Convolution-in-Time Approximations of Time Domain Boundary Integral Equations

We present a new temporal approximation scheme for the boundary integral formulation of time-dependent scattering problems which can be combined with either collocation or Galerkin approximation in space. It uses the backward-in-time framework introduced in [P. J. Davies and D. B. Duncan, Convolution Spline Approximations of Volterra Integral Equations, www.mathstat.strath.ac.uk/research/reports/2012 (2012)] with new temporal basis functions which share some properties with radial basis function multiquadrics. We analyze the stability and convergence properties of the new scheme for associated Volterra integral equations and perform extensive numerical tests for scattering from flat polygonal plates and open and closed cubes and spheres, which demonstrate effectiveness of this approach.

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

[2]  Holger Wendland,et al.  Scattered Data Approximation: Conditionally positive definite functions , 2004 .

[3]  Isabelle Terrasse,et al.  Resolution mathematique et numerique des equations de maxwell instationnaires par une methode de potentiels retardes , 1993 .

[4]  Lehel Banjai,et al.  An error analysis of Runge–Kutta convolution quadrature , 2011 .

[5]  T. Ha-Duong,et al.  On Retarded Potential Boundary Integral Equations and their Discretisation , 2003 .

[6]  Lehel Banjai,et al.  Multistep and Multistage Convolution Quadrature for the Wave Equation: Algorithms and Experiments , 2010, SIAM J. Sci. Comput..

[7]  P. Davies,et al.  Convolution spline approximations of Volterra integral equations , 2012 .

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

[9]  Ivan P. Gavrilyuk,et al.  Collocation methods for Volterra integral and related functional equations , 2006, Math. Comput..

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

[11]  Ernst P. Stephan,et al.  The hp-version of the boundary element method for Helmholtz screen problems , 1996, Computing.

[12]  Stefan A. Sauter,et al.  A Galerkin method for retarded boundary integral equations with smooth and compactly supported temporal basis functions , 2013, Numerische Mathematik.

[13]  C. Lubich.  Convolution quadrature and discretized operational calculus. I , 1988 .

[14]  A. Bamberger et T. Ha Duong,et al.  Formulation variationnelle espace‐temps pour le calcul par potentiel retardé de la diffraction d'une onde acoustique (I) , 1986 .

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

[16]  Ronald F. Boisvert,et al.  NIST Handbook of Mathematical Functions , 2010 .

[17]  T. Ha-Duong,et al.  On the transient acoustic scattering by a flat object , 1990 .

[18]  吴宗敏.  SHAPE PRESERVING PROPERTIES AND CONVERGENCE OF UNIVARIATE MULTIQUADRIC QUASI－INTERPOLATION , 1994 .

[19]  Jens Markus Melenk,et al.  Runge–Kutta convolution quadrature for operators arising in wave propagation , 2011, Numerische Mathematik.

[20]  Wolfgang Hackbusch,et al.  Sparse convolution quadrature for time domain boundary integral formulations of the wave equation , 2008 .