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Large time behavior of solutions to the non-isentropic compressible Navier-Stokes-Poisson system in $\mathbb{R}^{3}$
Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation
1. | Department of Mathematics at Faculty of Economics Sciences, National Research University Higher School of Economics, Myasnitskaya 20, 101000 Moscow, Russian Federation |
2. | Department of Mathematical Modelling, Moscow Power Engineering Institute, Krasnokazarmennaya 14, 111250 Moscow, Russian Federation |
References:
[1] |
X. Antoine, A. Arnold, C. Besse, M. Ehrhardt and A. Schädle, A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations, Commun. Comp. Phys., 4 (2008), 729-796. |
[2] |
X. Antoine and C. Besse, Unconditionally stable discretization schemes of non-reflecting boundary conditions for the one-dimensional Schrödinger equation, J. Comp. Phys., 188 (2003), 157-175.
doi: 10.1016/S0021-9991(03)00159-1. |
[3] |
A. Arnold, Numerically absorbing boundary conditions for quantum evolution equations, VLSI Design, 6 (1998), 313-319. |
[4] |
A. Arnold, M. Ehrhardt and I. Sofronov, Discrete transparent boundary conditions for the Schrödinger equation: Fast calculations, approximation, and stability, Comm. Math. Sci., 1 (2003), 501-556. |
[5] |
B. Ducomet and A. Zlotnik, On stability of the Crank-Nicolson scheme with approximate transparent boundary conditions for the Schrödinger equation. I, Comm. Math. Sci., 4 (2006), 741-766. |
[6] |
B. Ducomet and A. Zlotnik, On stability of the Crank-Nicolson scheme with approximate transparent boundary conditions for the Schrödinger equation. II, Comm. Math. Sci., 5 (2007), 267-298. |
[7] |
B. Ducomet, A. Zlotnik and I. Zlotnik, On a family of finite-difference schemes with approximate transparent boundary conditions for a generalized 1D Schrödinger equation, Kinetic and Related Models, 2 (2009), 151-179. |
[8] |
M. Ehrhardt and A. Arnold, Discrete transparent boundary conditions for the Schrödinger equation, Riv. Mat. Univ. Parma (6), 4 (2001), 57-108. |
[9] |
V. A. Gordin, "Mathematical Problems in Hydrodynamical Weather Forecasting. Computational Aspects," (in Russian), "Gidrometeoizdat," Leningrad, 1987; Abridged English version: "Mathematical Problems and Methods in Hydrodynamical Weather Forecasting," Gordon and Breach, Amsterdam, 2000. |
[10] |
R. A. Horn and C. R. Johnson, "Matrix Analysis," Cambridge University Press, Cambridge, 1985. |
[11] |
J. Jin and X. Wu, Analysis of finite element method for one-dimensional time-dependent Schrödinger equation on unbounded domains, J. Comp. Appl. Math., 220 (2008), 240-256.
doi: 10.1016/j.cam.2007.08.006. |
[12] |
C. A. Moyer, Numerov extension of transparent boundary conditions for the Schrödinger equation discretized in one dimension, Am. J. Phys., 72 (2004), 351-358.
doi: 10.1119/1.1619141. |
[13] |
F. Schmidt and D. Yevick, Discrete transparent boundary conditions for Schrödinger-type equations, J. Comp. Phys., 134 (1997), 96-107.
doi: 10.1006/jcph.1997.5675. |
[14] |
M. Schulte and A. Arnold, Discrete transparent boundary conditions for the Schrödinger equation-a compact higher order scheme, Kinetic and Related Models, 1 (2008), 101-125. |
[15] |
G. Strang and G. Fix, "An Analysis of the Finite Element Method," Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1973. |
[16] |
I. A. Zlotnik, Computer simulation of the tunnel effect, (in Russian), Moscow Power Engin. Inst. Bulletin, 6 (2010), 10-28. |
[17] |
I. A. Zlotnik, A family of difference schemes with approximate transparent boundary conditions for the generalized nonstationary Schrödinger equation in a half-strip, Comput. Math. Math. Phys., 51 (2011), 355-376.
doi: 10.1134/S0965542511030122. |
show all references
References:
[1] |
X. Antoine, A. Arnold, C. Besse, M. Ehrhardt and A. Schädle, A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations, Commun. Comp. Phys., 4 (2008), 729-796. |
[2] |
X. Antoine and C. Besse, Unconditionally stable discretization schemes of non-reflecting boundary conditions for the one-dimensional Schrödinger equation, J. Comp. Phys., 188 (2003), 157-175.
doi: 10.1016/S0021-9991(03)00159-1. |
[3] |
A. Arnold, Numerically absorbing boundary conditions for quantum evolution equations, VLSI Design, 6 (1998), 313-319. |
[4] |
A. Arnold, M. Ehrhardt and I. Sofronov, Discrete transparent boundary conditions for the Schrödinger equation: Fast calculations, approximation, and stability, Comm. Math. Sci., 1 (2003), 501-556. |
[5] |
B. Ducomet and A. Zlotnik, On stability of the Crank-Nicolson scheme with approximate transparent boundary conditions for the Schrödinger equation. I, Comm. Math. Sci., 4 (2006), 741-766. |
[6] |
B. Ducomet and A. Zlotnik, On stability of the Crank-Nicolson scheme with approximate transparent boundary conditions for the Schrödinger equation. II, Comm. Math. Sci., 5 (2007), 267-298. |
[7] |
B. Ducomet, A. Zlotnik and I. Zlotnik, On a family of finite-difference schemes with approximate transparent boundary conditions for a generalized 1D Schrödinger equation, Kinetic and Related Models, 2 (2009), 151-179. |
[8] |
M. Ehrhardt and A. Arnold, Discrete transparent boundary conditions for the Schrödinger equation, Riv. Mat. Univ. Parma (6), 4 (2001), 57-108. |
[9] |
V. A. Gordin, "Mathematical Problems in Hydrodynamical Weather Forecasting. Computational Aspects," (in Russian), "Gidrometeoizdat," Leningrad, 1987; Abridged English version: "Mathematical Problems and Methods in Hydrodynamical Weather Forecasting," Gordon and Breach, Amsterdam, 2000. |
[10] |
R. A. Horn and C. R. Johnson, "Matrix Analysis," Cambridge University Press, Cambridge, 1985. |
[11] |
J. Jin and X. Wu, Analysis of finite element method for one-dimensional time-dependent Schrödinger equation on unbounded domains, J. Comp. Appl. Math., 220 (2008), 240-256.
doi: 10.1016/j.cam.2007.08.006. |
[12] |
C. A. Moyer, Numerov extension of transparent boundary conditions for the Schrödinger equation discretized in one dimension, Am. J. Phys., 72 (2004), 351-358.
doi: 10.1119/1.1619141. |
[13] |
F. Schmidt and D. Yevick, Discrete transparent boundary conditions for Schrödinger-type equations, J. Comp. Phys., 134 (1997), 96-107.
doi: 10.1006/jcph.1997.5675. |
[14] |
M. Schulte and A. Arnold, Discrete transparent boundary conditions for the Schrödinger equation-a compact higher order scheme, Kinetic and Related Models, 1 (2008), 101-125. |
[15] |
G. Strang and G. Fix, "An Analysis of the Finite Element Method," Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1973. |
[16] |
I. A. Zlotnik, Computer simulation of the tunnel effect, (in Russian), Moscow Power Engin. Inst. Bulletin, 6 (2010), 10-28. |
[17] |
I. A. Zlotnik, A family of difference schemes with approximate transparent boundary conditions for the generalized nonstationary Schrödinger equation in a half-strip, Comput. Math. Math. Phys., 51 (2011), 355-376.
doi: 10.1134/S0965542511030122. |
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