Browsing by Author "Yang, X"

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    Convergence of frozen Gaussian approximation for high-frequency wave propagation
    (Communications on Pure and Applied Mathematics, 2012-06-01) Lu, J; Yang, X
    The frozen Gaussian approximation provides a highly efficient computational method for high-frequency wave propagation. The derivation of the method is based on asymptotic analysis. In this paper, for general linear strictly hyperbolic systems, we establish the rigorous convergence result for frozen Gaussian approximation. As a byproduct, higher-order frozen Gaussian approximation is developed. © 2011 Wiley Periodicals, Inc.
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    Effective Maxwell equations from time-dependent density functional theory
    (Acta Mathematica Sinica, English Series, 2011-01-20) E, W; Lu, J; Yang, X
    The behavior of interacting electrons in a perfect crystal under macroscopic external electric and magnetic fields is studied. Effective Maxwell equations for the macroscopic electric and magnetic fields are derived starting from time-dependent density functional theory. Effective permittivity and permeability coefficients are obtained. © 2011 Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg.
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    Frozen gaussian approximation for general linear strictly hyperbolic systems: Formulation and eulerian methods
    (Multiscale Modeling and Simulation, 2012-09-07) Lu, J; Yang, X
    The frozen Gaussian approximation, proposed in [J. Lu and X. Yang, Commun. Math. Sci., 9 (2011), pp. 663-683], is an efficient computational tool for high frequency wave propagation. We continue in this paper the development of frozen Gaussian approximation. The frozen Gaussian approximation is extended to general linear strictly hyperbolic systems. Eulerian methods based on frozen Gaussian approximation are developed to overcome the divergence problem of Lagrangian methods. The proposed Eulerian methods can also be used for the Herman-Kluk propagator in quantum mechanics. Numerical examples verify the performance of the proposed methods. © 2012 Society for Industrial and Applied Mathematics.
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    Frozen Gaussian approximation for high frequency wave propagation in periodic media
    (2017-04-26) Lu, J; Yang, X
    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 [Delgadillo, Lu and Yang, arXiv:1509.05552].
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    Gauge-invariant frozen Gaussian approximation method for the schrödinger equation with periodic potentials
    (SIAM Journal on Scientific Computing, 2016-01-01) Delgadillo, R; Lu, J; Yang, X
    © 2016 Society for Industrial and Applied Mathematics.We develop a gauge-invariant frozen Gaussian approximation (GIFGA) method for the Schrödinger equation (LSE) with periodic potentials in the semiclassical regime. The method generalizes the Herman-Kluk propagator for LSE to the case with periodic media. It provides an efficient computational tool based on asymptotic analysis on phase space and Bloch waves to capture the high-frequency oscillations of the solution. Compared to geometric optics and Gaussian beam methods, GIFGA works in both scenarios of caustics and beam spreading. Moreover, it is invariant with respect to the gauge choice of the Bloch eigenfunctions and thus avoids the numerical difficulty of computing gauge-dependent Berry phase. We numerically test the method by several one-dimensional examples; in particular, the first order convergence is validated, which agrees with our companion analysis paper [Frozen Gaussian Approximation for High Frequency Wave Propagation in Periodic Media, arXiv:1504.08051, 2015].