### Impacto

Carpio Rodríguez, Ana María and Tapiador Fernández, Bárbara
(2010)
*Nonreflecting boundary conditions for discrete waves.*
Journal of Computational Physics, 229
(5).
pp. 1879-1896.
ISSN 0021-9991

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Official URL: http://www.sciencedirect.com/science/article/pii/S0021999109006329

## Abstract

We introduce a new class of nonreflecting boundary conditions for lattice models, which minimizes reflections at artificial boundaries. Exact integrodifferential boundary conditions for finite chains and half-spaces are obtained by means of Green's functions for initial value problems. Truncating the resulting integrals in time, we obtain absorbing boundary conditions. Numerical tests illustrate the ability of these conditions to suppress reflections.

Item Type: | Article |
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Uncontrolled Keywords: | Molecular-dynamics; Hyperbolic systems; Absorbing layer; Simulations; Propagation; Equations; Nonreflecting boundary conditions; Green's functions; Lattice models; Discrete wave equations; Transparent boundary conditions; Absorbing boundary conditions; Dirichlet-to-Neumann operators; Molecular dynamics |

Subjects: | Sciences > Physics > Mathematical physics Sciences > Mathematics > Differential equations |

ID Code: | 14881 |

References: | [1] S.A. Adelman, J.D. Doll, Generalized Langevin equation approach for atom/solid-surface scattering: General formulation for classical scattering off harmonic solids, J. Chem. Phys. 64 (1976) 2375–2388. [2] A.R.A. Anderson, B.D. Sleeman, Wave front propagation and its failure in coupled systems of discrete bistable cells modelled by FitzHugh–Nagumo dynamics, Int. J. Bif. Chaos 5 (1995) 63–75. [3] A. Appeló, G. Kreiss, A new absorbing layer for elastic waves, J. Comput. Phys. 215 (2006) 642–660. [4] A. Arnold, M. Ehrhardt, Discrete transparent boundary conditions for wide angle parabolic equations in underwater acoustics, J. Comput. Phys. 145 (1998) 611–638. [5] J. Bafalui, J.M. Rubi, The harmonic liquid away from equilibrium I and II, Phys. A 153 (1988) 129–159. [6] A. Bayliss, E. Turkel, Radiation boundary conditions for wave-like equations, Commun. Pur. Appl. Math. XXXIII (1980) 707–725. [7] A. Bayliss, E. Turkel, Far field boundary conditions for compressible flows, J. Comput. Phys. 48 (1982) 182–199. [8] A. Bayliss, C.I. Goldstein, E. Turkel, On accuracy conditions for the numerical computation of waves, J. Comput. Phys. 59 (1985) 396–404. [9] E. Beaches, S. Fauqueux, P. Joly, Stability of perfectly matched layers, group velocities and anisotropic waves, J. Comput. Phys. 188 (2003) 399–433. [10] J.P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys. 114 (1994) 185–200. [11] L.L. Bonilla, Theory of nonlinear charge transport, wave propagation and self-oscillations in semiconductor superlattices, J. Phys. C 14 (2002) R341. [12] W. Cai, M. Koning, V.V. Bulatov, S. Yip, Minimizing boundary reflections in coupled-domain simulations, Phys. Rev. Lett. 85 (2000) 3213–3216. [13] A. Carpio, Nonlinear stability of oscillatory wave fronts in chains of coupled oscillators, Phys. Rev. E 67(2004) 046601. [14] P.M. Chaikin, T.C. Lubensky, Principles of condensed matter physics, Cambridge University Press, Cambridge, 1995 (Chapter 10). [15] F. Collino, C. Tsogka, Application of the PML absorbing layer model to the linear elastodynamic problem in anisotropic heterogenous media, Geophysics 66 (2001) 294–307. [16] P.A. Deymier, J.Q. Vasseur, Concurrent multiscale model of an atomic crystal coupled with elastic continua, Phys. Rev. B 66 (13) (2002) 134106. [17] W. E, Z. Huang, A dynamic atomistic-continuum method for the simulation of crystalline materials, J. Comput. Phys. 182 (1) (2002) 234–261. [18] W. E, Z. Huang, Matching conditions in atomistic continuum modeling of materials, Phys. Rev. Lett. 85 (2001) 135501. [19] B. Engquist, A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comput. 31 (1977) 629–651. [20] B. Engquist, A. Majda, Radiation boundary conditions for acoustic and elastic wave calculations, Commun. Pur. Appl. Math. 32 (1979) 312–358. [21] J. Frenkel, T. Kontorova, On the theory of plastic deformation and twinning, J. Phys. USSR 13 (1938) 1–10. [22] M.J. Grote, J.B. Keller, On nonreflecting boundary conditions, J. Comput. Phys. 122 (1995) 231–243. [23] M.J. Grote, J.B. Keller, Nonreflecting boundary conditions for the time dependent wave equations, SIAM J. Appl. Math. 55 (2) (1995) 280–297. [24] D. Givoli, I. Patlashenko, J.B. Keller, Discrete Dirichlet-to-Neumann maps for unbounded domains, Comput. Meth. Appl. Mech. Engrg. 164 (1998) 173–185. [25] B. Gustafsson, Far-field boundary conditions for time-dependent hyperbolic problems, SIAM J. Sci. Statist. Comput. 9 (1988) 812–828. [26] B. Gustafsson, Inhomogeneous conditions at open boundaries for wave propagation problems, Appl. Numer. Math. 4 (1988) 3–19. [27] R.L. Higdon, Absorbing boundary conditions for difference approximations to the multidimensional wave equation, Math. Comput. 47 (1986) 437–459. [28] R.L. Higdon, Radiation boundary conditions for dispersive waves, SIAM J. Numer. Anal. 31 (1994) 64–100. [29] B.L. Holian, R. Ravelo, Fracture simulations using large scale molecular dynamics, Phys. Rev. B 51 (1995) 11275–11288. [30] E.G. Karpov, G.J. Wagner, K.L. Wing, A Green’s function approach to deriving nonreflecting boundary conditions in molecular dynamics simulations, Int. J. Numer. Meth. 62 (9)(2005) 1250–1262. [31] J.P. Keener, J. Sneyd, Mathematical Physiology, Springer, New York, 1998. [32] Ch. Lubich, A. Schaedle, Fast convolution for nonreflecting boundary conditions, SIAM J. Sci. Comput. 24 (2002) 161–182. [33] M. Moseler, J. Nordiek, H. Haberland, Reduction of the reflected pressure wave in the molecular-dynamics simulation of energetic particle-solid collisions, Phys. Rev. B 56 (1997) 15439–15445. [34] C.W. Rowley, T. Colonius, Discretely nonreflecting boundary conditions for linear hyperbolic systems, J. Comput. Phys. 157 (2000) 500–538. [35] V.S. Ryaben’kii, S.V. Tsynkov, V.I. Turchaninov, Global discrete artificial boundary conditions for time-dependent wave propagation, J. Comput. Phys. 174 (2001) 712–758. [36] J.M. Sanz-Serna, M.P. Calvo, Numerical hamiltonian problems, Applied Mathematics and Mathematical Computation, vol. 7, Chapman & Hall, 1994. [37] F. Schmidt, D. Yevick, Transparent boundary conditions for Schrödinger-type equations, J. Comput. Phys. 134 (1997) 96–107. [38] E. Schrödinger, The dynamics of elastically coupled point systems, Annalen der Physik 44 (1914) 916–934. [39] B. Shiari, R.E. Miller, W.A. Curtin, Coupled atomistic/discrete dislocation simulations of nanoindentation at finite temperature, Trans. ASME 127(2005)358–368. [40] L.I. Slepyan, Dynamics of a crack in a lattice, Sov. Phys. Dokl. 26 (1981) 538–540. [41] L.N. Trefethen, L. Halpern, Well-posedness of one way wave equations and absorbing boundary conditions, Math. Comput. 47 (1986) 421–435. |

Deposited On: | 19 Apr 2012 08:44 |

Last Modified: | 06 Feb 2014 10:11 |

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