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Finite element analysis of time-dependent semi-infinite wave-guides with high-order boundary treatment. (English) Zbl 1034.78014

Summary: A new finite element (FE) scheme is proposed for the solution of time-dependent semi-infinite wave-guide problems, in dispersive or non-dispersive media. The semi-infinite domain is truncated via an artificial boundary \(\mathcal B\), and a high-order non-reflecting boundary condition (NRBC), based on the Higdon non-reflecting operators, is developed and applied on \(\mathcal B\). The new NRBC does not involve any high derivatives beyond second order, but its order of accuracy is as high as one desires. It involves some parameters which are chosen automatically as a pre-process. A \(C^0\) semi-discrete FE formulation incorporating this NRBC is constructed for the problem in the finite domain bounded by \(\mathcal B\). Augmented and split versions of this FE formulation are proposed. The semi-discrete system of equations is solved by the Newmark time-integration scheme. Numerical examples concerning dispersive waves in a semi-infinite wave-guide are used to demonstrate the performance of the new method.

MSC:

78M20 Finite difference methods applied to problems in optics and electromagnetic theory
78A55 Technical applications of optics and electromagnetic theory
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[1] Numerical Methods for Problems in Infinite Domains. Elsevier: Amsterdam, 1992.
[2] Engquist, Communications in Pure and Applied Mathematics 32 pp 313– (1979)
[3] Bayliss, Communications in Pure and Applied Mathematics 33 pp 707– (1980)
[4] Givoli, Journal of Computational Physics 94 pp 1– (1991)
[5] Bettess, International Journal for Numerical Methods in Engineering 11 pp 53– (1977)
[6] The boundary element method?some early history: a personal view. In Boundary Elements in Structural Analysis, (ed.). ASCE: New York, 1989; 1-16.
[7] Keller, Journal of Computational Physics 82 pp 172– (1989)
[8] Givoli, Wave Motion 12 pp 261– (1990)
[9] BĂ©renger, Journal of Computational Physics 114 pp 185– (1994)
[10] Burnett, Journal of the Acoustical Society of America 96 pp 2798– (1994)
[11] Astley, Journal of Sound and Vibration 192 pp 245– (1996) · Zbl 0855.65112
[12] Givoli, Computer Methods in Applied Mechanics and Engineering 164 pp 1– (1998)
[13] Turkel, Applied Numerical Mathematics 27 pp 331– (1998)
[14] Astley, Journal of Computational Acoustics 8 pp 1– (2000) · Zbl 1360.76121
[15] Tsynkov, Applied Numerical Mathematics 27 pp 465– (1998)
[16] Givoli, Applied Mechanics Reviews 52 pp 333– (1999)
[17] Hagstrom, Acta Numerica 8 pp 47– (1999)
[18] High order absorbing boundary conditions for wave propagation models. Straight Line Boundary and Corner Cases. In Proceedings of the Second International Conference on Mathematical and Numerical Aspects of Wave Propagation, et al. (eds). SIAM: Delaware, 1993; 161-171. · Zbl 0814.35065
[19] Grote, Journal of Computational Physics 127 pp 52– (1996)
[20] Sofronov, Russian Academy of Sciences Doklady Mathematics 46 pp 397– (1993)
[21] Hagstrom, Applied Numerical Mathematics 27 pp 403– (1998)
[22] Guddati, Journal of Computational Acoustic 8 pp 139– (2000) · Zbl 1360.76136
[23] Givoli, Journal of Computational Physics 170 pp 849– (2001)
[24] Givoli, International Journal for Numerical Methods in Engineering 53 pp 2389– (2002)
[25] Turkel, Applied Numerical Mathematics 27 pp 533– (1998)
[26] Collino, SIAM Journal of Scientific Computing 19 pp 2061– (1998)
[27] Ting, Journal of the Acoustical Society of America 80 pp 1825– (1986)
[28] Givoli, Journal of Computational Physics 117 pp 102– (1995)
[29] Safjan, Computer Methods in Applied Mechanics and Engineering 152 pp 175– (1998)
[30] Givoli, Computer Methods in Applied Mechanics and Engineering 95 pp 97– (1992)
[31] Patlashenko, Computer Methods in Applied Mechanics and Engineering 190 pp 5691– (2001)
[32] Thompson, Computational Methods in Applied Mechanics and Engineering 132 pp 229– (1996)
[33] Alpert, SIAM Journal on Numerical Analysis 37 pp 1138– (2000)
[34] Alpert, Journal of Computational Physics 162 pp 536– (2000)
[35] Thompson, International Journal for Numerical Methods in Engineering 45 pp 1607– (1999)
[36] Huan, International Journal for Numerical Methods in Engineering 47 pp 1569– (2000)
[37] Krenk, International Journal for Numerical Methods in Engineering 53 pp 275– (2002)
[38] A perfectly matched layer formulation for the nonlinear shallow water equations models: the split equation approach. To appear.
[39] Givoli, Archives of Computational Methods in Engineering 6 pp 71– (1999)
[40] Linear and Nonlinear Waves. Wiley: New York, 1974.
[41] Geophysical Fluid Dynamics. Springer: New York, 1987.
[42] Numerical Methods for Wave Equations in Geophysical Fluid Dynamics. Springer: New York, 1999.
[43] Higdon, SIAM Journal on Numerical Analysis 31 pp 64– (1994)
[44] Givoli, Wave Motion 37 pp 257– (2003)
[45] Givoli, Journal of Computational Physics 186 pp 24– (2003)
[46] The Finite Element Method. Prentice-Hall: Englewood Cliffs, NJ, 1987.
[47] Sommeijer, Applied Numerical Mathematics 2 pp 69– (1986)
[48] High-order higdon non-reflecting boundary conditions for the shallow water equations. Report NPS-MA-02-001. Naval Postgraduate School, Monterey, CA, 2002.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.