ; ^{elorin@math.carleton.ca" class="auth_mail" title="E-mail the corresponding author} ; X. Yang^c ; X. Antoine^d
• 关键词：Semiclassical approximation ; Absorbing boundary condition ; Domain decomposition ; Linear Schrö ; dinger equation ; Pseudodifferential operators
• 刊名：Journal of Computational Physics
• 出版年：2016
• 出版时间：15 June 2016
• 年：2016
• 卷：315
• 期：Complete
• 页码：221-237
• 全文大小：867 K

文摘

The paper is devoted to develop efficient domain decomposition methods for the linear Schrödinger equation beyond the semiclassical regime, which does not carry a small enough rescaled Planck constant for asymptotic methods (e.g. geometric optics) to produce a good accuracy, but which is too computationally expensive if direct methods (e.g. finite difference) are applied. This belongs to the category of computing middle-frequency wave propagation, where neither asymptotic nor direct methods can be directly used with both efficiency and accuracy. Motivated by recent works of the authors on absorbing boundary conditions (Antoine et al. (2014) [13] and Yang and Zhang (2014) [43]), we introduce Semiclassical Schwarz Waveform Relaxation methods (SSWR), which are seamless integrations of semiclassical approximation to Schwarz Waveform Relaxation methods. Two versions are proposed respectively based on Herman–Kluk propagation and geometric optics, and we prove the convergence and provide numerical evidence of efficiency and accuracy of these methods.