Gauge-Invariant Frozen Gaussian Approximation Method for the Schrödinger Equation with Periodic Potentials

Delgadillo, Ricardo; Lu, Jianfeng; Yang, Xu

doi:10.1137/15m1040384

Cited by 12 publications

(8 citation statements)

References 24 publications

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“…While for the analysis, it suffices to assume smooth dependence of u n on ξ (which is possible as the n-th band is separated from the rest of the spectrum), this gauge freedom makes numerical computation nontrivial. We will further address this by designing a gauge-invariant algorithm in a companion paper [5] on the numerical algorithms. Differentiating (2.2) with respect to ξ produces…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, based on Bloch decomposition and asymptotic analysis in the phase space, we derive the frozen Gaussian approximation for high-frequency wave propagation in periodic media and establish its converge to the true solution. The formulation leads to efficient numerical algorithms, which are presented in a companion paper [5].1 2 RICARDO DELGADILLO, JIANFENG LU, AND XU YANG avoid the boundary effects. Therefore the total number of grid points is huge, which usually leads to unaffordable computational cost, especially in high (d > 1) dimensions.An alternative efficient approach is to solve (1.1) asymptotically by the Bloch decomposition and modified WKB methods [3,4,7], which lead to eikonal and transport equations in the semi-classical regime.…”

mentioning

confidence: 99%

“…While the FGA works for general strictly hyperbolic equations, we focus in this paper the case of semiclassical Schrödinger equation with periodic media (1.1). The computational algorithm and numerical results will be presented in a separate paper [5]. The rest of the paper is organized as follows.…”

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Frozen Gaussian approximation for high frequency wave propagation in periodic media

Delgadillo

Yang

2018

ASY

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Propagation of high-frequency wave in periodic media is a challenging problem due to the existence of multiscale characterized by short wavelength, small lattice constant and large physical domain size. Conventional computational methods lead to extremely expensive costs, especially in high dimensions. In this paper, based on Bloch decomposition and asymptotic analysis in the phase space, we derive the frozen Gaussian approximation for high-frequency wave propagation in periodic media and establish its converge to the true solution. The formulation leads to efficient numerical algorithms, which are presented in a companion paper [5].1 2 RICARDO DELGADILLO, JIANFENG LU, AND XU YANG avoid the boundary effects. Therefore the total number of grid points is huge, which usually leads to unaffordable computational cost, especially in high (d > 1) dimensions.An alternative efficient approach is to solve (1.1) asymptotically by the Bloch decomposition and modified WKB methods [3,4,7], which lead to eikonal and transport equations in the semi-classical regime. An advantage of this method is that the computational cost is independent of ε. However, the eikonal equation can develop singularities which make the method break down at caustics. The Gaussian beam method (GBM) [31] was then introduced by Popov to overcome this drawback at caustics. The idea is to allow the phase function to be complex and choose the imaginary part properly so that the solution has a Gaussian profile; see [15-20, 28, 37, 38] for recent developments. Similar ideas can be also found in the Hagedorn wave packet method [8,9]. Unlike the geometric optics based method, the Gaussian beam method allows for accurate computation of wave function around caustics [6,33]. But the problem is that the constructed beam must stay near the geometric rays to maintain accuracy. This becomes a drawback when the solution spreads [23,28,32].The Herman-Kluk propagator [11,21,22] was proposed for Schrödinger equation without the oscillatory periodic background potential. The method was rigorously analyzed in [35,36] and further extended as the frozen Gaussian approximation (FGA) for general high frequency wave propagation in [23][24][25]. The FGA method uses Gaussian functions with fixed widths, instead of using those that might spread over time, to approximate the wave solution. Despite its superficial similarity with the Gaussian beam method, it is different at a fundamental level. FGA is based on phase plane analysis, while GBM is based on the asymptotic solution to a wave equation with Gaussian initial data. In FGA, the solution to the wave equation is approximated by a superposition of Gaussian functions living in phase space, and each function is not necessarily an asymptotic solution, while GBM uses Gaussian functions (called beams) in physical space, with each individual beam being an asymptotic solution to the wave equation. The main advantage of FGA over GBM is that the problem of beam spreading no longer exists.In this paper, we extend FGA for computation of hi...

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Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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Frozen Gaussian approximation for high frequency wave propagation in periodic media

Delgadillo

Yang

2018

ASY

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“…We remark here that for the numerical examples we proposed in this paper, the smooth Bloch waves are selected in advance. On the other hand, we refer to [14] for a technique in getting the gauge invariant approximation of (3.10e).…”

Section: The Bloch Decomposition-based Gaussian Beam Methodsmentioning

confidence: 99%

A Hybrid Method for Computing the Schrödinger Equations with Periodic Potential with Band-Crossings in the Momentum Space

Chai

Jin

Markowich

2018

CiCP

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We propose a hybrid method which combines the Bloch decomposition-based time splitting (BDTS) method and the Gaussian beam method to simulate the Schrödinger equation with periodic potentials in the case of band-crossings. With the help of the Bloch transformation, we develop a Bloch decomposition-based Gaussian beam (BDGB) approximation in the momentum space to solve the Schrödinger equation. Around the band-crossing a BDTS method is used to capture the inter-band transitions, and away from the crossing, a BDGB method is applied in order to improve the efficiency. Numerical results show that this method can capture the inter-band transitions accurately with a computational cost much lower than the direct solver. We also compare the Schrödinger equation with its Dirac approximation, and numerically show that, as the rescaled Planck number ε → 0, the Schrödinger equation converges to the Dirac equations when the external potential is zero or small, but for general external potentials there is an O(1) difference in between the solutions of the Schrödinger equation and its Dirac approximation.

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Frozen Gaussian Approximation-Based Artificial Boundary Conditions for One-Dimensional Nonlinear Schrödinger Equation in the Semiclassical Regime

2017

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Gauge-Invariant Frozen Gaussian Approximation Method for the Schrödinger Equation with Periodic Potentials

Cited by 12 publications

References 24 publications

Frozen Gaussian approximation for high frequency wave propagation in periodic media

Frozen Gaussian approximation for high frequency wave propagation in periodic media

A Hybrid Method for Computing the Schrödinger Equations with Periodic Potential with Band-Crossings in the Momentum Space

Frozen Gaussian Approximation-Based Artificial Boundary Conditions for One-Dimensional Nonlinear Schrödinger Equation in the Semiclassical Regime

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