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Non-reflecting boundary conditions. (English) Zbl 0731.65109
Past and recent research on the use of non-reflecting boundary conditions in the numerical solutions of wave propagation problems is reviewed. The paper is concerned with artificial boundary conditions. The need for those conditions arises when the spatial domain of the problem is unbounded and a numerical treatment usually requires the introduction of an artificial boundary \({\mathcal B}\) in order to make the computational domain finite. The appropriate boundary conditions to be used on \({\mathcal B}\) for various wave problems are discussed in this survey.

MSC:
65Z05 Applications to the sciences
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35L05 Wave equation
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References:
[1] Turkel, E., Comput. fluids, 11, 121, (1983)
[2] Sommerfeld, A., ()
[3] Roe, P.L., (), (unpublished)
[4] Halpern, L., Math. comput., 38, 415, (1982)
[5] Foreman, M.G.G., J. comput. phys., 64, 334, (1986)
[6] Engquist, B.; Majda, A., Math. comput., 31, 629, (1977)
[7] Engquist, B.; Majda, A., Commun. pure appl. math., 32, 313, (1979)
[8] Treves, F., ()
[9] Claerbout, J.F., Imaging the Earth’s interior, (1985), Blackwell London
[10] Berkhout, A.J., Geophysics, 46, 1638, (1981)
[11] Tappert, F.D., The parabolic approximation method, () · Zbl 0419.76061
[12] Trefethen, L.N.; Halpern, L., Math. comput., 47, 421, (1986)
[13] Halpern, L.; Trefethen, L.N., J. acoust. soc. amer., 84, 1397, (1988)
[14] Kreiss, H.-O., Commun. pure appl. math., 23, 277, (1970)
[15] Kolakowski, H., Math. methods appl. sci., 7, 486, (1985)
[16] Kolakowski, H., Math. methods appl. sci., 8, 41, (1986)
[17] Halpern, L.; Rauch, J., Numer. math., 51, 459, (1987)
[18] Lindman, E.L., J. comput. phys., 18, 66, (1975)
[19] Randall, C.J., Geophysics, 53, 611, (1988)
[20] Behrendt, L., A finite element model for water wave diffraction including boundary absorption and bottom friction, (), (unpublished)
[21] Hariharan, S.I.; Bayliss, A., SIAM J. sci. statist. comput., 6, 285, (1985)
[22] Wagatha, L., Numer. math., 42, 51, (1983)
[23] Clayton, R.; Engquist, B., Geophysics, 45, 895, (1980)
[24] Howell, L.H.; Trefethen, L.N., Geophysics, 53, 593, (1988)
[25] Reynolds, A.C., Geophysics, 43, 1099, (1978)
[26] Bayliss, A.; Turkel, E., Commun. pure appl. math., 33, 707, (1980)
[27] Bayliss, A.; Turkel, E., J. comput. phys., 48, 182, (1982)
[28] Bayliss, A.; Gunzburger, M.; Turkel, E., SIAM J. appl. math., 42, 430, (1982)
[29] Feng, K., (), 1439
[30] Higdon, R.L., Math. comput., 47, 437, (1986)
[31] Higdon, R.L., Math. comput., 49, 65, (1987)
[32] Keys, R.G., Geophysics, 50, 892, (1985)
[33] Lysmer, J.; Kuhlemeyer, R.L., J. eng. mech. div. ASCE, 95, 859, (1969)
[34] Kriegsmann, G.; Morawetz, C., SIAM J. sci. statist. comput., 1, 371, (1980)
[35] Kriegsmann, G., SIAM J. sci. statist. comput., 3, 318, (1982)
[36] Engquist, B.; Halpern, L., Appl. numer. methods, 4, 21, (1988)
[37] Sharan, S.K., Comput. struct., 26, 841, (1987)
[38] Zienkiewicz, O.C.; Newton, R.E., (), 359
[39] Bando, K.; Bettes, P.; Emson, C., Int. J numer. methods fluids, 4, 599, (1984)
[40] Sharan, S.K., Commun. appl. numer. methods, 4, 761, (1988)
[41] Pearson, R.A., J. atmos. sci., 31, 1418, (1974)
[42] Orlanski, I., J. comput. phys., 21, 251, (1976)
[43] Raymond, W.H.; Kuo, H.L., Q. J. roy. meteorol. soc., 110, 535, (1984)
[44] Miller, M.J.; Thorpe, A.J., Q. J. rov. meteorol. soc., 107, 615, (1981)
[45] Klemp, J.B.; Lilly, D.K., J. atmos. sci., 35, 78, (1978)
[46] Carpenter, K.M., Q. J. roy. meteorol. soc., 108, 717, (1982)
[47] Wurtele, M.; Paegle, J.; Sielecki, A., Mon. weather rev., 99, 537, (1971)
[48] Engquist, B.; Majda, A., J. comput. phys., 40, 91, (1981)
[49] Jiang, H.; Wong, Y.S., J. comput. phys., 88, 205, (1990)
[50] Hedstrom, G.W., J. comput. phys., 30, 222, (1979)
[51] Thompson, K.W., J. comput. phys., 68, 1, (1987)
[52] Thompson, K.W., J. comput. phys., 89, 439, (1990)
[53] Rudy, D.H.; Strikwerda, J.C., J. comput. phys., 36, 55, (1980)
[54] Rudy, D.H.; Strikwerda, J.C., Comput. fluids, 9, 327, (1981)
[55] Wilson, J.C., (), 1
[56] Hagstrom, T.; Hariharan, S.I., Math. comput., 51, 581, (1988)
[57] Kim, K.Y.; Reid, R.O.; Whitaker, R.E., J. comput. phys., 76, 327, (1988)
[58] Castellani, A., Meccanicca, 9, 199, (1974)
[59] White, W.; Valliappan, S.; Lee, I.K., J. eng. mech. div. ASCE, 103, 949, (1977)
[60] Clayton, R.; Engquist, B., Bull. seismol. soc. amer., 67, 1529, (1977)
[61] Emerman, S.H.; Stephen, R.A., Bull. seismol. soc. amer., 73, 661, (1983)
[62] Mahrer, K.D., Geophysics, 51, 1499, (1986)
[63] Sochacki, J., Appl. math. comput., 28, 1, (1988)
[64] Scandrett, C.L.; Kriegsmann, G.A.; Achenbach, J.D., SIAM J. sci. statist. comput., 7, 571, (1986)
[65] Higdon, R.L., SIAM J. numer. anal., 27, 831, (1990)
[66] Cohen, M.; Jennings, P.C., Silent boundary methods for transient analysis, (), 301
[67] Bamberger, A.; Chalinder, B.; Joly, P.; Roberts, J.E.; Teron, J.L., SIAM J. sci. statist. comput., 9, 1016, (1988)
[68] Robinson, A.R., The transmitting boundary-again, ()
[69] Barry, A.; Bielak, J.; MacCamy, R.C., J. comput. phys., 79, 449, (1988)
[70] Kriegsmann, G.; Taflove, A.; Umashankar, K.R., IEEE trans. antennas propag., AP-35, 153, (1987)
[71] Moore, T.G.; Blaschak, J.G.; Taflove, A.; Kriegsmann, G.A., IEEE trans. antennas propag., AP-36, 1797, (1988)
[72] Mur, G., IEEE trans. electromagn. comput., EMC-23, 377, (1981)
[73] Umashankar, K.R.; Taflove, A., IEEE trans. electromagn. compat., EMC-24, 392, (1982)
[74] Tajima, T., J. comput. phys., 42, 406, (1981)
[75] Blaschak, J.G.; Kriegsmann, G.A., J. comput. phys., 77, 109, (1988)
[76] Goldstein, C.I., Numerical methods for Helmholtz-type equations in unbounded regions, () · Zbl 0467.65058
[77] Grosch, C.E.; Orszag, S.A., J. comput. phys., 25, 273, (1977)
[78] Smith, W.D., J. comput. phys., 15, 492, (1974)
[79] Cerian, C.; Kosloff, D.; Kosloff, R.; Reshef, M., Geophysics, 50, 705, (1985)
[80] Sochacki, J.; Kubichek, R.; George, J.; Fletcher, W.R.; Smithson, S., Geophysics, 52, 60, (1987)
[81] Hanson, M.; Petschek, A., J. comput. phys., 21, 333, (1976)
[82] Kosloff, R.; Kosloff, D., J. comput. phys., 63, 363, (1986)
[83] Kurihara, Y.; Bender, M.A., Monl. weather rev., 111, 445, (1983)
[84] Israeli, M.; Orszag, S.A., J. comput. phys., 41, 115, (1981)
[85] \scS. Karni, “Far-Field Boundaries for Subsonic and Transonic Aerofoil Calculations,” College of Aeronautics, Cranfield Institute of Technology, England (unpublished).
[86] Goldstein, C.I., Math. comput., 36, 387, (1981)
[87] Day, S.M., (), (unpublished)
[88] Elvils, T.; Sundström, A., Tellus, 25, 132, (1973)
[89] Liao, Z.P.; Wong, H.L., Soil dyn. earthy. eng., 3, 174, (1984)
[90] Chu, C.K.; Sereny, A., J. comput. phys., 15, 476, (1974)
[91] Underwood, P.; Geers, T.L., Int. J. solids struct., 17, 687, (1981)
[92] Novak, M.; Mitwally, H., J. eng. mech. ASCE, 114, 181, (1988)
[93] Lysmer, J.; Wars, G., J. eng. mech. div. ASCE, 98, 85, (1972)
[94] Roesset, J.M.; Ettouney, M., Int. J. num. anal. meth. geomech., 1, 151, (1977)
[95] Greenspan, D.; Werner, P., Arch. rational mech. anal., 23, 288, (1967)
[96] Mei, K.K., IEEE trans. antennas propag., AP-22, 760, (1974)
[97] McDonald, B.H.; Wexler, A., IEEE trans. microwave theory tech., MTT-20, 841, (1972)
[98] Zienkiewicz, O.C.; Kelly, D.W.; Bettess, P., Int. J. numer. methods eng., 11, 355, (1977)
[99] Shaw, R.P.; Falby, W., Comput. fluids, 6, 153, (1978)
[100] Margulies, M., Combination of boundary element and finite element methods, (), 258
[101] Johnson, C.; Nedelec, J.C., Math. comput., 35, 1063, (1980)
[102] Hsiao, G.C., The coupling of BEM and FEM-A brief review, (), 431
[103] Cruse, T.; Rizzo, F., Boundary integral equation method: computational applications in applied mechanics, (1975), ASME, AMDII
[104] Brebbia, C.A., The boundary element method for engineers, (1978), Pentech London · Zbl 0414.65060
[105] Bielak, J.; MacCamy, R.C., Q. appl. math., 41, 143, (1983)
[106] Bettess, P., Int. J. numer. methods eng., 11, 53, (1977)
[107] Zienkiewicz, O.C., The finite element method, (1977), McGraw-Hill London · Zbl 0435.73072
[108] Trautenberg, E.A.; Gebauer, K.; Sachs, A., Numerical simulation of wave propagation in viscoelastic media with non-reflecting boundary conditions, (), 194
[109] Wagatha, L., Appl. numer. math., 1, 309, (1985)
[110] Beland, M.; Warn, T., J. atmos. sci., 32, 1873, (1975)
[111] Klemp, J.B.; Durran, D.R., Mon. weather rev., 111, 430, (1983)
[112] Bougeault, P., Mon weather rev., 111, 420, (1983)
[113] Bennett, A.F., J. atmos. sci., 33, 176, (1976)
[114] Gustafsson, B.; Kreiss, H.-O., J. comput. phys., 30, 333, (1979)
[115] Ferm, L.; Gustafsson, B., Comput. fluids, 10, 261, (1982)
[116] Hagstrom, T.; Keller, H.B., SIAM J. math. anal., 17, 322, (1986)
[117] Gustafsson, B., SIAM J. sci. statist. comput., 9, 812, (1988)
[118] Gustafsson, B., Appl. numer. math., 4, 3, (1988)
[119] Ting, L.; Miksis, M.J., J. acoust. soc. amer., 80, 1825, (1986)
[120] Fix, G.J.; Marin, S.P., J. comput. phys., 28, 253, (1978)
[121] MacCamy, R.C.; Marin, S.P., Int. J. math. math. sci., 3, 311, (1980)
[122] Marin, S.P., IEEE trans. antenn. and propagat., AP-30, 1045, (1982)
[123] Goldstein, C.I., Math. comput., 39, 309, (1982)
[124] Lenoir, M.; Tounsi, A., SIAM J. num. anal., 25, 729, (1988)
[125] Canuto, C.; Hariharan, S.I.; Lustman, L., Numer. math., 46, 505, (1985)
[126] Keller, J.B.; Givoll, D., J. comput. phys., 82, 172, (1988)
[127] \scD. Givoli and J. B. Keller, Wave Motion, to appear.
[128] Givoli, D.; Keller, J.B., Comput. methods. appl. mech. eng., 76, 41, (1989)
[129] Bayliss, A.; Goldstein, C.I.; Turkel, E., Comput. math. appl., 11, 655, (1984)
[130] Hariharan, S.I., Absorbing boundary conditions for exterior problems, () · Zbl 0611.65083
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