Nonreflecting boundary condition for time-dependent multiple scattering

An exact nonreflecting boundary condition (NBC) is derived for the numerical solution of time-dependent multiple scattering problems in three space dimensions, where the scatterer consists of several disjoint components. Because each sub-scatterer can be enclosed by a separate artificial boundary, the computational effort is greatly reduced and becomes independent of the relative distances between the different sub-domains. In fact, the computational work due to the NBC only requires a fraction of the computational work inside @W, due to any standard finite difference or finite element method, independently of the mesh size or the desired overall accuracy. Therefore, the overall numerical scheme retains the rate of convergence of the interior scheme without increasing the complexity of the total computational work. Moreover, the extra storage required depends only on the geometry and not on the final time. Numerical examples show that the NBC for multiple scattering is as accurate as the NBC for a single convex artificial boundary [M.J. Grote, J.B. Keller, Nonreflecting boundary conditions for time-dependent scattering, J. Comput. Phys. 127(1) (1996), 52-65], while being more efficient due to the reduced size of the computational domain.

[1]  T. Ha-Duong,et al.  A Galerkin BEM for transient acoustic scattering by an absorbing obstacle , 2003 .

[2]  Peter Monk,et al.  The Perfectly Matched Layer in Curvilinear Coordinates , 1998, SIAM J. Sci. Comput..

[3]  Christian Lubich,et al.  Fast Convolution for Nonreflecting Boundary Conditions , 2002, SIAM J. Sci. Comput..

[4]  Wave-Motions with Discontinuities at Wave-Fronts , 1904 .

[5]  Thomas Hagstrom,et al.  A new auxiliary variable formulation of high-order local radiation boundary conditions: corner compatibility conditions and extensions to first-order systems , 2004 .

[6]  D. Givoli,et al.  High-order non-reflecting boundary scheme for time-dependent waves , 2003 .

[7]  E. T. Copson,et al.  The mathematical theory of Huygens' principle , 1939 .

[8]  Calvin H. Wilcox,et al.  A generalization of theorems of Rellich and Atkinson , 1956 .

[9]  Lehel Banjai,et al.  Rapid Solution of the Wave Equation in Unbounded Domains , 2008, SIAM J. Numer. Anal..

[10]  Marcus J. Grote,et al.  Local nonreflecting boundary condition for Maxwell's equations , 2006 .

[11]  Martin J. Mohlenkamp.  A fast transform for spherical harmonics , 1997 .

[12]  E. Tadmor,et al.  Hyperbolic Problems: Theory, Numerics, Applications , 2003 .

[13]  E. Hobson.  The Theory of Spherical and Ellipsoidal Harmonics , 1955 .

[14]  Lonny L. Thompson,et al.  Accurate radiation boundary conditions for the time‐dependent wave equation on unbounded domains , 2000 .

[15]  L. Ting,et al.  Exact boundary conditions for scattering problems , 1986 .

[16]  Marcus J. Grote,et al.  Exact Nonreflecting Boundary Condition For Elastic Waves , 2000, SIAM J. Appl. Math..

[17]  Marcus J. Grote,et al.  Exact Nonreflecting Boundary Conditions for the Time Dependent Wave Equation , 1995, SIAM J. Appl. Math..

[18]  R. J. Astley,et al.  Transient spheroidal elements for unbounded wave problems , 1998 .

[19]  D. Givoli.  High-order local non-reflecting boundary conditions: a review☆ , 2004 .

[20]  Thomas Hagstrom,et al.  A formulation of asymptotic and exact boundary conditions using local operators , 1998 .

[21]  T. Hagstrom.  Radiation boundary conditions for the numerical simulation of waves , 1999, Acta Numerica.

[22]  Marcus J. Grote,et al.  On nonreflecting boundary conditions , 1995 .

[23]  Jean-Pierre Berenger,et al.  A perfectly matched layer for the absorption of electromagnetic waves , 1994 .

[24]  Marcus J. Grote,et al.  Nonreflecting Boundary Conditions for Maxwell's Equations , 1998 .

[25]  S. Tsynkov.  Numerical solution of problems on unbounded domains. a review , 1998 .

[26]  Marcus J. Grote,et al.  Dirichlet-to-Neumann boundary conditions for multiple scattering problems , 2004 .

[27]  Marcus J. Grote,et al.  Far-field Evaluation via Nonreflecting Boundary Conditions , 2003 .

[28]  E. Michielssen,et al.  Fast Evaluation of Three-Dimensional Transient Wave Fields Using Diagonal Translation Operators , 1998 .

[29]  Ivan Sofronov,et al.  Artificial boundary conditions of absolute transparency for two- and three-dimensional external time-dependent scattering problems , 1998, European Journal of Applied Mathematics.

[30]  Zhen-huan Teng,et al.  Exact boundary condition for time-dependent wave equation based on boundary integral , 2003 .

[31]  Robert L. Higdon,et al.  Numerical absorbing boundary conditions for the wave equation , 1987 .

[32]  J. Desanto.  Mathematical and numerical aspects of wave propagation , 1998 .

[33]  Hiroshi Akima,et al.  A New Method of Interpolation and Smooth Curve Fitting Based on Local Procedures , 1970, JACM.

[34]  D. Givoli,et al.  Nonreflecting boundary conditions based on Kirchhoff-type formulae , 1995 .

[35]  P. A. Martin.  Multiple Scattering: an Invitation , 1995 .

[36]  Patrick Joly,et al.  Stability of perfectly matched layers, group velocities and anisotropic waves , 2003 .

[37]  M. Grote.  Nonreflecting Boundary Conditions for Elastodynamic Scattering , 2000 .

[38]  Leslie Greengard,et al.  Rapid Evaluation of Nonreflecting Boundary Kernels for Time-Domain Wave Propagation , 2000, SIAM J. Numer. Anal..

[39]  A. Bayliss,et al.  Radiation boundary conditions for wave-like equations , 1980 .

[40]  D. Givoli.  Numerical Methods for Problems in Infinite Domains , 1992 .

[41]  J. Keller,et al.  Non-reflecting boundary conditions for elastic waves , 1990 .

[42]  A. Majda,et al.  Absorbing boundary conditions for the numerical simulation of waves , 1977 .

[43]  Tao Tang,et al.  A FAST NUMERICAL METHOD FOR INTEGRAL EQUATIONS OF THE FIRST KIND WITH LOGARITHMIC KERNEL USING MESH GRADING , 2004 .

[44]  Marcus J. Grote,et al.  Nonreflecting Boundary Conditions for Time-Dependent Scattering , 1996 .