Hostname: page-component-745bb68f8f-g4j75 Total loading time: 0 Render date: 2025-02-05T04:03:25.138Z Has data issue: false hasContentIssue false
  • English
  • Français

Radiation boundary conditions for the numerical simulation of waves

Published online by Cambridge University Press:  07 November 2008

Thomas Hagstrom
Thomas Hagstrom
Affiliation:
Department of Mathematics and Statistics, The University of New Mexico, Albuquerque, NM 87131, USA E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the efficient evaluation of accurate radiation boundary conditions for time domain simulations of wave propagation on unbounded spatial domains. This issue has long been a primary stumbling block for the reliable solution of this important class of problems. In recent years, a number of new approaches have been introduced which have radically changed the situation. These include methods for the fast evaluation of the exact nonlocal operators in special geometries, novel sponge layers with reflectionless interfaces, and improved techniques for applying sequences of approximate conditions to higher order. For the primary isotropic, constant coefficient equations of wave theory, these new developments provide an essentially complete solution of the numerical radiation condition problem.

Type
Research Article
Information
Acta Numerica , Volume 8 , January 1999 , pp. 47 - 106
Copyright
Copyright © Cambridge University Press 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abarbanel, S. and Gottlieb, D. (1997), ‘A mathematical analysis of the PML method’, J. Comput. Phys. 134, 357–363.Google Scholar
Abramowitz, M. and Stegun, I., eds (1972), Handbook of Mathematical Functions, Dover, New York.Google Scholar
Alpert, B., Greengard, L. and Hagstrom, T. (1999 a), ‘Accurate solution of the wave equation on unbounded domains’. In preparation.Google Scholar
Alpert, B., Greengard, L. and Hagstrom, T. (1999 b), ‘Rapid evaluation of nonreflecting boundary kernels for time-domain wave propagation’, SIAM J. Numer. Anal. To appear.Google Scholar
Anderson, C. R. (1992), ‘An implementation of the fast multipole method without multipoles’, SIAM J. Sci. Statist. Comput. 13, 923–947.Google Scholar
Barry, A., Bielak, J. and MacCamy, R. (1988), ‘On absorbing boundary conditions for wave propagation’, J. Comput. Phys. 79, 449–468.Google Scholar
Bayliss, A. and Turkel, E. (1980), ‘Radiation boundary conditions for wave-like equations’, Comm. Pure Appl. Math. 33, 707–725.CrossRefGoogle Scholar
Berenger, J.-P. (1994), ‘A perfectly matched layer for the absorption of electromagnetic waves’, J. Comput. Phys. 114, 185–200.Google Scholar
Bettess, P. (1992), Infinite Elements, Penshaw Press, Sunderland, UK.Google Scholar
Chew, W. and Weedon, W. (1994), ‘A 3-D perfectly matched medium from modified Maxwell's equations with stretched coordinates’, Microwave Optical Technol. Lett. 7, 599–604.CrossRefGoogle Scholar
Collino, F. (1993), Conditions d'ordre élevé pour des modèles de propagation d'ondes dans des domaines rectangulaires, Technical Report 1790, INRIA.Google Scholar
Collino, F. and Monk, P. (1998), ‘Optimizing the perfectly matched layer’. Preprint.CrossRefGoogle Scholar
Demkowicz, L. and Gerdes, K. (1999), ‘Convergence of the infinite element methods for the Helmholtz equation in separable domains’, Numer. Math. To appear.Google Scholar
Doetsch, G. (1974), Introduction to the Theory and Application of the Laplace Transformation, Springer, New York.Google Scholar
Driscoll, J., Healy, D. and Rockmore, D. (1997), ‘Fast discrete polynomial transforms with applications to data analysis for distance transitive graphs’, SIAM J. Comput. 26, 1066–1099.Google Scholar
Engquist, B. and Majda, A. (1977), ‘Absorbing boundary conditions for the numerical simulation of waves’, Math. Comput. 31, 629–651.Google Scholar
Engquist, B. and Majda, A. (1979), ‘Radiation boundary conditions for acoustic and elastic wave calculations’, Comm. Pure Appl. Math. 32, 313–357.Google Scholar
Eringen, A. and Şuhubi, E. (1975), Elastodynamics, Vol. 2, Academic Press, New York.Google Scholar
Geers, T. (1998), Benchmark problems, in Computational Methods for Unbounded Domains (Geers, T., ed.), Kluwer Academic Publishers, Dordrecht, Netherlands, pp. 1–10.Google Scholar
Giles, M. (1990), ‘Nonreflecting boundary conditions for Euler equation calculations’, AIAA Journal 28, 2050–2058.CrossRefGoogle Scholar
Givoli, D. (1991), ‘Non-reflecting boundary conditions’, J. Comput. Phys. 94, 1–29.CrossRefGoogle Scholar
Givoli, D. (1992), Numerical Methods for Problems in Infinite Domains, Vol. 33 of Studies in Applied Mechanics, Elsevier, Amsterdam.Google Scholar
Givoli, D. and Kohen, D. (1995), ‘Non-reflecting boundary conditions based on Kirchoff-type formulae’, J. Comput. Phys. 117, 102–113.CrossRefGoogle Scholar
Goodrich, J. and Hagstrom, T. (1999), ‘High-order radiation boundary conditions for computational aeroacoustics’. In preparation.Google Scholar
Greengard, L. and Lin, P. (1998), ‘On the numerical solution of the heat equation on unbounded domains (Part I)’. Preprint.Google Scholar
Greengard, L. and Rokhlin, V. (1997), A new version of the fast multipole method for the Laplace equation in three dimensions, in Acta Numerica, Vol. 6, Cambridge University Press, pp. 229–269.Google Scholar
Grote, M. and Keller, J. (1995), ‘Exact nonreflecting boundary conditions for the time dependent wave equation’, SIAM J. Appl. Math. 55, 280–297.CrossRefGoogle Scholar
Grote, M. and Keller, J. (1996), ‘Nonreflecting boundary conditions for time dependent scattering’, J. Comput. Phys. 127, 52–81.Google Scholar
Grote, M. and Keller, J. (1998), Exact nonreflecting boundary conditions for elastic waves, Technical Report 1998–08, ETH, Zürich.Google Scholar
Grote, M. and Keller, J. (1999), ‘Nonreflecting boundary conditions for Maxwell's equations’, J. Comput. Phys. To appear.Google Scholar
Gustafsson, B. and Kreiss, H.-O. (1979), ‘Boundary conditions for time-dependent problems with an artificial boundary’, J. Comput. Phys. 30, 333–351.Google Scholar
Hagstrom, T. (1983), Reduction of Unbounded Domains to Bounded Domains for Partial Differential Equation Problems, PhD thesis, California Institute of Technology.Google Scholar
Hagstrom, T. (1991a), ‘Asymptotic boundary conditions for dissipative waves: General theory’, Math. Comput. 56, 589–606.Google Scholar
Hagstrom, T. (1991b), ‘Conditions at the downstream boundary for simulations of viscous, incompressible flow’, SIAM J. Sci. Statist. Comput. 12, 843–858.Google Scholar
Hagstrom, T. (1995), On the convergence of local approximations to pseudodifferential operators with applications, in Proc. 3rd Int. Conf. on Math, and Num. Aspects of Wave Prop. Phen. (Bécache, E., Cohen, G., Joly, P. and Roberts, J., eds), SIAM, pp. 474–482.Google Scholar
Hagstrom, T. (1996), On high-order radiation boundary conditions, in IMA Volume on Computational Wave Propagation (Engquist, B. and Kriegsmann, G., eds), Springer, New York, pp. 1–22.Google Scholar
Hagstrom, T. and Goodrich, J. (1998), ‘Experiments with approximate radiation boundary conditions for computational aeroacoustics’, Appl. Numer. Math. 27, 385–402.Google Scholar
Hagstrom, T. and Hariharan, S. (1996), Progressive wave expansions and open boundary problems, in IMA Volume on Computational Wave Propagation (Engquist, B. and Kriegsmann, G., eds), Springer, New York, pp. 23–43.Google Scholar
Hagstrom, T. and Hariharan, S. (1998), ‘A formulation of asymptotic and exact boundary conditions using local operators’, Appl. Numer. Math. 27, 403–416.Google Scholar
Hagstrom, T. and Keller, H. B. (1986), ‘Exact boundary conditions at an artificial boundary for partial differential equations in cylinders’, SIAM J. Math. Anal. 17, 322–341.Google Scholar
Hagstrom, T. and Lorenz, J. (1994), Boundary conditions and the simulation of low Mach number flows, in Proceedings of the First International Conference on Theoretical and Computational Acoustics (Lee, D. and Schultz, M., eds), World Scientific, Singapore, pp. 657–668.Google Scholar
Hairer, E., Lubich, C. and Schlichte, M. (1985), ‘Fast numerical solution of nonlinear Volterra convolutional equations’, SIAM J. Sci. Statist. Comput. 6, 532–541.Google Scholar
Halpern, L. (1986), ‘Artificial boundary conditions for the linear advection diffusion equation’, Math. Comput. 46, 425–438.Google Scholar
Halpern, L. (1991), ‘Artificial boundary conditions for incompletely parabolic perturbations of hyperbolic systems’, SIAM J. Math. Anal. 22, 1256–1283.Google Scholar
Halpern, L. and Rauch, J. (1987), ‘Error analysis for absorbing boundary conditions’, Numer. Math. 51, 459–467.Google Scholar
Halpern, L. and Rauch, J. (1995), ‘Absorbing boundary conditions for diffusion equations’, Numer. Math. 71, 185–224.Google Scholar
Halpern, L. and Schatzman, M. (1989), ‘Artificial boundary conditions for viscous incompressible flows’, SIAM J. Math. Anal. 20, 308–353.Google Scholar
He, S. and Weston, V. (1996), Wave-splitting and absorbing boundary conditions for Maxwell's equations on a curved surface, Technical Report TRITA-TET-96–14, KTH, Stockholm.Google Scholar
Higdon, R. (1986), ‘Absorbing boundary conditions for difference approximations to the multidimensional wave equation’, Math. Comput. 47, 437–459.Google Scholar
Higdon, R. (1987), ‘Numerical absorbing boundary conditions for the wave equation’, Math. Comput. 49, 65–90.Google Scholar
Higdon, R. (1991), ‘Absorbing boundary conditions for elastic waves’, Geophysics 56, 231–254.Google Scholar
Higdon, R. (1992), ‘Absorbing boundary conditions for acoustic and elastic waves in stratified media’, J. Comput. Phys. 101, 386–418.Google Scholar
Higdon, R. (1994), ‘Radiation boundary conditions for dispersive waves’, SIAM J. Numer. Anal. 31, 64–100.Google Scholar
Holford, R. (1999), ‘A multipole expansion for the acoustic field exterior to a prolate or oblate spheroid’. Submitted to J. Acoust. Soc. Amer.Google Scholar
Johansson, C. (1993), ‘Boundary conditions for open boundaries for the incompressible Navier-Stokes equations’, J. Comput. Phys. 105, 233–251.Google Scholar
Kato, T. (1976), Perturbation Theory for Linear Operators, Springer, New York.Google Scholar
Kreiss, H.-O. and Lorenz, J. (1989), Initial-Boundary Value Problems and the Navier–Stokes Equations, Academic Press, New York.Google Scholar
Lindman, E. (1975), ‘Free space boundary conditions for the time dependent wave equation’, J. Comput. Phys. 18, 66–78.CrossRefGoogle Scholar
Lohéac, J.-P. (1991), ‘An artificial boundary condition for an advection-diffusion equation’, Math. Meth. Appl. Sci. 14, 155–175.Google Scholar
Ludwig, D. (1960), ‘Exact and asymptotic solutions of the Cauchy problem’, Comm. Pure Appl. Math. 13, 473–508.Google Scholar
Mohlenkamp, M. (1997), ‘A fast transform for spherical harmonics’. Preprint.Google Scholar
Newton, R. (1966), Scattering Theory of Waves and Particles, McGraw-Hill, New York.Google Scholar
Nordström, J. (1995), ‘Accurate solutions of the Navier–Stokes equations despite unknown outflow boundary data’, J. Comput. Phys 120, 184–205.Google Scholar
Nordström, J. (1997), ‘On extrapolation procedures at artificial outflow boundaries for the time-dependent Navier–Stokes equations’, Appl. Numer. Math. 23, 457–468.Google Scholar
Oberhettinger, F. and Badii, L. (1970), Tables of Laplace Transforms, Springer, New York.Google Scholar
Olver, F. (1954), ‘The asymptotic expansion of Bessel functions of large order’, Philos. Trans. Royal Soc. London A247, 328–368.Google Scholar
Petropoulos, P. (1999), ‘Reflectionless sponge layers as absorbing boundary conditions for the numerical solution of Maxwell's equations in rectangular, cylindrical and spherical coordinates’. Submitted to SIAM J. Appl. Math.Google Scholar
Radvogin, Y. and Zaitsev, N. (1998), Absolutely transparent boundary conditions for time-dependent wave problems, in Seventh International Conference on Hyperbolic Problems.Google Scholar
Ramm, A. (1986), Scattering by Obstacles, D. Reidel, Dordrecht, Netherlands.Google Scholar
Rokhlin, V. (1990), ‘Rapid solution of integral equations of scattering theory in two dimensions’, J. Comput. Phys. 86, 414–439.Google Scholar
Ryabeńkii, V. (1985), ‘Boundary equations with projections’, Russian Math. Surveys 40, 147–183.Google Scholar
Schwartz, M. (1987), Principles of Electrodynamics, Dover, New York.Google Scholar
Sofronov, I. (1993), ‘Conditions for complete transparency on the sphere for the three-dimensional wave equation’, Russian Acad. Sci. Dokl. Math. 46, 397–401.Google Scholar
Sofronov, I. (1999), ‘Artificial boundary conditions of absolute transparency for two-and three-dimensional external time-dependent scattering problems’, Euro. J. Appl. Math. To appear.Google Scholar
Ting, L. and Miksis, M. (1986), ‘Exact boundary conditions for scattering problems’, J. Acoust. Soc. Amer. 80, 1825–1827.Google Scholar
Trefethen, L. and Halpern, L. (1986), ‘Well-posedness of one-way wave equations and absorbing boundary conditions’, Math. Comput. 47, 421–435.Google Scholar
Trefethen, L. and Halpern, L. (1988), ‘Wide-angle one-way wave equations’, J. Acoust. Soc. Amer. 84, 1397–1404.Google Scholar
Tsynkov, S. (1998), ‘Numerical solution of problems on unbounded domains. A review’, Appl. Numer. Math. 27, 465–532.CrossRefGoogle Scholar
Turkel, E. and Yefet, A. (1998), ‘Absorbing PML boundary layers for wave-like equations’, Appl. Numer. Math. 27, 533–557.Google Scholar
Vacus, O. (1996), Singularités de frontière et conditions limites absorbantes: le problème du coin, Technical Report 2851, INRIA.Google Scholar
Xu, L. and Hagstrom, T. (1999), ‘On convergent sequences of approximate radiation boundary conditions and reflectionless sponge layers’. In preparation.Google Scholar