High-order discretization of a stable time-domain integral equation for 3D acoustic scattering. (English) Zbl 1453.65447
Summary: We develop a high-order, explicit method for acoustic scattering in three space dimensions based on a combined-field time-domain integral equation. The spatial discretization, of Nyström type, uses Gaussian quadrature on panels combined with a special treatment of the weakly singular kernels arising in near-neighbor interactions. In time, a new class of convolution splines is used in a predictor-corrector algorithm. Experiments on a torus and a perturbed torus are used to explore the stability and accuracy of the proposed scheme. This involved around one thousand solver runs, at up to 8th order and up to around 20,000 spatial unknowns, demonstrating 5–9 digits of accuracy. In addition we show that parameters in the combined field formulation, chosen on the basis of analysis for the sphere and other convex scatterers, work well in these cases.
MSC:
| 65R20 | Numerical methods for integral equations |
| 35P25 | Scattering theory for PDEs |
| 35L05 | Wave equation |
Software:
ARPACKReferences:
| [1] | Banjai, L., Multistep and multistage convolution quadrature for the wave equation: algorithms and experiments, SIAM J. Sci. Comput., 32, 2964-2994 (2010) · Zbl 1216.65122 |
| [2] | J.W. Banks, T. Hagstrom, On difference splines, in preparation. · Zbl 1349.65264 |
| [3] | Bleszynski, E.; Bleszynski, M.; Jaroszewicz, T., A new fast time domain integral equation solution algorithm, (Proc. 2001 IEEE Antennas and Propagation Society Int. Symp. (2001)), 176-179 |
| [4] | Bochev, P. B.; Gunzburger, M. D.; Shadid, J. N., On inf-sup stabilized finite element methods for transient problems, Comput. Methods Appl. Mech. Eng., 193, 1471-1489 (2004) · Zbl 1079.76577 |
| [5] | Bremer, J.; Gimbutas, Z., A Nyström method for weakly singular integral operators on surfaces, J. Comput. Phys., 231, 4885-4903 (2012) · Zbl 1245.65177 |
| [6] | Bruno, O. P.; Kunyansky, L. A., A fast, high-order algorithm for the solution of surface scattering problems: basic implementation, tests, and applications, J. Comput. Phys., 169, 80-110 (2001) · Zbl 1052.76052 |
| [7] | Burman, E.; Fernández, M. A., Analysis of the PSPG method for the transient Stokes’ problem, Comput. Methods Appl. Mech. Eng., 200, 2882-2890 (2011) · Zbl 1230.76022 |
| [8] | Cheng, Hongwei; Crutchfield, William Y.; Gimbutas, Zydrunas; Greengard, Leslie; Ethridge, J. Frank; Huang, Jingfang; Rokhlin, Vladimir; Yarvin, Norman; Zhao, Junsheng, A wideband fast multipole method for the Helmholtz equation in three dimensions, J. Comput. Phys., 216, 300-325 (2006) · Zbl 1093.65117 |
| [9] | Chew, W. C.; Michielssen, E.; Song, J. M.; Jin, J. M., Fast and Efficient Algorithms in Computational Electromagnetics (2001), Artech House, Inc.: Artech House, Inc. Norwood, MA |
| [10] | Colton, David; Kress, Rainer, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, vol. 93 (1998), Springer-Verlag: Springer-Verlag Berlin · Zbl 0893.35138 |
| [11] | Davies, P.; Duncan, D., Convolution-in-time approximations of time domain boundary integral equations, SIAM J. Sci. Comput., 35, B43-B61 (2013) · Zbl 1272.65074 |
| [12] | Davies, P.; Duncan, D., Convolution spline approximations of Volterra integral equations, J. Integral Equ. Appl., 3, 369-410 (2014) · Zbl 1307.65127 |
| [13] | Davis, P. J.; Rabinowitz, P., Methods of Numerical Integration (1984), Academic Press: Academic Press San Diego · Zbl 0537.65020 |
| [14] | Dyatlov, S.; Zworski, M., Mathematical Theory of Scattering Resonances, Graduate Studies in Mathematics, vol. 200 (2019), American Mathematical Society · Zbl 1454.58001 |
| [15] | Epstein, C.; Greengard, L., Debye sources and the numerical solution of the time harmonic Maxwell equations, Commun. Pure Appl. Math., 63, 413-463 (2010) · Zbl 1190.35215 |
| [16] | Epstein, C.; Greengard, L.; Hagstrom, T., On the stability of time-domain integral equations for acoustic wave propagation, Discrete Contin. Dyn. Syst., 36, 4367-4382 (2016) · Zbl 1333.65117 |
| [17] | Ergin, A.; Shanker, B.; Michielssen, E., Fast evaluation of three-dimensional transient wave fields using diagonal translation operators, J. Comput. Phys., 146, 157-180 (1998) · Zbl 0916.65092 |
| [18] | Ganesh, M.; Graham, I. G., A high-order algorithm for obstacle scattering in three dimensions, J. Comput. Phys., 198, 211-424 (2004) · Zbl 1052.65108 |
| [19] | Gimbutas, Z.; Veerapaneni, S., A fast algorithm for spherical grid rotations and its application to singular quadrature, SIAM J. Sci. Comput., 5, 6, A2738-A2751 (2013) · Zbl 1285.65089 |
| [20] | Greengard, L.; Hagstrom, T.; Jiang, S., Extension of the Lorenz-Mie-Debye method for electromagnetic scattering to the time domain, J. Comput. Phys., 299, 98-105 (2015) · Zbl 1351.78020 |
| [21] | Guenther, Ronald B.; Lee, John W., Partial Differential Equations of Mathematical Physics and Integral Equations (1988), Prentice Hall: Prentice Hall Englewood Cliffs, New Jersey · Zbl 0663.34017 |
| [22] | Ha-Duong, T., On retarded potential boundary integral equations and their discretisations, (Ainsworth, M.; Davies, P.; Duncan, D.; Martin, P.; Rynne, B., Topics in Computational Wave Propagation (2003), Springer-Verlag), 301-336 · Zbl 1051.78018 |
| [23] | Klöckner, A.; Barnett, A.; Greengard, L.; O’Neil, M., Quadrature by expansion: a new method for the evaluation of layer potentials, J. Comput. Phys., 252, 332-349 (2013) · Zbl 1349.65094 |
| [24] | Kress, R., Boundary integral equations in time-harmonic acoustic scattering, Math. Comput. Model., 15, 229-243 (1991) · Zbl 0731.76077 |
| [25] | Küther, M., Error estimates for the staggered Lax-Friedrichs scheme on unstructured grids, SIAM J. Numer. Anal., 39, 4, 1269-1301 (2001) · Zbl 1020.65061 |
| [26] | Lehoucq, R. B.; Sorensen, D. C.; Yang, C., ARPACK Users’ Guide. Solution of Large-Scale Eigenvalue Problems With Implicitly Restarted Arnoldi Methods (1998), Society for Industrial and Applied Mathematics (SIAM, 3600 Market Street, Floor 6, Philadelphia, PA 19104): Society for Industrial and Applied Mathematics (SIAM, 3600 Market Street, Floor 6, Philadelphia, PA 19104) Philadelphia, Pa. · Zbl 0901.65021 |
| [27] | Lubich, C., On the multistep time discretization of linear initial-boundary value problems and their boundary integral equations, Numer. Math., 67, 365-389 (1994) · Zbl 0795.65063 |
| [28] | Meng, J.; Boag, A.; Lomakin, V.; Michielssen, E., A multilevel Cartesian non-uniform grid time-domain algorithm, J. Comput. Phys., 229, 8430-8444 (2010) · Zbl 1202.78025 |
| [29] | Morawetz, C.; Ralston, J.; Strauss, W., Decay of solutions of the wave equation outside nontrapping obstacles, Commun. Pure Appl. Math., 30, 447-508 (1977) · Zbl 0372.35008 |
| [30] | Nonnenmacher, S.; Zworski, M., Quantum decay rates in chaotic scattering, Acta Math., 203, 2, 149-233 (2009) · Zbl 1226.35061 |
| [31] | Rokhlin, V., Diagonal forms of translation operators for the Helmholtz equation in three dimensions, Appl. Comput. Harmon. Anal., 1, 82-93 (1993) · Zbl 0795.35021 |
| [32] | Sauter, S.; Veit, A., A Galerkin method for retarded boundary integral equations with smooth and compactly supported temporal basis functions, Numer. Math., 123, 145-176 (2013) · Zbl 1262.65125 |
| [33] | Sayas, F.-J., Retarded Potentials and Time Domain Boundary Integral Equations: A Road Map, Springer Series in Computational Mathematics, vol. 50 (2016), Springer-Verlag: Springer-Verlag Switzerland · Zbl 1346.65047 |
| [34] | Shanker, B.; Ergin, A. A.; Aygun, K.; Michielssen, E., Analysis of transient electromagnetic scattering from closed surfaces using a combined field integral equation, IEEE Trans. Antennas Propag., 48, 1064-1074 (2000) · Zbl 1368.78061 |
| [35] | Shanker, B.; Ergin, A. A.; Mingyu, L.; Michielssen, E., Fast analysis of transient electromagnetic scattering phenomena using the multilevel plane wave time domain algorithm, IEEE Trans. Antennas Propag., 51, 628-641 (2003) |
| [36] | Trefethen, L. N., Approximation Theory and Approximation Practice (2013), SIAM · Zbl 1264.41001 |
| [37] | Ülkü, Hüseyin Arda; Bagci, Hakan; Michielssen, Eric, Marching on-in-time solution of the time domain magnetic field integral equation using a predictor-corrector scheme, IEEE Trans. Antennas Propag., 61, 8, 4120-4131 (2013) · Zbl 1370.78410 |
| [38] | Valdés, Felipe; Ghaffari-miab, Mohsen; Andriulli, Francesco P.; Cools, Kristof; Michielssen, Eric, High-order Calderón preconditioned time domain integral equation solvers, IEEE Trans. Antennas Propag., 61, 5, 2570-2588 (2013) · Zbl 1372.65354 |
| [39] | Weile, D.; Pisharody, G.; Chen, N.-W.; Shanker, B.; Michielssen, E., A novel scheme for the solution of the time-domain integral equations of electromagnetics, IEEE Trans. Antennas Propag., 52, 283-295 (2004) |
| [40] | Wienert, L., Die Numerische approximation von Randintegraloperatoren für die Helmholtzgleichung im \(R^3 (1990)\), University of Göttingen, PhD thesis · Zbl 0741.65085 |
| [41] | Yilmaz, A. E.; Jian-Ming, J.; Michielssen, E., Time domain adaptive integral method for surface integral equations, IEEE Trans. Antennas Propag., 52, 2692-2708 (2004) · Zbl 1368.78198 |
| [42] | Ying, L.; Biros, G.; Zorin, D., A high-order 3D boundary integral equation solver for elliptic PDEs in smooth domains, J. Comput. Phys., 216, 247-275 (2006) · Zbl 1105.65115 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
