The Gaussian beam method for the wigner equation withdiscontinuous potentials

Dongsheng Yin; Min Tang; Shi Jin

doi:10.3934/ipi.2013.7.1051

Article Contents

2013, Volume 7, Issue 3: 1051-1074. Doi: 10.3934/ipi.2013.7.1051

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The Gaussian beam method for the wigner equation with discontinuous potentials

1.
Department of Mathematical Sciences, Tsinghua University, Beijing, 100084
2.
Department of Mathematics, Institute of Nature Science, and Ministry of Education Key Laboratory in Scientific and Engineering Computing, Shanghai Jiao Tong University, Shanghai 200240
3.
Department of Mathematics, Institute of Natural Sciences, and MOE Key Lab in Scientific and Engineering Computing, Shanghai Jiao Tong University, Shanghai 200240

Received: May 31, 2012

Revised: December 31, 2012

Published: July 2013

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Abstract

For the Wigner equation with discontinuous potentials, a phase space Gaussian beam (PSGB) summation method is proposed in this paper. We first derive the equations satisfied by the parameters for PSGBs and establish the relations for parameters of the Gaussian beams between the physical space (GBs) and the phase space, which motivates an efficient initial data preparation thus a reduced computational cost than previous method in the literature. The method consists of three steps: 1) Decompose the initial value of the wave function into the sum of GBs and use the parameter relations to prepare the initial values of PSGBs; 2) Solve the evolution equations for each PSGB; 3) Sum all the PSGBs to construct the approximate solution of the Wigner equation. Additionally, in order to connect PSGBs at the discontinuous points of the potential, we provide interface conditions for a single phase space Gaussian beam. Numerical examples are given to verify the validity and accuracy of method.

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Mathematics Subject Classification: Primary: 65M75, 81Q20, 35Q41; Secondary: 65Z05, 34E05.

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