Sparse convolution quadrature for time domain boundary integral formulations of the wave equation

Many important physical applications are governed by the wave equation. The formulation as time domain boundary integral equations involves retarded potentials. For the numerical solution of this problem, we employ the convolution quadrature method for the discretization in time and the Galerkin boundary element method for the space discretization. We introduce a simple a priori cut-off strategy where small entries of the system matrices are replaced by zero. The threshold for the cut-off is determined by an a priori analysis which will be developed in this paper. This analysis will also allow to estimate the effect of additional perturbations such as panel clustering and numerical integration on the overall discretization error. This method reduces the storage complexity for time domain integral equations from O(M 2 N) to O(M 2 N 1/2 log M), where N denotes the number of time steps and M is the dimension of the boundary element space.

[1]  Martin Costabel,et al.  Developments in Boundary Element Methods for Time-Dependent Problems , 1994 .

[2]  C. Lubich,et al.  On the multistep time discretization of linear\newline initial-boundary value problems and their boundary integral equations , 1994 .

[3]  Penny J. Davies,et al.  Numerical stability and convergence of approximations of retarded potential integral equations , 1994 .

[4]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[5]  M. B. Friedman,et al.  Diffraction of Pulses by Cylindrical Obstacles of Arbitrary Cross Section , 1962 .

[6]  Rolf Schneider,et al.  Time discretization of parabolic boundary integral equations , 1992 .

[7]  B. P. Rynne,et al.  Stability of Time Marching Algorithms for the Electric Field Integral Equation , 1990 .

[8]  A. Forestier,et al.  A Galerkin scheme for the time domain integral equation of acoustic scattering from a hard surface , 1989 .

[9]  Wolfgang Dahmen,et al.  Inverse inequalities on non-quasi-uniform meshes and application to the mortar element method , 2003, Math. Comput..

[10]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[11]  Björn Birgisson,et al.  Elastodynamic direct boundary element methods with enhanced numerical stability properties , 1999 .

[12]  Joseph Lipka,et al.  A Table of Integrals , 2010 .

[13]  Stefan A. Sauter,et al.  Numerical Treatment of Retarded Boundary Integral Equations by Sparse Panel Clustering (extended version) , 2006 .

[14]  P. Henrici.  Fast Fourier Methods in Computational Complex Analysis , 1979 .

[15]  Ergin,et al.  Fast analysis of transient acoustic wave scattering from rigid bodies using the multilevel plane wave time domain algorithm , 2000, The Journal of the Acoustical Society of America.

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

[17]  Simon P. Walker,et al.  ANALYSIS OF THREE-DIMENSIONAL TRANSIENT ACOUSTIC WAVE PROPAGATION USING THE BOUNDARY INTEGRAL EQUATION METHOD , 1996 .

[18]  Dugald B. Duncan,et al.  Averaging techniques for time-marching schemes for retarded potential integral equations , 1997 .

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

[20]  Ernst Hairer,et al.  FAST NUMERICAL SOLUTION OF NONLINEAR VOLTERRA CONVOLUTION EQUATIONS , 1985 .

[21]  Wolfgang Hackbusch,et al.  Sparse Convolution Quadrature for Time Domain Boundary Integral Formulations of the Wave Equation by Cutoff and Panel-Clustering , 2007 .

[22]  Dugald B. Duncan,et al.  Stability and Convergence of Collocation Schemes for Retarded Potential Integral Equations , 2004, SIAM J. Numer. Anal..

[23]  W. Hackbusch.  Elliptic Differential Equations , 1992 .

[24]  C. Lubich.  Convolution quadrature and discretized operational calculus. II , 1988 .

[25]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[26]  W. McLean.  Strongly Elliptic Systems and Boundary Integral Equations , 2000 .