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 - A, 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.  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.  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, (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.  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.  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, (2003), 301.  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.  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.  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, (1989).   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.  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.  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.  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.  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, (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.  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.  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, (2003), 301.  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.  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.  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, (1989).   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.  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.  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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