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Convolution Quadrature for Wave Simulations

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Book cover Numerical Simulation in Physics and Engineering

Part of the book series: SEMA SIMAI Springer Series ((SEMA SIMAI,volume 9))

Abstract

These notes develop the algorithmic aspects of convolution equations and their discretization by Convolution Quadrature, with an emphasis on the convolution equations that occur in the boundary integral equation formulation of wave scattering problems. The authors explore the development of CQ from a number of different perspectives. Clear algorithms for implementation of CQ are presented. A final example brings together the entire course to demonstrate the full discretization of a time domain boundary integral equation using Convolution Quadrature in time and a simple to program Nyström flavored method in space.

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References

  1. Bamberger, A., Duong, T.H.: Formulation variationnelle espace-temps pour le calcul par potentiel retardé de la diffraction d’une onde acoustique. I. Math. Methods Appl. Sci. 8 (3), 405–435 (1986)

    MATH  Google Scholar 

  2. Bamberger, A., Duong, T.H.: Formulation variationnelle pour le calcul de la diffraction d’une onde acoustique par une surface rigide. Math. Methods Appl. Sci. 8 (4), 598–608 (1986)

    MathSciNet  MATH  Google Scholar 

  3. Banjai, L.: Multistep and multistage convolution quadrature for the wave equation: algorithms and experiments. SIAM J. Sci. Comput. 32 (5), 2964–2994 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Banjai, L., Lubich, C.: An error analysis of Runge-Kutta convolution quadrature. BIT 51 (3), 483–496 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Banjai, L., Sauter, S.: Rapid solution of the wave equation in unbounded domains. SIAM J. Numer. Anal. 47 (1), 227–249 (2008/2009)

    Google Scholar 

  6. Banjai, L., Schanz, M.: Wave propagation problems treated with convolution quadrature and BEM. In: Fast Boundary Element Methods in Engineering and Industrial Applications. Lecture Notes in Applied and Computational Mechanics, vol. 63, pp. 145–184. Springer, Heidelberg (2012)

    Google Scholar 

  7. Banjai, L., Lubich, C., Melenk, J.M.: Runge-Kutta convolution quadrature for operators arising in wave propagation. Numer. Math. 119 (1), 1–20 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Banjai, L., Messner, M., Schanz, M.: Runge-Kutta convolution quadrature for the boundary element method. Comput. Methods Appl. Mech. Eng. 245/246, 90–101 (2012)

    Google Scholar 

  9. Banjai, L., Laliena, A.R., Sayas, F.J.: Fully discrete Kirchhoff formulas with CQ-BEM. IMA J. Numer. Anal. 35 (2), 859–884 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Betcke, T., Salles, N., Smigaj, W.: Exponentially accurate evaluation of time-stepping methods for the wave equation via convolution quadrature type schemes (2016). arXiv:1603.01761

    Google Scholar 

  11. Calvo, M.P., Cuesta, E., Palencia, C.: Runge-Kutta convolution quadrature methods for well-posed equations with memory. Numer. Math. 107 (4), 589–614 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dautray, R., Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 3. Springer, Berlin (1990)

    Book  MATH  Google Scholar 

  13. Dautray, R., Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 5. Springer, Berlin (1992)

    MATH  Google Scholar 

  14. Domínguez, V., Sayas, F.J.: Some properties of layer potentials and boundary integral operators for the wave equation. J. Integral Equ. Appl. 25 (2), 253–294 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Domínguez, V., Lu, S., Sayas, F.J.: A fully discrete Calderón calculus for two dimensional time harmonic waves. Int. J. Numer. Anal. Model. 11 (2), 332–345 (2014)

    MathSciNet  MATH  Google Scholar 

  16. Hackbusch, W., Kress, W., Sauter, S.A.: Sparse convolution quadrature for time domain boundary integral formulations of the wave equation. SIAM J. Numer. Anal. 29 (1), 158–179 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations. II. Springer Series in Computational Mathematics, vol. 14. Springer, Berlin (2010)

    Google Scholar 

  18. Hairer, E., Lubich, C., Schlichte, M.: Fast numerical solution of nonlinear Volterra convolution equations. SIAM J. Sci. Stat. Comput. 6 (3), 532–541 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  19. Henrici, P.: Applied and computational complex analysis, vol. 1. Wiley Classics Library. Wiley, New York (1988)

    MATH  Google Scholar 

  20. Laliena, A.R.: Theoretical and algorithmic aspects of the convolution quadrature method applied to scattering of acoustic waves. Ph.D. thesis, Universidad de Zaragoza (2011)

    Google Scholar 

  21. Laliena, A.R., Sayas, F.J.: A distributional version of Kirchhoff’s formula. J. Math. Anal. Appl. 359 (1), 197–208 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Laliena, A.R., Sayas, F.J.: Theoretical aspects of the application of convolution quadrature to scattering of acoustic waves. Numer. Math. 112 (4), 637–678 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lubich, C.: Convolution quadrature and discretized operational calculus. I. Numer. Math. 52 (2), 129–145 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lubich, C.: Convolution quadrature and discretized operational calculus. II. Numer. Math. 52 (4), 413–425 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lubich, C.: On the multistep time discretization of linear initial-boundary value problems and their boundary integral equations. Numer. Math. 67 (3), 365–389 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lubich, C., Ostermann, A.: Runge-Kutta methods for parabolic equations and convolution quadrature. Math. Comput. 60 (201), 105–131 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lubich, C., Schneider, R.: Time discretization of parabolic boundary integral equations. Numer. Math. 63 (4), 455–481 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  28. McLean, W.: Strongly elliptic systems and boundary integral equations. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  29. Sayas, F.: Retarded Potentials and Time Domain Boundary Integral Equations: A Road-Map. Springer Series in Computational Mathematics, vol. 50. Springer, New York (2016). http://math.udel.edu/~fjsayas/TDBIEclassnotes2012.pdf (2013)

    Google Scholar 

  30. Sayas, F.J.: Energy estimates for Galerkin semidiscretizations of time domain boundary integral equations. Numer. Math. 124 (1), 121–149 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. Schanz, M.: Wave Propagation in Viscoelastic and Poroelastic Continua: A Boundary Element Approach. Springer, Berlin/New York (2001)

    Book  MATH  Google Scholar 

  32. Schwartz, L.: Méthodes mathématiques pour les sciences physiques. Enseignement des Sciences. Hermann, Paris (1961)

    MATH  Google Scholar 

  33. Trèves, F.: Topological vector spaces, distributions and kernels. Academic, New York/London (1967)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Delaware, Newark, DE, USA

    Matthew Hassell & Francisco-Javier Sayas

Authors
  1. Matthew Hassell
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  2. Francisco-Javier Sayas
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Corresponding author

Correspondence to Matthew Hassell .

Editor information

Editors and Affiliations

  1. Ingeniería Matemática e Informática, Universidad Pública de Navarra, Pamplona, Spain

    Inmaculada Higueras

  2. Ingeniería Matemática e Informática, Universidad Pública de Navarra, Pamplona, Spain

    Teo Roldán

  3. Ingeniería Matemática e Informática, Universidad Pública de Navarra, Pamplona, Spain

    Juan José Torrens

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Hassell, M., Sayas, FJ. (2016). Convolution Quadrature for Wave Simulations. In: Higueras, I., Roldán, T., Torrens, J. (eds) Numerical Simulation in Physics and Engineering. SEMA SIMAI Springer Series, vol 9. Springer, Cham. https://doi.org/10.1007/978-3-319-32146-2_2

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