Well-posedness of one-way wave equations and absorbing boundary conditions

Authors:
Lloyd N. Trefethen and Laurence Halpern

Journal:
Math. Comp. **47** (1986), 421-435

MSC:
Primary 65N99; Secondary 65D05, 65M05

DOI:
https://doi.org/10.1090/S0025-5718-1986-0856695-2

MathSciNet review:
856695

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Abstract: A one-way 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 one-way 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 well-posedness 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 one-way wave equations derived by Chebyshev or least-squares approximation.

**[1]**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.**[2]**A. Bamberger et al.,*Construction et Analyse d'Approximations Paraxiales en Milieu Hétérogène*I and II, Rapps. Internes Nos. 114 and 128, Centre de Mathématiques Appliquées, Ecole Polytechnique, 1984-85.**[3]**Alvin Bayliss and Eli Turkel,*Radiation boundary conditions for wave-like equations*, Comm. Pure Appl. Math.**33**(1980), no. 6, 707–725. MR**596431**, https://doi.org/10.1002/cpa.3160330603**[4]**A. J. Berkhout, "Wave field extrapolation techniques in seismic migration, a tutorial,"*Geophysics*, v. 46, 1981, pp. 1638-1656.**[5]**E. W. Cheney,*Introduction to approximation theory*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0222517****[6]**E. W. Cheney and A. A. Goldstein,*Mean-square approximation by generalized rational functions*, Math. Z.**95**(1967), 232–241. MR**0219952**, https://doi.org/10.1007/BF01111526**[7]**J. F. Claerbout,*Imaging the Earth's Interior*, Blackwell Publ. Co., 1985.**[8]**R. W. Clayton,*Wavefield Inversion Methods for Refraction and Reflection Data*, Ph.D. diss., Dept. of Geophysics, Stanford University, 1981.**[9]**R. W. Clayton & B. Engquist, "Absorbing boundary conditions for wave-equation migration,"*Geophysics*, v. 45, 1980, pp. 895-904.**[10]**Bjorn Engquist and Andrew Majda,*Absorbing boundary conditions for the numerical simulation of waves*, Math. Comp.**31**(1977), no. 139, 629–651. MR**0436612**, https://doi.org/10.1090/S0025-5718-1977-0436612-4**[11]**Björn Engquist and Andrew Majda,*Radiation boundary conditions for acoustic and elastic wave calculations*, Comm. Pure Appl. Math.**32**(1979), no. 3, 314–358. MR**517938**, https://doi.org/10.1002/cpa.3160320303**[12]**W. B. Gragg and G. D. Johnson,*The Laurent-Padé table*, Information processing 74 (Proc. IFIP Congress, Stockholm, 1974) North-Holland, Amsterdam, 1974, pp. 632–637. MR**0411123****[13]**Robert L. Higdon,*Initial-boundary value problems for linear hyperbolic systems*, SIAM Rev.**28**(1986), no. 2, 177–217. MR**839822**, https://doi.org/10.1137/1028050**[14]**Robert L. Higdon,*Absorbing boundary conditions for difference approximations to the multidimensional wave equation*, Math. Comp.**47**(1986), no. 176, 437–459. MR**856696**, https://doi.org/10.1090/S0025-5718-1986-0856696-4**[15]**Lars Hörmander,*The analysis of linear partial differential operators. II*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 257, Springer-Verlag, Berlin, 1983. Differential operators with constant coefficients. MR**705278****[16]**Arieh Iserles,*A note on Padé approximations and generalized hypergeometric functions*, BIT**19**(1979), no. 4, 543–545. MR**559965**, https://doi.org/10.1007/BF01931272**[17]**Arieh Iserles and Gilbert Strang,*The optimal accuracy of difference schemes*, Trans. Amer. Math. Soc.**277**(1983), no. 2, 779–803. MR**694388**, https://doi.org/10.1090/S0002-9947-1983-0694388-9**[18]**Moshe Israeli and Steven A. Orszag,*Approximation of radiation boundary conditions*, J. Comput. Phys.**41**(1981), no. 1, 115–135. MR**624739**, https://doi.org/10.1016/0021-9991(81)90082-6**[19]**F. John,*Partial Differential Equations*, 3rd ed., Springer-Verlag, Berlin and New York, 1978.**[20]**Heinz-Otto Kreiss,*Initial boundary value problems for hyperbolic systems*, Comm. Pure Appl. Math.**23**(1970), 277–298. MR**0437941**, https://doi.org/10.1002/cpa.3160230304**[21]**G. Lamprecht,*Zur Mehrdeutigkeit bei der Approximation in der 𝐿_{𝑝}-Norm mit Hilfe rationaler Funktionen*, Computing (Arch. Elektron. Rechnen)**5**(1970), 349–355 (German, with English summary). MR**0281319****[22]**D. Lee,*The State-of-the-Art Parabolic Equation Approximation as Applied to Underwater Acoustic Propagation with Discussions on Intensive Computations*, Tech. Doc. 7247, U.S. Naval Underwater Systems Center, New London, Conn., 1984.**[23]**E. L. Lindman, "'Free-space' boundary conditions for the time dependent wave equation,"*J. Comput. Phys.*, v. 18, 1975, pp. 66-78.**[24]**S. T. McDaniel, "Parabolic approximations for underwater sound propagation,"*J. Acoust. Soc. Amer.*, v. 58, 1975, pp. 1178-1185.**[25]**D. J. Newman,*Rational approximation to |𝑥|*, Michigan Math. J.**11**(1964), 11–14. MR**0171113****[26]**R. H. Stolt, "Migration by Fourier transform,"*Geophysics*, v. 43, 1978, pp. 23-48.**[27]**Fred D. Tappert,*The parabolic approximation method*, Wave propagation and underwater acoustics (Workshop, Mystic, Conn., 1974), Springer, Berlin, 1977, pp. 224–287. Lecture Notes in Phys., Vol. 70. MR**0475274****[28]**Lloyd N. Trefethen,*Instability of difference models for hyperbolic initial-boundary value problems*, Comm. Pure Appl. Math.**37**(1984), no. 3, 329–367. MR**739924**, https://doi.org/10.1002/cpa.3160370305**[29]**Lloyd N. Trefethen,*Square blocks and equioscillation in the Padé, Walsh, and CF tables*, Rational approximation and interpolation (Tampa, Fla., 1983) Lecture Notes in Math., vol. 1105, Springer, Berlin, 1984, pp. 170–181. MR**783272**, https://doi.org/10.1007/BFb0072410**[30]**Ludwig Wagatha,*Approximation of pseudodifferential operators in absorbing boundary conditions for hyperbolic equations*, Numer. Math.**42**(1983), no. 1, 51–64. MR**716473**, https://doi.org/10.1007/BF01400917**[31]**G. Wanner, E. Hairer, and S. P. Nørsett,*Order stars and stability theorems*, BIT**18**(1978), no. 4, 475–489. MR**520756**, https://doi.org/10.1007/BF01932026**[32]**Guan Quan Zhang,*High order approximation of one-way wave equations*, J. Comput. Math.**3**(1985), no. 1, 90–97. MR**815413****[33]**Laurence Halpern and Lloyd N. Trefethen,*Wide-angle one-way wave equations*, J. Acoust. Soc. Amer.**84**(1988), no. 4, 1397–1404. MR**965847**, https://doi.org/10.1121/1.396586

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1986-0856695-2

Keywords:
Well-posedness,
one-way wave equation,
paraxial approximation,
parabolic approximation,
rational approximation,
Padé table,
absorbing boundary condition,
migration,
underwater acoustics

Article copyright:
© Copyright 1986
American Mathematical Society