Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Asymptotic boundary conditions for dissipative waves: general theory


Author: Thomas Hagstrom
Journal: Math. Comp. 56 (1991), 589-606
MSC: Primary 65N99; Secondary 65P05
DOI: https://doi.org/10.1090/S0025-5718-1991-1068817-3
MathSciNet review: 1068817
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: An outstanding issue in the computational analysis of time-dependent problems is the imposition of appropriate radiation boundary conditions at artificial boundaries. In this work we develop accurate conditions based on the asymptotic analysis of wave propagation over long ranges. Employing the method of steepest descent, we identify dominant wave groups and consider simple approximations to the dispersion relation in order to derive local boundary operators. The existence of a small number of dominant wave groups may be expected for systems with dissipation. Estimates of the error as a function of domain size are derived under general hypotheses, leading to convergence results. Some practical aspects of the numerical construction of the asymptotic boundary operators are also discussed.


References [Enhancements On Off] (What's this?)

  • [1] A. Barry, J. Bielak, and R. C. MacCamy, On absorbing boundary conditions for wave propagation, J. Comp. Phys. 79 (1988), 449-468. MR 973337 (90e:73046)
  • [2] A. Bayliss and E. Turkel, Radiation boundary conditions for wave-like equations, Comm. Pure Appl. Math. 33 (1980), 707-725. MR 596431 (82b:65091)
  • [3] J. E. Dennis, Jr. and R. B. Schnabel, Numerical methods for unconstrained optimization and nonlinear equations, Prentice-Hall, Englewood Cliffs, N.J., 1983. MR 702023 (85j:65001)
  • [4] S. D. Eidel'man, Parabolic systems, North-Holland, Amsterdam, 1969. MR 0252806 (40:6023)
  • [5] B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp. 31 (1977), 629-651. MR 0436612 (55:9555)
  • [6] G. H. Golub and C. F. van Loan, Matrix computations, 2nd ed., Johns Hopkins Univ. Press, Baltimore, 1989. MR 1002570 (90d:65055)
  • [7] B. Gustafsson and H.-O. Kreiss, Boundary conditions for time dependent problems with an artificial boundary, J. Comput. Phys. 30 (1979), 333-351. MR 529999 (80i:65096)
  • [8] T. Hagstrom, Boundary conditions at outflow for a problem with transport and diffusion, J. Comput. Phys. 69 (1987), 69-80. MR 892254 (88c:76006)
  • [9] -, Asymptotic expansions and boundary conditions for time-dependent problems, SIAM J. Numer. Anal. 23 (1986), 948-958. MR 859007 (87k:35035)
  • [10] -, Conditions at the downstream boundary for simulations of viscous, incompressible flow, SIAM J. Sci. Statist. Comput. (submitted).
  • [11] T. Hagstrom and H. B. Keller, Numerical calculation of traveling wave solutions of nonlinear parabolic equations, SIAM J. Sci. Statist. Comput. 7 (1986), 978-988. MR 848573 (88b:65099)
  • [12] L. Halpern, Artificial boundary conditions for the linear advection diffusion equation, Math. Comp. 46 (1986), 425-438. MR 829617 (87e:76005)
  • [13] L. Halpern and M. Schatzman, Artificial boundary conditions for viscous incompressible flows, SIAM J. Math. Anal. 20 (1989), 308-353. MR 982662 (90c:35164)
  • [14] S. I. Hariharan and T. Hagstrom, Far field expansion for anisotropic waves, Computational Acoustics, vol. 2 (D. Lee, A. Cakmak, and R. Vichnevetsky, eds.), North-Holland, Amsterdam, 1990, pp. 283-294. MR 1095065 (92c:76056)
  • [15] R. L. Higdon, Initial-boundary value problems for linear hyperbolic systems, SIAM Rev. 28 (1986), 177-217. MR 839822 (88a:35138)
  • [16] -, Numerical absorbing boundary conditions for the wave equation, Math. Comp. 49 (1987), 65-90. MR 890254 (88f:65168)
  • [17] H.-O. Kreiss, Difference approximations for boundary and eigenvalue problems for ordinary differential equations, Math. Comp. 26 (1972), 605-624. MR 0373296 (51:9496)
  • [18] L. Sirovich, Techniques of asymptotic analysis, Appl. Math. Sci., Vol. 2, Springer-Verlag, New York, 1971. MR 0275034 (43:792)
  • [19] J. C. Strikwerda, Initial boundary value problems for incompletely parabolic systems, Comm. Pure Appl. Math. 30 (1977), 797-822. MR 0460893 (57:884)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65N99, 65P05

Retrieve articles in all journals with MSC: 65N99, 65P05


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1991-1068817-3
Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society