Journal of Scientific Computing

, Volume 61, Issue 1, pp 90–118 | Cite as

Acoustic Wave Propagation in Complicated Geometries and Heterogeneous Media

  • Kristoffer VirtaEmail author
  • Ken Mattsson


We construct finite difference discretizations of the acoustic wave equation in complicated geometries and heterogeneous media. Particular emphasis is placed on the accurate treatment of interfaces at which the underlying media parameters have jump discontinuities. Discontinuous media is treated by subdividing the domain into blocks with continuous media. The equation on each block is then discretized with finite difference operators satisfying a summation-by-parts property and patched together via the simultaneous approximation term method. The energy method is used to estimate a semi-norm of the numerical solution in terms of data, showing that the discretization is stable. Numerical experiments in two and three spatial dimensions verifies the accuracy and stability properties of the schemes.


Acoustic wave equation Curvilinear grids High order methods  Strong stability Energy estimates SBP–SAT 


  1. 1.
    Appelö, D., Petersson, N.A.: A fourth-order accurate embedded boundary method for the wave equation. SIAM J. Sci. Comput. 34(6), 2982–3008 (2012)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Browning, G., Kreiss, H.-O., Oliger, J.: Mesh refinement. Math. Comput. 27, 29–39 (1973)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Carpenter, M., Gottlieb, D., Abarbanel, S.: Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: methodology and application to high-order compact schemes. J. Comput. Phys. 111, 220–236 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Gustafsson, B., Kreiss, H.-O., Oliger, J.: Time Dependent Problems and Difference Methods. Wiley, London (1995)zbMATHGoogle Scholar
  5. 5.
    Henshaw, W.D.: A high-order accurate parallel solver for Maxwellś equations on overlapping grids. SIAM J. Sci. Comput. 5(28), 1730–1765 (2006)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Jensen, F.B., Ferla, C.M.: Numerical solutions of range-dependent benchmark problems in ocean acoustics. J. Acoust. Soc. Am. 87, 1499–1510 (1990)CrossRefGoogle Scholar
  7. 7.
    Jensen, F.B., Kuperman, W.A., Porter, M.B., Schmidt, H.: Computational Ocean Acoustics. Springer, New York (2000)Google Scholar
  8. 8.
    Knupp, P., Steinberg, S.: Fundamentals of Grid Generation. CRC Press, Boca Raton, FL (1993)Google Scholar
  9. 9.
    Kreiss, H.-O., Oliger, J.: Comparison of accurate methods for the integration of hyperbolic equations. Tellus 24, 199–215 (1972)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Kreiss, H.-O., Petersson, N.A.: An embedded boundary method for the wave equation with discontinuous coefficients. SIAM J. Sci. Comput. 28(6), 2054–2074 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Kreiss, H.-O., Petersson, N.A.: A second order accurate embedded boundary method for the wave equation with Dirichlet data. SIAM J. Sci. Comput. 27(4), 1141–1167 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Kreiss, H.-O., Petersson, N.A., Ystrom, J.: Difference approximations of the Neumann problem for the second order wave equation. SIAM J. Numer. Anal. 42(3), 1292–1323 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Kreiss, H.-O., Petersson, N.A.: Boundary estimates for the elastic wave equation in almost incompressible materials. SIAM J. Numer. Anal. 50(3), 1556–1580 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Mattsson, K., Svärd, M., Carpenter, M., Nordström, J.: High order accurate computations for unsteady aeordynamics. Comput. Fluids 36(3), 636–649 (2007)CrossRefzbMATHGoogle Scholar
  15. 15.
    Mattsson, K.: Summation by parts operators for finite difference approximations of second-derivatives with variable coefficients. J. Sci. Comput. 51(3), 650–682 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Mattsson, K., Carpenter, M.: Stable and accurate interpolation operators for high-order multiblock finite difference methods. SIAM J. Sci. Comput. 32(4), 2298–2320 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Mattsson, K., Ham, F., Iccarino, G.: Stable boundary treatment for the wave equation on second order form. J. Sci. Comput. 41(3), 366–383 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Mattsson, K., Ham, F., Iaccarino, G.: Stable and accurate wave propagation in discontinuous media. J. Comput. Phys. 227(19), 8753–8767 (2008)Google Scholar
  19. 19.
    Mattsson, K., Nordström, J.: Summation by parts operators for finite difference approximations of second derivatives. J. Comput. Phys. 199(2), 503–540 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Nissen, A., Kreiss, G., Gerritsen, M.: High order stable finite difference methods for the Schrödinger equation. J. Sci. Comput. 55(1), 173–199 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Nordström, J., Carpenter, M.H.: High-order finite difference methods, multidimensional linear problems and curvilinear coordinates. J. Comput. Phys. 173(26), 149–174 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Nordström, J., Gong, J., Wiede, E., Svärd, M.: A stable and high order multi-block method for the compressible Navier–Stokes equations. J. Comput. Phys. 228, 9020–9035 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Petersson, N.A., Sjögreen, N.: Stable grid refinement and singular source discretization for seismic wave simulations. Commun. Comput. Phys. 8(5), 1074–1110 (2010)Google Scholar
  24. 24.
    Strand, B.: Summation by parts for finite difference approximations for d/dx. J. Comput. Phys. 110(1), 47–67 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Sturm, F.: Investigation of 3-D benchmark problems in underwater acoustics: a uniform approach. In: Proceedings of the 9th European Conference on Underwater Acoustics vol. 2, pp. 759–764 (2008)Google Scholar
  26. 26.
    Sturm, F., Ivansson, S., Jiang, Y., Chapman, N.R.: Numerical investigation of out-of-plane sound propagation in a shallow water experiment. J. Acoust. Soc. Am. 124(6), 341–346 (2008)CrossRefGoogle Scholar
  27. 27.
    Virta, K., Duru, K.: High Order Finite Difference Schemes for the Elastic Wave Equation in Discontinuous Media. Available as preprint arXiv:1309.5768 (2013)Google Scholar
  28. 28.
    Waldén, J.: On the approximation of singular source terms in differential equations. Numer. Methods Partial Differ. Equ. 15(4), 503–520 (1999)CrossRefzbMATHGoogle Scholar
  29. 29.
    Whitham, G.B.: Linear and Nonlinear Waves. Wiley, London (1974)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Division of Scientific Computing, Department of Information TechnologyUppsala UniversityUppsalaSweden

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