Numerical solution of a two-dimensional nonlocal wave equation on unbounded domains. (English) Zbl 1392.82036
Summary: We are concerned with the numerical solution of a nonlocal wave equation in an infinite two-dimensional space. The contribution of this paper is the derivation of an absorbing boundary condition which allows the wave field defined on the finite computational domain to retain the same feature as that defined on the original infinite domain. We resort to the idea of a first-kind integral equation method and develop a solution formulation in terms of a potential summation on a surrounding ghost region. This new formulation can be taken as an absorbing boundary condition of generalized Dirichlet-to-Dirichlet type. The accuracy and effectiveness of our approach are illustrated by some numerical examples.
MSC:
| 82C21 | Dynamic continuum models (systems of particles, etc.) in time-dependent statistical mechanics |
| 65R20 | Numerical methods for integral equations |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 46N20 | Applications of functional analysis to differential and integral equations |
| 45A05 | Linear integral equations |
Keywords:
nonlocal wave equation; nonlocal models; discrete absorbing boundary condition; artificial boundary method; Green’s function; Dirichlet-to-Dirichlet mappingReferences:
| [1] | O. Weckner and R. Abeyaratne, {\it The effect of long-range forces on the dynamics of a bar}, J. Mech. Phys. Solids, 53 (2005), pp. 705-728. · Zbl 1122.74431 |
| [2] | B. Alpert, L. Greengard, and T. Hagstrom, {\it Rapid evaluation of nonreflecting boundary kernels for time-domain wave propagation}, SIAM J. Numer. Anal., 37 (2000), pp. 1138-1164. · Zbl 0963.65104 |
| [3] | A. Arnold, M. Ehrhardt, and I. Sofronov, {\it Approximation, stability and fast calculation of nonlocal boundary conditions for the Schrödinger equation}, Commun. Math. Sci., 1 (2003), pp. 501-556. · Zbl 1085.65513 |
| [4] | F. Bobaru and M. Duangpanya, {\it The peridynamic formulation for transient heat conduction}, Int. J. Heat Mass Transf., 53 (2010), pp. 4047-4059. · Zbl 1194.80010 |
| [5] | X. Chen and M. Gunzburger, {\it Continuous and discontinuous finite element methods for a peridynamics model of mechanics}, Comput. Methods Appl. Mech. Engrg., 200 (2011), pp. 1237-1250. · Zbl 1225.74082 |
| [6] | Q. Du, M. Gunzburger, R.B. Lehoucq, and K. Zhou, {\it Analysis and approximation of nonlocal diffusion problems with volume constraints}, SIAM Rev., 54 (2012), pp. 667-696. · Zbl 1422.76168 |
| [7] | Q. Du, M. Gunzburger, R.B. Lehoucq, and K. Zhou, {\it A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws}, Math. Models Method Appl. Sci., 23 (2013), pp. 493-540. · Zbl 1266.26020 |
| [8] | Q. Du, Y. Tao, X. Tian, and J. Yang, {\it Asymptotically compatible discretization of multidimensional nonlocal diffusion models and nonlocal gradient recovery}, IMA J. Numer. Amal., to appear. · Zbl 1466.65222 |
| [9] | Q. Du, J. Zhang, and C. Zheng, {\it Nonlocal wave propagation in unbounded multi-scale mediums}, Commun. Comput. Phys., to appear. · Zbl 1475.65156 |
| [10] | E. Emmrich and O. Weckner, {\it Analysis and numerical approximation of an integro-differential equation modeling non-local effects in linear elasticity}, Math. Mech. Solids, 12 (2007), pp. 363-384. · Zbl 1175.74013 |
| [11] | D. Givoli, {\it High-order local non-reflecting boundary conditions: A review}, Wave Motion 39 (2004), pp. 319-326. · Zbl 1163.74356 |
| [12] | I. S. Gradshteyn and I. M. Ryzhik, {\it Tables of Integrals, Series, and Products}, 6th ed., Academic Press, San Diego, CA, 2000, p. 1090. · Zbl 0981.65001 |
| [13] | M. J. Grote and J. B. Keller, {\it Exact nonreflecting boundary conditions for the time dependent wave equation}, SIAM J. Appl. Math., 55 (1995), pp. 280-297. · Zbl 0817.35049 |
| [14] | T. Hagstrom, {\it New results on absorbing layers and radiation boundary conditions, in Topics in Computational Wave Propagation,} M. Ainsworth et al., eds., Springer, New York, 2003, pp. 1-42. · Zbl 1059.78040 |
| [15] | T. Hagstrom, A. Mar-Or, and D. Givoli, {\it High-order local absorbing conditions for the wave equation: Extensions and improvements}, J. Comput. Phys., 227 (2008), pp. 3322-3357. · Zbl 1136.65081 |
| [16] | H. Han and Z. Huang, {\it A class of artificial boundary conditions for heat equation in unbounded domains,} Comput. Math. Appl., 43 (2002), pp. 889-900. · Zbl 0999.65086 |
| [17] | H. Han and X. Wu, {\it Artificial Boundary Method}, Springer, Heidelberg, 2013. · Zbl 1277.65105 |
| [18] | H. D. Han and C. Zheng, {\it Exact nonreflecting boundary conditions for an acoustic problem in three dimensions}, J. Comput. Math., 21 (2003), pp. 15-24. · Zbl 1027.76048 |
| [19] | S. G. Krantz, {\it Handbook of Complex Variables}, Birkhäuser, Boston, MA, 1999. · Zbl 0946.30001 |
| [20] | R. W. Macek and S. Silling, {\it Peridynamics via finite element analysis}, Finite Elem. Anal. Des., 43 (2007), pp. 1169-1178. |
| [21] | G. Pang, Y. Yang, and S. Tang, {\it Exact boundary condition for semi-discretized Schrödinger equation and heat equation in a rectangular domain}, J. Sci. Comput., 72 (2017), pp. 1-13. · Zbl 1371.65099 |
| [22] | S. Silling, {\it Reformulation of elasticity theory for discontinuities and long-range forces}, J. Mech. Phys. Solids, 48 (2000), pp. 175-209. · Zbl 0970.74030 |
| [23] | Z. H. Teng, {\it Exact boundary condition for time-dependent wave equation based on boundary integral}, J. Comput. Phys., 190 (2003), pp. 398-418. · Zbl 1031.65098 |
| [24] | X. Tian and Q. Du, {\it Analysis and comparison of different approximations to nonlocal diffusion and linear peridynamic equations}, SIAM J. Numer. Anal., 51 (2013), pp. 3458-3482. · Zbl 1295.82021 |
| [25] | X. Tian and Q. Du, {\it Asymptotically compatible schemes and applications to robust discretization of nonlocal models}, SIAM J. Numer. Anal., 52 (2014), pp. 1641-1665. · Zbl 1303.65098 |
| [26] | H. Tian, L. Ju, and Q. Du, {\it A conservative nonlocal convection-diffusion model and asymptotically compatible finite difference discretization}, Comput. Methods Appl. Mech. Engrg., 320 (2017), pp. 46-67. · Zbl 1439.65134 |
| [27] | L. Ting and M. J. Miksis, {\it Exact boundary conditions for scattering problems}, J. Acoust. Soc. Amer., 80 (1986), 1825-1827. |
| [28] | S. V. Tsynkov, {\it Numerical solution of problems on unbounded domains: A review}, Appl. Numer. Math., 27 (1998), pp. 465-532. · Zbl 0939.76077 |
| [29] | L. Verlet, {\it Computer experiments on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules}, Phys. Rev. (2), 159 (1967), pp. 98-103. |
| [30] | V. Villamizar, S. Acosta, and B. Dastrup, {\it High order local absorbing boundary conditions for acoustic waves in terms of farfield expansions,} J. Comput. Phys., 333 (2017), pp. 331-351. · Zbl 1375.35290 |
| [31] | X. Wang and S. Tang, {\it Matching boundary conditions for lattice dynamics,} Internat. J. Numer Methods Engrg., 93 (2013), pp. 1255-1285. · Zbl 1352.74015 |
| [32] | R. A. Wildman and G. A. Gazonas, {\it A perfectly matched layer for peridynamics in two dimensions}, J. Mech. Mater. Struct., 7 (2012), pp. 765-781. |
| [33] | X. Wu and J. Zhang, {\it High-order local absorbing boundary conditions for heat equation in unbounded domains}, J. Comput. Math., 29 (2011), pp. 74-90. · Zbl 1249.65178 |
| [34] | W. Zhang, J. Yang, J. Zhang, and Q. Du, {\it Absorbing boundary conditions for nonlocal heat equations on unbounded domain}, Commun. Comput. Phys., 21 (2017), pp. 16-39. · Zbl 1373.45011 |
| [35] | C. Zheng, J. Hu, Q. Du, and J. Zhang, {\it Numerical Solution of the Nonlocal Diffusion Equation on the Real Line}, SIAM J. Sci. Comput., 39 (2017), pp. A1951-A1968. · Zbl 1376.82053 |
| [36] | K. Zhou and Q. Du, {\it Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions}, SIAM J. Numer. Anal., 48 (2010), pp. 1759-1780. · Zbl 1220.82074 |
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