August  2016, 36(8): 4367-4382. doi: 10.3934/dcds.2016.36.4367

On the stability of time-domain integral equations for acoustic wave propagation

1. 

209 S. 33rd Street, Department of Mathematics, Philadelphia, PA, 19104-6395, United States

2. 

251 Mercer St, Courant Institute, NYU, New York, NY, 10012, United States

3. 

Dept. of Mathematics, Southern Methodist University, PO Box 750156, Dallas, TX 75275-0156, United States

Received  April 2015 Revised  October 2015 Published  March 2016

We give a principled approach for the selection of a boundary integral, retarded potential representation for the solution of scattering problems for the wave equation in an exterior domain.
Citation: Charles L. Epstein, Leslie Greengard, Thomas Hagstrom. On the stability of time-domain integral equations for acoustic wave propagation. Discrete & Continuous Dynamical Systems, 2016, 36 (8) : 4367-4382. doi: 10.3934/dcds.2016.36.4367
References:
[1]

D. Baskin, E. Spence and J. Wunsch, Sharp high-frequency estimates for the Helmholtz equation and applications to boundary integral equations, SIAM J. Math. Anal., 48 (2016), 229-267, arXiv:1504.01037v1. doi: 10.1137/15M102530X.  Google Scholar

[2]

S. N. Chandler-Wilde and P. Monk, Wave-number-explicit bounds in time- harmonic scattering, SIAM Journal on Mathematical Analysis, 39 (2008), 1428-1455. doi: 10.1137/060662575.  Google Scholar

[3]

M. Costabel, Time-dependent Problems with the Boundary Integral Equation Method, in Encyclopedia of Computational Mechanics (eds. E. Stein, R. Borst and T. Hughes), Wiley, New York, 2004. doi: 10.1002/0470091355.ecm022.  Google Scholar

[4]

V. Dominguez and F. Sayas, Some properties of layer potentials and boundary integral operators for the wave equation, J. Int. Equations Appl., 25 (2013), 253-294. doi: 10.1216/JIE-2013-25-2-253.  Google Scholar

[5]

T. Ha-Duong, B. Ludwig and I. Terrasse, A Galerkin BEM for transient acoustic scattering by an absorbing obstacle, Internat. J. Numer. Methods Engrg., 57 (2003), 1845-1882. doi: 10.1002/nme.745.  Google Scholar

[6]

T. Ha-Duong, On retarded potential boundary integral equations and their discretisation, in Topics in computational wave propagation, vol. 31 of Lect. Notes Comput. Sci. Eng., Springer, Berlin, 2003, 301-336, URL http://dx.doi.org/10.1007/978-3-642-55483-4_8. doi: 10.1007/978-3-642-55483-4_8.  Google Scholar

[7]

R. Kress, Minimizing the condition number of boundary integral-operators in acoustic and electromagnetic scattering, Q. J. Mech. Appl. Math., 38 (1985), 323-341. doi: 10.1093/qjmam/38.2.323.  Google Scholar

[8]

P. D. Lax, C. S. Morawetz and R. S. Phillips, Exponential decay of solutions of the wave equation in the exterior of a star-shaped obstacle, Comm. Pure Appl. Math., 16 (1963), 477-486. doi: 10.1002/cpa.3160160407.  Google Scholar

[9]

P. D. Lax and R. S. Phillips, Scattering Theory, vol. 26 of Pure and Applied Mathematics, 2nd edition, Academic Press, Inc., Boston, MA, 1989, With appendices by Cathleen S. Morawetz and Georg Schmidt.  Google Scholar

[10]

A. Ludwig and Y. Leviatan, Towards a stable two-dimensional time-domain source-model solution by use of a combined source formulation, IEEE Trans. Antennas Propag., 54 (2006), 3010-3021. doi: 10.1109/TAP.2006.882169.  Google Scholar

[11]

B. Shanker, A. A. Ergin, K. Aygün and E. Michielssen, Analysis of transient electromagnetic scattering from closed surfaces using a combined field integral equation, IEEE Trans. Antennas and Propagation, 48 (2000), 1064-1074. doi: 10.1109/8.876325.  Google Scholar

[12]

B. R. Vainberg, Asymptotic Methods in Equations of Mathematical Physics, CRC Press, 1989. Google Scholar

show all references

References:
[1]

D. Baskin, E. Spence and J. Wunsch, Sharp high-frequency estimates for the Helmholtz equation and applications to boundary integral equations, SIAM J. Math. Anal., 48 (2016), 229-267, arXiv:1504.01037v1. doi: 10.1137/15M102530X.  Google Scholar

[2]

S. N. Chandler-Wilde and P. Monk, Wave-number-explicit bounds in time- harmonic scattering, SIAM Journal on Mathematical Analysis, 39 (2008), 1428-1455. doi: 10.1137/060662575.  Google Scholar

[3]

M. Costabel, Time-dependent Problems with the Boundary Integral Equation Method, in Encyclopedia of Computational Mechanics (eds. E. Stein, R. Borst and T. Hughes), Wiley, New York, 2004. doi: 10.1002/0470091355.ecm022.  Google Scholar

[4]

V. Dominguez and F. Sayas, Some properties of layer potentials and boundary integral operators for the wave equation, J. Int. Equations Appl., 25 (2013), 253-294. doi: 10.1216/JIE-2013-25-2-253.  Google Scholar

[5]

T. Ha-Duong, B. Ludwig and I. Terrasse, A Galerkin BEM for transient acoustic scattering by an absorbing obstacle, Internat. J. Numer. Methods Engrg., 57 (2003), 1845-1882. doi: 10.1002/nme.745.  Google Scholar

[6]

T. Ha-Duong, On retarded potential boundary integral equations and their discretisation, in Topics in computational wave propagation, vol. 31 of Lect. Notes Comput. Sci. Eng., Springer, Berlin, 2003, 301-336, URL http://dx.doi.org/10.1007/978-3-642-55483-4_8. doi: 10.1007/978-3-642-55483-4_8.  Google Scholar

[7]

R. Kress, Minimizing the condition number of boundary integral-operators in acoustic and electromagnetic scattering, Q. J. Mech. Appl. Math., 38 (1985), 323-341. doi: 10.1093/qjmam/38.2.323.  Google Scholar

[8]

P. D. Lax, C. S. Morawetz and R. S. Phillips, Exponential decay of solutions of the wave equation in the exterior of a star-shaped obstacle, Comm. Pure Appl. Math., 16 (1963), 477-486. doi: 10.1002/cpa.3160160407.  Google Scholar

[9]

P. D. Lax and R. S. Phillips, Scattering Theory, vol. 26 of Pure and Applied Mathematics, 2nd edition, Academic Press, Inc., Boston, MA, 1989, With appendices by Cathleen S. Morawetz and Georg Schmidt.  Google Scholar

[10]

A. Ludwig and Y. Leviatan, Towards a stable two-dimensional time-domain source-model solution by use of a combined source formulation, IEEE Trans. Antennas Propag., 54 (2006), 3010-3021. doi: 10.1109/TAP.2006.882169.  Google Scholar

[11]

B. Shanker, A. A. Ergin, K. Aygün and E. Michielssen, Analysis of transient electromagnetic scattering from closed surfaces using a combined field integral equation, IEEE Trans. Antennas and Propagation, 48 (2000), 1064-1074. doi: 10.1109/8.876325.  Google Scholar

[12]

B. R. Vainberg, Asymptotic Methods in Equations of Mathematical Physics, CRC Press, 1989. Google Scholar

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