Error estimates for finite difference methods for a wide-angle "parabolic" equation

SIAM Journal on Numerical Analysis

Volumn 33, Issue 6, 1996, Pages 2488-2509

  Akrivis, G D ^a   Dougalis, V A ^b,d   Zouraris, G E ^c,d  

^a UNIVERSITY OF IOANNINA   (Greece)

^b UNIVERSITY OF ATHENS   (Greece)

^c UNIVERSITY OF CRETE   (Greece)

^d INSTITUTE OF COMPUTER SCIENCE   (Greece)

Author keywords

Finite difference error estimates; Interface problems; Underwater acoustics; Wide angle "parabolic" equation

Indexed keywords

EID: 0040580194     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142994266352     Document Type: Article

References (32)

1. G. D. AKRIVIS AND V. A. DOUGALIS, Finite difference discretizations of some initial and boundary value problems with interface, Math. Comp., 56 (1991), pp. 505-522.
2. _, On a conservative finite difference method for the third-order, wide-angle parabolic equation, in Computational Acoustics: Acoustic Propagation 2, D. Lee, R. Vichnevetsky, and A. R. Robinson, eds., North-Holland, Amsterdam, 1993, pp. 209-220.
3. G. D. AKRIVIS, V. A. DOUGALIS, AND N. A. KAMPANIS, On Galerkin methods for the wide-angle parabolic equation, J. Comput. Acoustics, 2 (1994), pp. 99-112.
4. _, Error estimates for finite element methods for a wide-angle parabolic equation, Appl. Numer. Math., 16 (1994), pp. 81-100.
5. D. N. ARNOLD, J. DOUGLAS JR., AND V. THOMÉE, Superconvergence of a finite element approximation to the solution of a Sobolev equation in a single space variable, Math. Comp., 36 (1981), pp. 53-63.
6. A. BAMBERGER, B. ENGQUIST, L. HALPERN, AND P. JOLY, Parabolic wave equation approximations in heterogeneous media, SIAM J. Appl. Math., 48 (1988), pp. 99-128.
7. _, Higher order paraxial wave equation approximations in heterogeneous media, SIAM J. Appl. Math., 48 (1988), pp. 129-154.
8. G. BOTSEAS, D. LEE, AND K. E. GILBERT, IFD: Wide Angle Capability, Technical Report 6905, Naval Underwater Systems Center, New London Laboratory, New London, CT, 1983.
9. G. BOTSEAS, D. LEE, AND D. KING, FOR3D: A Computer Model for Solving the LSS Three-Dimensional, Wide Angle Wave Equation, Technical Report 7943, Naval Underwater Systems Center, New London Laboratory, New London, CT, 1987.
10. H. K. BROCK, The AESD Parabolic Equation Model, Technical Note 12, Naval Ocean Research and Development Activity, NSTL Station, Mississippi, 1978.
11. J. F. CLAERBOUT, Fundamentals of Geophysical Data Processing, with Applications to Petroleum Prospecting, McGraw-Hill, New York, 1976.
12. M. D. COLLINS, The time-domain solution of the wide angle parabolic equation including the effects of sediment dispersion, J. Acoust. Soc. Amer., 84 (1988), pp. 2114-2125.
13. _, Applications and time-domain solution of higher order parabolic equations in underwater acoustics, J. Acoust. Soc. Amer., 86 (1989), pp. 1097-1102.
14. M. D. COLLINS AND S. A. CHIN-BING, Three-dimensional sound propagation in shallow water including the effects of rough surfaces, in Computational Acoustics: Seismo-Ocean Acoustics and Modeling 3, D. Lee, A. Cakmak, and R. Vichnevetsky, eds., North-Holland, Amsterdam, 1990, pp. 247-259.
15. J. A. DAVIS, D. WHITE, AND R. C. CANAVAGH, NORDA Parabolic Equation Workshop 1981, Technical Note 143, Naval Ocean Research and Development Activity, NSTL Station, Mississippi, 1982.
16. R. R. GREENE, The rational approximation to the acoustic wave equation with bottom interaction, J. Acoust. Soc. Amer, 76 (1984), pp. 1764-1773.
17. L. HALPERN AND L. N. TREFETHEN, Wide-angle one-way wave equations, J. Acoust. Soc. Amer., 84 (1988), pp. 1397-1404.
18. F. B. JENSEN AND H. KROL, The Use of the Parabolic Equation Method in Sound Propagation Modelling, SACLANTCEN Memorandum 72, North Atlantic Treaty Organization, SACLANT ASW Research Centre, La Spezia, Italy, 1975.
19. N. A. KAMPANIS, Galerkin-Finite Element Methods for Interface Problems in Underwater Acoustics, Ph.D. Thesis, University of Crete, Heraklion, Greece, 1992.
20. J. E. LAGNESE, General boundary value problems for differential equations of Sobolev type, SIAM J. Math. Anal., 3 (1972), pp. 105-119.
21. D. LEE AND S. T. MCDANIEL, Ocean acoustic propagation by finite difference methods, Comput. Math. Appl., 14 (1987), pp. 305-423.
22. D. LEE, Y. SAAD, AND M. H. SCHULTZ, An efficient method for solving the three-dimensional wide angle wave equation, in Computational Acoustics: Wave Propagation, D. Lee, R. L. Sternberg, and M. H. Schultz, eds., North-Holland, Amsterdam, 1988, pp. 75-89.
23. S. T. MCDANIEL, Applications of energy methods to finite-difference solutions of the parabolic wave equation, Comput. Math. Appl., 11 (1985), pp. 823-829.
24. S. T. MCDANIEL AND D. LEE, A finite difference treatment of interface conditions for the parabolic wave equation: Horizontal interface, J. Acoust. Soc. Amer., 71 (1982), pp. 855-858.
25. W. L. SIEGMANN, G. A. KRIEGSMANN, AND D. LEE, A wide-angle three-dimensional parabolic wave equation, J. Acoust, Soc. Amer., 78 (1985), pp. 659-664.
26. D. F. ST. MARY, Analysis of an implicit finite-difference scheme for a third-order partial differential equation in three dimensions, Comput. Math. Appl., 11 (1985), pp. 873-885.
27. D. F. ST. MARY AND D. LEE, Analysis of an implicit finite difference solution to an underwater wave propagation problem, J. Comput. Phys., 57 (1985), pp. 378-390.
28. _, Accurate computation of the wide angle wave equation, in Computational Acoustics: Wave Propagation, D. Lee, R. L. Sternberg, and M. H. Schultz, eds., North-Holland, Amsterdam, 1988, pp. 409-422.
29. F. D. TAPPERT, The parabolic approximation method, Lecture Notes in Physics 70, in Wave Propagation and Underwater Acoustics, J. B. Keller and J. S. Papadakis, eds., Springer-Verlag, Berlin, 1977, pp. 224-287.
30. D. J. THOMSON, The parabolic equation method in underwater acoustics, Lecture Notes, Advanced Course on Acoustical Oceanography, Commission of the European Communities DG-XII, Marine Sciences and Technologies (M.A.S.T.) Programs, and Institute of Applied and Computational Mathematics, Foundation for Research and Technology-Hellas, Heraklion, Greece, 1993.
31. D. J. THOMSON AND N. R. CHAPMAN, A wide angle split-step algorithm for the parabolic equation, J. Acoust. Soc. Amer., 74 (1983), pp. 1848-1854.
32. C. H. WILCOX, Scattering theory for the D'Alembert equation in exterior domains, Lecture Notes in Mathematics 442, Springer-Verlag, Berlin, 1975.