Skip to main content
    • Aa
    • Aa

Radiation boundary conditions for the numerical simulation of waves

  • Thomas Hagstrom (a1)

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.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

S. Abarbanel and D. Gottlieb (1997), ‘A mathematical analysis of the PML method’, J. Comput. Phys. 134, 357363.

C. R. Anderson (1992), ‘An implementation of the fast multipole method without multipoles’, SIAM J. Sci. Statist. Comput. 13, 923947.

A. Barry , J. Bielak and R. MacCamy (1988), ‘On absorbing boundary conditions for wave propagation’, J. Comput. Phys. 79, 449468.

A. Bayliss and E. Turkel (1980), ‘Radiation boundary conditions for wave-like equations’, Comm. Pure Appl. Math. 33, 707725.

J.-P. Berenger (1994), ‘A perfectly matched layer for the absorption of electromagnetic waves’, J. Comput. Phys. 114, 185200.

W. Chew and W. Weedon (1994), ‘A 3-D perfectly matched medium from modified Maxwell's equations with stretched coordinates’, Microwave Optical Technol. Lett. 7, 599604.

G. Doetsch (1974), Introduction to the Theory and Application of the Laplace Transformation, Springer, New York.

J. Driscoll , D. Healy and D. Rockmore (1997), ‘Fast discrete polynomial transforms with applications to data analysis for distance transitive graphs’, SIAM J. Comput. 26, 10661099.

B. Engquist and A. Majda (1977), ‘Absorbing boundary conditions for the numerical simulation of waves’, Math. Comput. 31, 629651.

B. Engquist and A. Majda (1979), ‘Radiation boundary conditions for acoustic and elastic wave calculations’, Comm. Pure Appl. Math. 32, 313357.

T. Geers (1998), Benchmark problems, in Computational Methods for Unbounded Domains ( T. Geers , ed.), Kluwer Academic Publishers, Dordrecht, Netherlands, pp. 110.

M. Giles (1990), ‘Nonreflecting boundary conditions for Euler equation calculations’, AIAA Journal 28, 20502058.

D. Givoli (1991), ‘Non-reflecting boundary conditions’, J. Comput. Phys. 94, 129.

D. Givoli and D. Kohen (1995), ‘Non-reflecting boundary conditions based on Kirchoff-type formulae’, J. Comput. Phys. 117, 102113.

L. Greengard and V. Rokhlin (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. 229269.

M. Grote and J. Keller (1995), ‘Exact nonreflecting boundary conditions for the time dependent wave equation’, SIAM J. Appl. Math. 55, 280297.

M. Grote and J. Keller (1996), ‘Nonreflecting boundary conditions for time dependent scattering’, J. Comput. Phys. 127, 5281.

B. Gustafsson and H.-O. Kreiss (1979), ‘Boundary conditions for time-dependent problems with an artificial boundary’, J. Comput. Phys. 30, 333351.

T. Hagstrom (1991a), ‘Asymptotic boundary conditions for dissipative waves: General theory’, Math. Comput. 56, 589606.

T. Hagstrom (1991b), ‘Conditions at the downstream boundary for simulations of viscous, incompressible flow’, SIAM J. Sci. Statist. Comput. 12, 843858.

T. Hagstrom and J. Goodrich (1998), ‘Experiments with approximate radiation boundary conditions for computational aeroacoustics’, Appl. Numer. Math. 27, 385402.

T. Hagstrom and S. Hariharan (1998), ‘A formulation of asymptotic and exact boundary conditions using local operators’, Appl. Numer. Math. 27, 403416.

T. Hagstrom and H. B. Keller (1986), ‘Exact boundary conditions at an artificial boundary for partial differential equations in cylinders’, SIAM J. Math. Anal. 17, 322341.

E. Hairer , C. Lubich and M. Schlichte (1985), ‘Fast numerical solution of nonlinear Volterra convolutional equations’, SIAM J. Sci. Statist. Comput. 6, 532541.

L. Halpern (1986), ‘Artificial boundary conditions for the linear advection diffusion equation’, Math. Comput. 46, 425438.

L. Halpern (1991), ‘Artificial boundary conditions for incompletely parabolic perturbations of hyperbolic systems’, SIAM J. Math. Anal. 22, 12561283.

L. Halpern and J. Rauch (1987), ‘Error analysis for absorbing boundary conditions’, Numer. Math. 51, 459467.

L. Halpern and J. Rauch (1995), ‘Absorbing boundary conditions for diffusion equations’, Numer. Math. 71, 185224.

L. Halpern and M. Schatzman (1989), ‘Artificial boundary conditions for viscous incompressible flows’, SIAM J. Math. Anal. 20, 308353.

R. Higdon (1986), ‘Absorbing boundary conditions for difference approximations to the multidimensional wave equation’, Math. Comput. 47, 437459.

R. Higdon (1987), ‘Numerical absorbing boundary conditions for the wave equation’, Math. Comput. 49, 6590.

R. Higdon (1991), ‘Absorbing boundary conditions for elastic waves’, Geophysics 56, 231254.

R. Higdon (1992), ‘Absorbing boundary conditions for acoustic and elastic waves in stratified media’, J. Comput. Phys. 101, 386418.

R. Higdon (1994), ‘Radiation boundary conditions for dispersive waves’, SIAM J. Numer. Anal. 31, 64100.

C. Johansson (1993), ‘Boundary conditions for open boundaries for the incompressible Navier-Stokes equations’, J. Comput. Phys. 105, 233251.

T. Kato (1976), Perturbation Theory for Linear Operators, Springer, New York.

E. Lindman (1975), ‘Free space boundary conditions for the time dependent wave equation’, J. Comput. Phys. 18, 6678.

J.-P. Lohéac (1991), ‘An artificial boundary condition for an advection-diffusion equation’, Math. Meth. Appl. Sci. 14, 155175.

D. Ludwig (1960), ‘Exact and asymptotic solutions of the Cauchy problem’, Comm. Pure Appl. Math. 13, 473508.

J. Nordström (1995), ‘Accurate solutions of the Navier–Stokes equations despite unknown outflow boundary data’, J. Comput. Phys 120, 184205.

J. Nordström (1997), ‘On extrapolation procedures at artificial outflow boundaries for the time-dependent Navier–Stokes equations’, Appl. Numer. Math. 23, 457468.

A. Ramm (1986), Scattering by Obstacles, D. Reidel, Dordrecht, Netherlands.

V. Rokhlin (1990), ‘Rapid solution of integral equations of scattering theory in two dimensions’, J. Comput. Phys. 86, 414439.

V. Ryabeńkii (1985), ‘Boundary equations with projections’, Russian Math. Surveys 40, 147183.

L. Ting and M. Miksis (1986), ‘Exact boundary conditions for scattering problems’, J. Acoust. Soc. Amer. 80, 18251827.

L. Trefethen and L. Halpern (1986), ‘Well-posedness of one-way wave equations and absorbing boundary conditions’, Math. Comput. 47, 421435.

S. Tsynkov (1998), ‘Numerical solution of problems on unbounded domains. A review’, Appl. Numer. Math. 27, 465532.

E. Turkel and A. Yefet (1998), ‘Absorbing PML boundary layers for wave-like equations’, Appl. Numer. Math. 27, 533557.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Acta Numerica
  • ISSN: 0962-4929
  • EISSN: 1474-0508
  • URL: /core/journals/acta-numerica
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 36 *
Loading metrics...

Abstract views

Total abstract views: 203 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 24th May 2017. This data will be updated every 24 hours.