Welcome to the IKCEST

Journal of Scientific Computing | Vol.83, Issue.1 | 2020-04-03 | Pages 1-21

Journal of Scientific Computing

Optimal Error Estimate of the Extended-WKB Approximation to the High Frequency Wave-Type Equation in the Semi-classical Regime

Chunxiong Zheng   Jiashun Hu  
Abstract

For the Cauchy problem of the high frequency wave-type equation with Wentzel–Kramers–Brillouin (WKB) type initial data, the extended Wentzel

Original Text (This is the original text for your reference.)

Optimal Error Estimate of the Extended-WKB Approximation to the High Frequency Wave-Type Equation in the Semi-classical Regime

For the Cauchy problem of the high frequency wave-type equation with Wentzel–Kramers–Brillouin (WKB) type initial data, the extended Wentzel

+More

Cite this article
APA

APA

MLA

Chicago

Chunxiong Zheng,Jiashun Hu,.Optimal Error Estimate of the Extended-WKB Approximation to the High Frequency Wave-Type Equation in the Semi-classical Regime. 83 (1),1-21.

References

Chai, L., Tong, P., Yang, X.: Frozen Gaussian approximation for 3-D seismic wave propagation. Geophys. J. Int. 208(1), 59–74 (2017)

Tanushev, N.M.: Superpositions and higher order Gaussian beams. Commun. Math. Sci. 6(2), 449–475 (2008)

Wu, H., Huang, Z., Jin, S., Yin, D.: Gaussian beam methods for the Dirac equation in the semi-classical regime. Commun. Math. Sci. 10, 1301–1315 (2012)

Herman, M., Kluk, E.: A semiclassical justification for the use of non-spreading wavepackets in dynamics calculations. Chem. Phys. 91, 27–34 (1984)

Jin, S., Wu, H., Yang, X.: Gaussian beam methods for the Schrödinger equation in the semi-classical regime: Lagrangian and Eulerian formulations. Commun. Math. Sci. 6, 995–1020 (2008)

Karasev, M.V.: Connections on Lagrangian submanifolds and some quasiclassical approximation problems I. J. Sov. Math. 59(5), 1053–1062 (1992)

Liu, H., Runborg, O., Tanushev, N.: Error estimates for Gaussian beam superpositions. Math. Comput. 82(282), 919–952 (2013)

Delgadillo, R., Lu, J., Yang, X.: Frozen Gaussian approximation for high frequency wave propagation in periodic media. Asymptot. Anal. 110(3–4), 113–135 (2018)

Heller, E.: Cellular dynamics: a new semiclassical approach to time-dependent quantum mechanics. J. Chem. Phys. 94(4), 2723–2729 (1991)

Lu, J., Yang, X.: Frozen Gaussian approximation for high frequency wave propagation. Commun. Math. Sci. 9(3), 663–683 (2011)

Delgadillo, R., Lu, J., Yang, X.: Gauge-invariant frozen Gaussian approximation method for the Schrödinger equation in periodic media. SIAM J. Sci. Comput. 38(4), A2440–A2463 (2018)

Popov, M.M.: A new method of computation of wave fields using Gaussian beams. Wave Motion 4, 85–97 (1982)

Ralston, J.: Gaussian beams and the propagation of singularities. Stud. Partial Differ. Equ. MAA Stud. Math. 23, 206–248 (1982)

Lu, J., Yang, X.: Convergence of frozen Gaussian approximation for high-frequency wave propagation. Commun. Pure Appl. Math. 65(6), 759–789 (2012)

Zheng, C., Hu, J.: Extended WKB analysis for the linear vectorial wave equation in the high-frequency regime. Commun. Math. Sci (accepted)

Guillemin, V., Sternberg, S.: Symplectic Techniques in Physics. Cambridge University Press, Cambridge (1990)

Jin, S., Wei, D., Yin, D.: Gaussian beam methods for the Schrödinger equation with discontinuous potentials. J. Comput. Appl. Math. 265(1), 199–219 (2014)

Malenova, G., Motamed, M., Runborg, O., Tempone, R.: A sparse stochastic collocation technique for high-frequency wave propagation with uncertainty. SIAM/ASA J. Uncertain. Quantif. 4, 1084–1110 (2016)

Maslov, V.P., Fedoryuk, M.V.: Semi-classical Approximation in Quantum Mechanics. Reidel, Dordrecht (1982)

Popov, M.M.: Ray Theory and Gaussian Beams for Geophysics. EDUFBA, Salvador (2002)

Zheng, C.: Global geometrical optics method. Commun. Math. Sci. 11(1), 105–140 (2013)

Chai, L., Lorin, E., Yang, X.: Frozen Gaussian approximation for the Dirac equation in semiclassical regime. SIAM J. Numer. Anal. 57(5), 2383–2412 (2019)

Folland, G.B.: Harmonic Analysis in Phase Space. Princeton University Press, Princeton (1989)

Hall, B.C.: Quantum Theory for Mathematicians. Springer, New York (2013)

Motamed, M., Runborg, O.: Taylor expansion and discretization errors in Gaussian beam superposition. Wave Motion 47(7), 421–439 (2010)

Bao, W., Jin, S., Markowich, P.: On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime. J. Comput. Phys. 175(2), 487–524 (2002)

Engquist, B., Runborg, O.: Multi-phase computations in geometrical optics. J. Comput. Appl. Math. 74(1–2), 175–192 (1996)

Hill, N.R.: Gaussian beam migration. Geophysics 55, 1416–1428 (1990)

Lu, J., Zhou, Z.: Improved sampling and validation of frozen Gaussian approximation with surface hopping algorithm for nonadiabatic dynamics. J. Chem. Phys. 145(12), 124109 (2016)

Jin, S., Wu, H., Yang, X.: Semi-Eulerian and high order Gaussian beam methods for the Schrödinger equation in the semiclassical regime. Commun. Comput. Phys. 9(3), 668–687 (2011)

Zheng, C.: Optimal error estimates for first-order Gaussian beam approximations to the Schrödinger equation. SIAM J. Numer. Anal. 52(6), 2905–2930 (2014)

Cheng, L.T., Liu, H.L., Osher, S.: Computational high-frequency wave propagation in Schrödinger equations using the level set method, with applications to the semi-classical limit of Schrödinger equations. Commun. Math. Sci. 1(3), 593–621 (2003)

Hörmander, L.: On the existence and the regularity of solutions of linear pseudo-differential equations. Enseign. Math. 17, 99–163 (1971)

Jin, S., Markowich, P., Sparber, C.: Mathematical and computational methods for semiclassical Schrödinger equations. Acta Numer. 20, 121–209 (2011)

Liu, H., Ralston, J.: Recovery of high frequency wave fields from phase space-based measurements. Multiscale Model. Sim. 8(2), 622–644 (2010)

Disclaimer: The translated content is provided by third-party translation service providers, and IKCEST shall not assume any responsibility for the accuracy and legality of the content.
Translate engine
Article's language
English
中文
Pусск
Français
Español
العربية
Português
Kikongo
Dutch
kiswahili
هَوُسَ
IsiZulu
Action
Recommended articles

Report

Select your report category*



Reason*



By pressing send, your feedback will be used to improve IKCEST. Your privacy will be protected.

Submit
Cancel