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	GAUSSIAN BEAM METHODS FOR THE SCHRODINGER EQUATION IN THE SEMI-CLASSICAL REGIME: LAGRANGIAN AND EULERIAN FORMULATIONS
	Jin, Shi ; Wu, Hao ; Yang, Xu
	2010-10-12 ; 2010-10-12
关键词	Schrodinger equation Gaussian beam method Liouville equation LEVEL SET METHOD GEOMETRIC OPTICS WAVE FIELDS TRAVEL-TIME COMPUTATION LIMIT OBSERVABLES MIGRATION WIGNER Mathematics, Applied
中文摘要	The solution to the Schrodinger equation is highly oscillatory when the rescaled Planck constant epsilon is small in the semiclassical regime. A direct numerical simulation requires the mesh size to be O (epsilon). The Gaussian beam method is an efficient way to solve the high frequency wave equations asymptotically, outperforming the geometric optics method in that the Gaussian beam method is accurate even at caustics. In this paper, we solve the Schrodinger equation using both the Lagrangian and Eulerian formulations of the Gaussian beam methods. A new Eulerian Gaussian beam method is developed using the level set method based only on solving the (complex-valued) homogeneous Liouville equations. A major contribution here is that we are able to construct the Hessian matrices of the beams by using the level set function's first derivatives. This greatly reduces the computational cost in computing the Hessian of the phase function in the Eulerian framework, yielding an Eulerian Gaussian beam method with computational complexity comparable to that of the geometric optics but with a much better accuracy around caustics. We verify through several numerical experiments that our Gaussian beam solutions are good approximations to Schrodinger solutions even at caustics. We also numerically study the optimal relation between the number of beams and there scaled Planck constant epsilon in the Gaussian beam summation.
语种	英语 ; 英语
出版者	INT PRESS ; SOMERVILLE ; PO BOX 43502, SOMERVILLE, MA 02143 USA
内容类型	期刊论文
源URL	[http://hdl.handle.net/123456789/81270]
专题	清华大学
推荐引用方式 GB/T 7714	Jin, Shi,Wu, Hao,Yang, Xu. GAUSSIAN BEAM METHODS FOR THE SCHRODINGER EQUATION IN THE SEMI-CLASSICAL REGIME: LAGRANGIAN AND EULERIAN FORMULATIONS[J],2010, 2010.
APA	Jin, Shi,Wu, Hao,&Yang, Xu.(2010).GAUSSIAN BEAM METHODS FOR THE SCHRODINGER EQUATION IN THE SEMI-CLASSICAL REGIME: LAGRANGIAN AND EULERIAN FORMULATIONS..
MLA	Jin, Shi,et al."GAUSSIAN BEAM METHODS FOR THE SCHRODINGER EQUATION IN THE SEMI-CLASSICAL REGIME: LAGRANGIAN AND EULERIAN FORMULATIONS".(2010).

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