Wellposedness of oneway wave equations and absorbing boundary conditions
Authors:
Lloyd N. Trefethen and Laurence Halpern
Journal:
Math. Comp. 47 (1986), 421435
MSC:
Primary 65N99; Secondary 65D05, 65M05
MathSciNet review:
856695
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Abstract: A oneway wave equation is a partial differential equation which, in some approximate sense, behaves like the wave equation in one direction but permits no propagation in the opposite one. Such equations are used in geophysics, in underwater acoustics, and as numerical "absorbing boundary conditions". Their construction can be reduced to the approximation of on by a rational function . This paper characterizes those rational functions r for which the corresponding oneway wave equation is well posed, both as a partial differential equation and as an absorbing boundary condition for the wave equation. We find that if interpolates at sufficiently many points in , then wellposedness is assured. It follows that absorbing boundary conditions based on Padé approximation are well posed if and only if (m, n) lies in one of two distinct diagonals in the Padé table, the two proposed by Engquist and Majda. Analogous results also hold for oneway wave equations derived by Chebyshev or leastsquares approximation.
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 A. Bamberger et al., "The paraxial approximation for the wave equation: some new results," in Advances in Computer Methods for Partial Differential Equations V (R. Vichnevetsky and R. S. Stepleman, eds.), IMACS, 1984.
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 [3]
 A. Bayliss & E. Turkel, "Radiation boundary conditions for wavelike equations," Comm. Pure Appl. Math., v. 33, 1980, pp. 707725. MR 596431 (82b:65091)
 [4]
 A. J. Berkhout, "Wave field extrapolation techniques in seismic migration, a tutorial," Geophysics, v. 46, 1981, pp. 16381656.
 [5]
 E. W. Cheney, Introduction to Approximation Theory, McGrawHill, New York, 1966. MR 0222517 (36:5568)
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 E.W. Cheney & A. A. Goldstein, "Meansquare approximation by generalized rational functions," Math. Z., v. 95, 1967, pp. 232241. MR 0219952 (36:3022)
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 J. F. Claerbout, Imaging the Earth's Interior, Blackwell Publ. Co., 1985.
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 R. W. Clayton, Wavefield Inversion Methods for Refraction and Reflection Data, Ph.D. diss., Dept. of Geophysics, Stanford University, 1981.
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 R. W. Clayton & B. Engquist, "Absorbing boundary conditions for waveequation migration," Geophysics, v. 45, 1980, pp. 895904.
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 R. L. Higdon, "Initialboundary value problems for linear hyperbolic systems," SIAM Rev., v. 28, 1986, pp. 177217. MR 839822 (88a:35138)
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 L. Hörmander, The Analysis of Linear Partial Differential Operators. II, SpringerVerlag, Berlin and New York, 1983. MR 705278 (85g:35002b)
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 A. Iserles, "A note on Padé approximation and generalized hypergeometric functions," BIT, v. 19, 1979, pp. 543545. MR 559965 (81a:41028)
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 A. Iserles & G. Strang, "The optimal accuracy of difference schemes," Trans. Amer. Math. Soc., v. 277, 1983, pp. 779803. MR 694388 (84f:65070)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198608566952
PII:
S 00255718(1986)08566952
Keywords:
Wellposedness,
oneway wave equation,
paraxial approximation,
parabolic approximation,
rational approximation,
Padé table,
absorbing boundary condition,
migration,
underwater acoustics
Article copyright:
© Copyright 1986
American Mathematical Society
