Sixth-order compact finite difference method for 2D Helmholtz equations with singular sources and reduced pollution effect. (English) Zbl 1538.65454
Summary: Due to its highly oscillating solution, the Helmholtz equation is numerically challenging to solve. To obtain a reasonable solution, a mesh size that is much smaller than the reciprocal of the wavenumber is typically required (known as the pollution effect). High-order schemes are desirable, because they are better in mitigating the pollution effect. In this paper, we present a high-order compact finite difference method for 2D Helmholtz equations with singular sources, which can also handle any possible combinations of boundary conditions (Dirichlet, Neumann, and impedance) on a rectangular domain. Our method is sixth-order consistent for a constant wavenumber, and fifth-order consistent for a piecewise constant wavenumber. To reduce the pollution effect, we propose a new pollution minimization strategy that is based on the average truncation error of plane waves. Our numerical experiments demonstrate the superiority of our proposed finite difference scheme with reduced pollution effect to several state-of-the-art finite difference schemes, particularly in the critical pre-asymptotic region where \(kh\) is near 1 with \(k\) being the wavenumber and \(h\) the mesh size.
MSC:
| 65N06 | Finite difference methods for boundary value problems involving PDEs |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 65F20 | Numerical solutions to overdetermined systems, pseudoinverses |
Keywords:
Helmholtz equation; finite difference; pollution effect; interface; pollution minimization; mixed boundary conditions; corner treatmentSoftware:
OASESReferences:
| [1] | M. Ainsworth, Discrete dispersion relation for hp-version finite element approximation at high wave number. SIAM J. Numer. Anal. 42 (2004), no. 2, 553-575. · Zbl 1074.65112 |
| [2] | I. M. Babuška and S. A. Sauter, Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? SIAM Rev. 42 (2000), no. 3, 451-484. · Zbl 0956.65095 |
| [3] | G. Bao and W. Sun, A fast algorithm for the electromagnetic scattering from a large cavity. SIAM J. Sci. Comput. 27 (2005), no. 2, 553-574. · Zbl 1089.78024 |
| [4] | S. Britt, S. Tsynkov, and E. Turkel, A compact fourth order scheme for the Helmholtz equa-tion in polar coordinates. J. Sci. Comput. 45 (2010), 26-47. · Zbl 1203.65218 |
| [5] | S. Britt, S. Tsynkov, and E. Turkel, Numerical simulation of time-harmonic waves in inho-mogeneous media using compact high order schemes. Commun. Comput. Phys. 9 (2011), no. 3, 520-541. · Zbl 1364.78034 |
| [6] | S. Britt, S. Tsynkov, and E. Turkel, A high order numerical method for the Helmholtz equa-tion with nonstandard boundary conditions. SIAM J. Sci. Comput. 35 (2013), no. 5, A2255-A2292. · Zbl 1281.65135 |
| [7] | T. Chaumont-Frelet, Approximations par l’elements finis de problemes d’Helmholtz pour la propagation d’ondes sismiques, PhD Thesis at Inria, (2015). |
| [8] | Z. Chen, D. Cheng, W. Feng, and T. Wu, An optimal 9-point finite difference scheme for the Helmholtz equation with PML. Int. J. Numer. Anal. Mod. 10 (2013), no. 2, 389-410. · Zbl 1272.65081 |
| [9] | Z. Chen, T. Wu, and H. Yang, An optimal 25-point finite difference scheme for the Helmholtz equation with PML. J. Comput. Appl. Math. 236 (2011), 1240-1258. · Zbl 1233.65076 |
| [10] | P.-H. Cocquet, M. J. Gander, and X. Xiang, Closed form dispersion corrections including a real shifted wavenumber for finite difference discretizations of 2D constant coefficient Helmholtz problems. SIAM J. Sci. Comput. 43 (2021), no. 1, A278-A308. · Zbl 1464.35064 |
| [11] | H. Dastour and W. Liao, A fourth-order optimal finite difference scheme for the Helmholtz equation with PML. Comput. Math. Appl. 78 (2019), no. 6, 2147-2165. · Zbl 1442.65323 |
| [12] | H. Dastour and W. Liao, An optimal 13-point finite difference scheme for a 2D Helmholtz equation with a perfectly matched layer boundary condition. Numer. Algorithms 86 (2021), 1109-1141. · Zbl 1458.65133 |
| [13] | Y. Du and H. Wu, Preasymptotic error analysis of higher order FEM and CIP-FEM for Helmholtz equation with high wave number. SIAM J. Numer. Anal. 53 (2015), no. 2, 782-804. · Zbl 1312.65189 |
| [14] | V. Dwarka and C. Vuik, Pollution and accuracy of solutions of the Helmholtz equation: a novel perspective from the eigenvalues. J. Comput. Appl. Math. 395 (2021), 1-21. · Zbl 1466.35092 |
| [15] | Y. A. Erlangga, C. W. Oosterlee, and C. Vuik, A novel multigrid based preconditioner for heterogeneous Helmholtz problems. SIAM J. Sci. Comput. 27 (2006), no. 4, 1471-1492. · Zbl 1095.65109 |
| [16] | O. G. Ernst and M. J. Gander, Why is it difficult to solve Helmholtz problems with classical iterative methods. Numerical analysis of multiscale problems, Lecture Notes in Computational Science and Engineering 83, Springer, Berlin, Heidelberg, 2011, 325-363. · Zbl 1248.65128 |
| [17] | Q. Feng, B. Han, and P. Minev, Sixth order compact finite difference schemes for Poisson interface problems with singular sources. Comp. Math. Appl. 99 (2021), 2-25. · Zbl 1524.65711 |
| [18] | Q. Feng, B. Han, and P. Minev, A high order compact finite difference scheme for elliptic interface problems with discontinuous and high-contrast coefficients. Appl. Math. Comput. 431 (2022), 127314. · Zbl 1510.76109 |
| [19] | X. Feng, Z. Li, and Z. Qiao, High order compact finite difference schemes for the Helmholtz equation with discontinuous coefficients. J. Comput. Math. 29 (2011), no. 3, 324-340. · Zbl 1249.65214 |
| [20] | S. Fu and K. Gao, A fast solver for the Helmholtz equation based on the generalized multi-scale finite-element method. Geophys. J. Int. 211 (2017), no. 2, 797-813. |
| [21] | S. Fu, G. Li, R. Craster, and S. Guenneau, Wavelet-based edge multiscale finite element method for Helmholtz problems in perforated domains. Multiscale Model. Simul. 19 (2021), no. 4, 1684-1709. · Zbl 1478.65124 |
| [22] | X. Feng and H. Wu, Discontinuous Galerkin methods for the Helmholtz equation with large wave number. SIAM J. Numer. Anal. 47 (2009), no. 4, 2872-2896. · Zbl 1197.65181 |
| [23] | X. Feng and H. Wu, hp-discontinuous Galerkin methods for the Helmholtz equation with large wave number. Math. Comp. 80 (2011), no. 276, 1997-2024. · Zbl 1228.65222 |
| [24] | M. J. Gander and H. Zhang, A class of iterative solvers for the Helmholtz equation: factor-izations, sweeping preconditioners, source transfer, single layer potentials, polarized traces, and optimized Schwarz methods. SIAM Rev. 61 (2019), no. 1, 3-76. · Zbl 1417.65216 |
| [25] | I. G. Graham and S. A. Sauter, Stability and finite element error analysis for the Helmholtz equation with variable coefficients. Math. Comp. 89 (2020), no. 321, 105-138. · Zbl 1427.35016 |
| [26] | B. Han, M. Michelle, and Y. S. Wong, Dirac assisted tree method for 1D heterogeneous Helmholtz equations with arbitrary variable wave numbers. Comput. Math. Appl. 97 (2021), 416-438. · Zbl 1524.35159 |
| [27] | B. Han and M. Michelle, Sharp wavenumber-explicit stability bounds for 2D Helmholtz equations, SIAM J. Numer. Anal. 60 (2022), no. 4, 1985-2013. · Zbl 1497.35103 |
| [28] | U. Hetmaniuk, Stability estimates for a class of Helmholtz problems. Commun. Math. Sci. 5 (2007), no. 3, 665-678. · Zbl 1135.35323 |
| [29] | R. Hiptmair, A. Moiola, and I. Perugia, A survey of Trefftz methods for the Helmholtz equation. Building bridges: connections and challenges in modern approaches to numer-ical partial differential equations, Lecture Notes in Computational Science and Engineering 114, Springer, Cham, 2016, 237-279. · Zbl 1357.65282 |
| [30] | F. Ihlenburg and I. M. Babuška, Finite element solution of the Helmholtz equation with high wave number part II: the hp version of the FEM. SIAM J. Numer. Anal. 34 (2006), no. 1, 315-358. · Zbl 0884.65104 |
| [31] | F. B. Jensen, W. A. Kuperman, M. B. Porter, and H. Schmidt, Computational Ocean Acoustics, Modern Acoustics and Signal Processing. Springer, New York, 2011. xviii+794. · Zbl 1234.76003 |
| [32] | Z. Li and K. Pan, High order compact schemes for ux type BCs. SIAM J. Sci. Comput. 45 (2023), A646-A674. · Zbl 1516.65115 |
| [33] | M. Medvinsky, S. Tsynkov, and E. Turkel, The method of difference potentials for the Helmholtz equation using compact high order schemes. J.Sci. Comput. 53 (2012), 150-193. · Zbl 1254.65118 |
| [34] | J. M. Melenk and S. Sauter, Wavenumber explicit convergence analysis for Galerkin dis-cretizations of the Helmholtz equation. SIAM J. Numer. Anal. 49 (2011), no. 3, 1210-1243. · Zbl 1229.35038 |
| [35] | J.-C. Nédélec, Acoustic and electromagnetic equations. Integral representations for har-monic problems, Applied Mathematical Sciences 144. Springer-Verlag, New York, 2001. x+316. · Zbl 0981.35002 |
| [36] | K. Pan, D. He, and Z. Li, A high order compact FD framework for elliptic BVPs involving singular sources, interfaces, and irregular domains. J. Sci. Comput. 88 (2021), no. 67, 1-25. · Zbl 1480.65315 |
| [37] | D. Peterseim, Eliminating the pollution effect in Helmholtz problems by local subscale cor-rection. Math. Comp. 86 (2017), no. 305, 1005-1036. · Zbl 1359.65265 |
| [38] | C. C. Stolk, M. Ahmed, and S. K. Bhowmik, A multigrid method for the Helmholtz equation with optimized coarse grid corrections. SIAM J. Sci. Comput. 36 (2014), no. 6, A2819-A2841. · Zbl 1310.65157 |
| [39] | E. Turkel, D. Gordon, R. Gordon, and S. Tsynkov, Compact 2D and 3D sixth order schemes for the Helmholtz equation with variable wave number. J. Comp. Phys. 232 (2013), no. 1, 272-287. · Zbl 1291.65273 |
| [40] | K. Wang and Y. S. Wong, Pollution-free finite difference schemes for non-homogeneous Helmholtz equation. Int. J. Numer. Anal. Mod. 11 (2014), no. 4, 787-815. · Zbl 1499.65614 |
| [41] | K. Wang and Y. S. Wong, Is pollution effect of finite difference schemes avoidable for multi-dimensional Helmholtz equations with high wave numbers? Commun. Comput. Phys. 21 (2017), no. 2, 490-514. · Zbl 1488.65562 |
| [42] | T. Wu and R. Xu, An optimal compact sixth-order finite difference scheme for the Helmholtz equation. Comput. Math. Appl. 75 (2018), no. 7, 2520-2537. · Zbl 1409.78008 |
| [43] | Y. Zhang, K. Wang, and R. Guo, Sixth-order finite difference scheme for the Helmholtz equa-tion with inhomogeneous Robin boundary condition. Adv. Differ. Equ. 362 (2019), 1-15. · Zbl 1459.65211 |
| [44] | S. Zhao, High order matched interface and boundary methods for the Helmholtz equation in media with arbitrarily curved interfaces. J. Comput. Phys. 229 (2010), 3155-3170. · Zbl 1187.78044 |
| [45] | S. Zhao and G. W. Wei, Matched interface and boundary (MIB) for the implementation of boundary conditions in high-order central finite differences. Int. J. Numer. Methods Eng. 77 (2009), no. 12, 1690-1730. · Zbl 1161.65080 |
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.
