Britt, S.; Tsynkov, S.; Turkel, E. Numerical solution of the wave equation with variable wave speed on nonconforming domains by high-order difference potentials. (English) Zbl 1380.65146 J. Comput. Phys. 354, 26-42 (2018). Summary: We solve the wave equation with variable wave speed on nonconforming domains with fourth order accuracy in both space and time. This is accomplished using an implicit finite difference (FD) scheme for the wave equation and solving an elliptic (modified Helmholtz) equation at each time step with fourth order spatial accuracy by the method of difference potentials (MDP). High-order MDP utilizes compact FD schemes on regular structured grids to efficiently solve problems on nonconforming domains while maintaining the design convergence rate of the underlying FD scheme. Asymptotically, the computational complexity of high-order MDP scales the same as that for FD. Cited in 17 Documents MSC: 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35L05 Wave equation Keywords:method of difference potentials; compact finite difference scheme; implicit scheme; regular structured grid; nonconforming boundary; high-order accuracy; high-order method of difference potentials PDFBibTeX XMLCite \textit{S. Britt} et al., J. Comput. Phys. 354, 26--42 (2018; Zbl 1380.65146) Full Text: DOI References: [1] Britt, D. S.; Tsynkov, S. V.; Turkel, E., A high-order numerical method for the Helmholtz equation with nonstandard boundary conditions, SIAM J. Sci. Comput., 35, 5, A2255-A2292 (2013) · Zbl 1281.65135 [2] Magura, S.; Petropavlovsky, S.; Tsynkov, S.; Turkel, E., High-order numerical solution of the Helmholtz equation for domains with reentrant corners, Appl. Numer. Math., 118, 87-116 (2017) · Zbl 1367.65155 [3] Bayliss, A.; Goldstein, C. I.; Turkel, E., On accuracy conditions for the numerical computation of waves, J. Comput. Phys., 59, 3, 396-404 (1985) · Zbl 0647.65072 [4] Babuska, Ivo M.; Sauter, Stefan A., Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?, SIAM J. Numer. Anal., 34, 6, 2392-2423 (1997) · Zbl 0894.65050 [5] Ihlenburg, Frank, Finite Element Analysis of Acoustic Scattering, Appl. Math. Sci., vol. 132 (1998), Springer · Zbl 0908.65091 [6] Kreiss, Heinz-Otto; Oliger, Joseph, Comparison of accurate methods for the integration of hyperbolic equations, Tellus, 24, 199-215 (1972) [7] Tam, Christopher K. W.; Webb, Jay C., Dispersion-relation-preserving finite difference schemes for computational acoustics, J. Comput. Phys., 107, 2, 262-281 (1993) · Zbl 0790.76057 [8] Tam, Christopher K. W.; Webb, Jay C., Radiation boundary condition and anisotropy correction for finite difference solutions of the Helmholtz equation, J. Comput. Phys., 113, 1, 122-133 (1994) · Zbl 0810.65094 [9] Britt, S.; Tsynkov, S.; Turkel, E., A compact fourth order scheme for the Helmholtz equation in polar coordinates, J. Sci. Comput., 45, 1-3, 26-47 (2010) · Zbl 1203.65218 [10] Britt, Steven; Tsynkov, Semyon; Turkel, Eli, Numerical simulation of time-harmonic waves in inhomogeneous media using compact high order schemes, Commun. Comput. Phys., 9, 3, 520-541 (March 2011) [11] Turkel, Eli; Gordon, Dan; Gordon, Rachel; Tsynkov, Semyon, Compact 2D and 3D sixth order schemes for the Helmholtz equation with variable wave number, J. Comput. Phys., 232, 1, 272-287 (2013) · Zbl 1291.65273 [12] Lele, S. K., Compact finite difference schemes with spectral like resolution, J. Comput. Phys., 103, 16-42 (1992) · Zbl 0759.65006 [13] Harari, I.; Turkel, E., Accurate finite difference methods for time-harmonic wave propagation, J. Comput. Phys., 119, 2, 252-270 (1995) · Zbl 0848.65072 [14] Singer, I.; Turkel, E., High-order finite difference methods for the Helmholtz equation, Comput. Methods Appl. Mech. Eng., 163, 1-4, 343-358 (1998) · Zbl 0940.65112 [15] Singer, I.; Turkel, E., Sixth-order accurate finite difference schemes for the Helmholtz equation, J. Comput. Acoust., 14, 3, 339-351 (2006) · Zbl 1198.65210 [16] Britt, S.; Tsynkov, S.; Turkel, E., A high order compact time/space finite difference scheme for the wave equation in variable media, J. Sci. Comput. (2017), Submitted for publication [17] Irons, B. M.; Zienkiewicz, O. C., The Isoparametric Finite Element System: A New Concept in Finite Element Analysis (1969), Royal Aeronautical Society: Royal Aeronautical Society London [18] Kleinman, R. E.; Roach, G. F., Boundary integral equations for the three-dimensional Helmholtz equation, SIAM Rev., 16, 214-236 (1974) · Zbl 0253.35023 [19] Biros, George; Ying, Lexing; Zorin, Denis, A fast solver for the Stokes equations with distributed forces in complex geometries, J. Comput. Phys., 193, 1, 317-348 (2004) · Zbl 1047.76065 [20] Jiang, Shidong; Veerapaneni, Shravan; Greengard, Leslie, Integral equation methods for unsteady Stokes flow in two dimensions, SIAM J. Sci. Comput., 34, 4, A2197-A2219 (2012) · Zbl 1254.35179 [21] Preston, Mark D.; Chamberlain, Peter G.; Chandler-Wilde, Simon N., An integral equation method for a boundary value problem arising in unsteady water wave problems, J. Integral Equ. Appl., 20, 1, 121-152 (2008) · Zbl 1134.76007 [22] Tuong, Ha-Duong, On retarded potential boundary integral equations and their discretisation, (Topics in Computational Wave Propagation. Topics in Computational Wave Propagation, Lect. Notes Comput. Sci. Eng., vol. 31 (2003), Springer: Springer Berlin), 301-336 · Zbl 1051.78018 [23] Sayas, Francisco-Javier, Retarded Potentials and Time Domain Boundary Integral EquationsA Road-Map (March 2013), University of Delaware [24] Weile, Daniel S.; Pisharody, G.; Chen, Nan-Wei; Shanker, B.; Michielssen, E., A novel scheme for the solution of the time-domain integral equations of electromagnetics, IEEE Trans. Antennas Propag., 52, 1, 283-295 (Jan 2004) [25] Kobidze, G.; Gao, Jun; Shanker, B.; Michielssen, E., A fast time domain integral equation based scheme for analyzing scattering from dispersive objects, IEEE Trans. Antennas Propag., 53, 3, 1215-1226 (March 2005) [26] Abboud, Toufic; Joly, Patrick; Rodríguez, Jerónimo; Terrasse, Isabelle, Coupling discontinuous Galerkin methods and retarded potentials for transient wave propagation on unbounded domains, J. Comput. Phys., 230, 15, 5877-5907 (2011) · Zbl 1416.74081 [27] Domínguez, Víctor; Sayas, Francisco-Javier, Some properties of layer potentials and boundary integral operators for the wave equation, J. Integral Equ. Appl., 25, 2, 253-294 (2013) · Zbl 1277.35082 [28] Epstein, Charles L.; Greengard, Leslie; Hagstrom, Thomas, On the stability of time-domain integral equations for acoustic wave propagation, Discrete Contin. Dyn. Syst., 36, 8, 4367-4382 (2016) · Zbl 1333.65117 [29] Lubich, C., Convolution quadrature and discretized operational calculus. I, Numer. Math., 52, 2, 129-145 (1988) · Zbl 0637.65016 [30] Lubich, C., Convolution quadrature and discretized operational calculus. II, Numer. Math., 52, 4, 413-425 (1988) · Zbl 0643.65094 [31] Falletta, Silvia; Monegato, Giovanni, An exact non-reflecting boundary condition for 2D time-dependent wave equation problems, Wave Motion, 51, 1, 168-192 (2014) · Zbl 1349.65413 [32] Falletta, Silvia; Monegato, Giovanni, Exact non-reflecting boundary condition for 3D time-dependent multiple scattering-multiple source problems, Wave Motion, 58, 281-302 (2015) · Zbl 1467.35210 [33] Medvinsky, M.; Tsynkov, S.; Turkel, E., The method of difference potentials for the Helmholtz equation using compact high order schemes, J. Sci. Comput., 53, 1, 150-193 (2012) · Zbl 1254.65118 [34] Medvinsky, M.; Tsynkov, S.; Turkel, E., High order numerical simulation of the transmission and scattering of waves using the method of difference potentials, J. Comput. Phys., 243, 305-322 (2013) · Zbl 1349.78097 [35] Britt, S.; Petropavlovsky, S.; Tsynkov, S.; Turkel, E., Computation of singular solutions to the Helmholtz equation with high order accuracy, Appl. Numer. Math., 93, 215-241 (July 2015) [36] Ryaben’kii, V. S., Boundary equations with projections, Russ. Math. Surv., 40, 2, 147-183 (1985) · Zbl 0594.35035 [37] Ryaben’kii, V. S., Method of Difference Potentials and Its Applications, Springer Ser. Comput. Math., vol. 30 (2002), Springer-Verlag: Springer-Verlag Berlin · Zbl 0994.65107 [38] Ryaben’kii, V. S., Difference potentials method and its applications, Math. Nachr., 177, 251-264 (1996) · Zbl 0851.65091 [39] Ryaben’kii, V. S., On the method of difference potentials, J. Sci. Comput., 28, 2-3, 467-478 (2006) · Zbl 1100.76046 [40] Calderon, A. P., Boundary-value problems for elliptic equations, (Proceedings of the Soviet-American Conference on Partial Differential Equations in Novosibirsk (1963), Fizmatgiz: Fizmatgiz Moscow), 303-304 [41] Seeley, R. T., Singular integrals and boundary value problems, Am. J. Math., 88, 781-809 (1966) · Zbl 0178.17601 [42] Petrowsky, I., On the diffusion of waves and the lacunas for hyperbolic equations, Mat. Sb., 17(59), 3, 289-370 (1945) · Zbl 0061.21309 [43] Petropavlovsky, S.; Tsynkov, S.; Turkel, E., An efficient numerical algorithm for the 3D wave equation in domains of complex shape, (Mathematical and Numerical Aspects of Wave Propagation WAVES 2017. Mathematical and Numerical Aspects of Wave Propagation WAVES 2017, The 13th International Conference, Minneapolis, MN, USA, May 15-19, 2017, Minneapolis, MN, USA. Mathematical and Numerical Aspects of Wave Propagation WAVES 2017. Mathematical and Numerical Aspects of Wave Propagation WAVES 2017, The 13th International Conference, Minneapolis, MN, USA, May 15-19, 2017, Minneapolis, MN, USA, Book of Abstracts (2017)), 365-366 [44] Ryaben’kii, V. S.; Tsynkov, S. V.; Turchaninov, V. I., Long-time numerical computation of wave-type solutions driven by moving sources, Appl. Numer. Math., 38, 187-222 (2001) · Zbl 0987.65080 [45] Ryaben’kii, V. S.; Tsynkov, S. V.; Turchaninov, V. I., Global discrete artificial boundary conditions for time-dependent wave propagation, J. Comput. Phys., 174, 2, 712-758 (2001) · Zbl 0991.65100 [46] Tsynkov, S. V., Artificial boundary conditions for the numerical simulation of unsteady acoustic waves, J. Comput. Phys., 189, 2, 626-650 (August 2003) [47] Tsynkov, S. V., On the application of lacunae-based methods to Maxwell’s equations, J. Comput. Phys., 199, 1, 126-149 (September 2004) [48] Qasimov, H.; Tsynkov, S., Lacunae based stabilization of PMLs, J. Comput. Phys., 227, 7322-7345 (2008) · Zbl 1145.65082 [49] Petropavlovsky, S. V.; Tsynkov, S. V., A non-deteriorating algorithm for computational electromagnetism based on quasi-lacunae of Maxwell’s equations, J. Comput. Phys., 231, 2, 558-585 (2012) · Zbl 1241.78013 [50] Petropavlovsky, S. V.; Tsynkov, S. V., Quasi-lacunae of Maxwell’s equations, SIAM J. Appl. Math., 71, 4, 1109-1122 (2011) · Zbl 1229.35291 [51] Petropavlovsky, S. V.; Tsynkov, S. V., Non-deteriorating time domain numerical algorithms for Maxwell’s electrodynamics, J. Comput. Phys., 336, 1-35 (May 2017) [52] Soffer, Avy; Stucchio, Chris, Stable and accurate outgoing wave filters for anisotropic and nonlocal waves, (Frontiers of Applied and Computational Mathematics (2008), World Sci. Publ.: World Sci. Publ. Hackensack, NJ), 240-247 · Zbl 1176.65116 [53] Epshteyn, Yekaterina, Algorithms composition approach based on difference potentials method for parabolic problems, Commun. Math. Sci., 12, 4, 723-755 (2014) · Zbl 1305.65184 [54] Albright, Jason; Epshteyn, Yekaterina; Steffen, Kyle R., High-order accurate difference potentials methods for parabolic problems, Appl. Numer. Math., 93, 87-106 (2015) · Zbl 1326.65103 [55] Medvinsky, M.; Tsynkov, S.; Turkel, E., Solving the Helmholtz equation for general smooth geometry using simple grids, Wave Motion, 62, 75-97 (2016) · Zbl 1469.65159 [56] Chabassier, Juliette; Imperiale, Sébastien, Introduction and study of fourth order theta schemes for linear wave equations, J. Comput. Appl. Math., 245, 194-212 (2013) · Zbl 1262.65119 [57] Liang, Hui; Liu, M. Z.; Lv, Wanjin, Stability of \(θ\)-schemes in the numerical solution of a partial differential equation with piecewise continuous arguments, Appl. Math. Lett., 23, 2, 198-206 (2010) · Zbl 1209.65091 [58] Reznik, A. A., Approximation of surface potentials of elliptic operators by difference potentials, Dokl. Akad. Nauk SSSR, 263, 6, 1318-1321 (1982) [59] Reznik, A. A., Approximation of the Surface Potentials of Elliptic Operators by Difference Potentials and Solution of Boundary-Value Problems (1983), Moscow Institute for Physics and Technology: Moscow Institute for Physics and Technology Moscow, USSR, (in Russian) [60] Nikkar, Samira; Nordström, Jan, Fully discrete energy stable high order finite difference methods for hyperbolic problems in deforming domains, J. Comput. Phys., 291, 82-98 (2015) · Zbl 1349.65326 [61] Canuto, Claudio; Quarteroni, Alfio, Variational methods in the theoretical analysis of spectral approximations, (Spectral Methods for Partial Differential Equations. Spectral Methods for Partial Differential Equations, Hampton, Va., 1982 (1984), SIAM: SIAM Philadelphia, PA), 55-78 · Zbl 0539.65080 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. 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