Stable Boundary Treatment for the Wave Equation on Second-Order Form



A stable and accurate boundary treatment is derived for the second-order wave equation. The domain is discretized using narrow-diagonal summation by parts operators and the boundary conditions are imposed using a penalty method, leading to fully explicit time integration. This discretization yields a stable and efficient scheme. The analysis is verified by numerical simulations in one-dimension using high-order finite difference discretizations, and in three-dimensions using an unstructured finite volume discretization.


High-order finite difference methods Unstructured finite volume method Wave equation Numerical stability Second derivatives Boundary conditions Complex geometries 


  1. 1.
    Bamberger, A., Glowinski, R., Tran, Q.H.: A domain decomposition method for the acoustic wave equation with discontinuous coefficients and grid change. SIAM J. Numer. Anal. 34(2), 603–639 (1997) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bayliss, A., Jordan, K.E., Lemesurier, B.J., Turkel, E.: A fourth order accurate finite difference scheme for the computation of elastic waves. Bull. Seismol. Soc. Am. 76(4), 1115–1132 (1986) Google Scholar
  3. 3.
    Calabrese, G.: Finite differencing second order systems describing black holes. Phys. Rev. D 71, 027501 (2005) MathSciNetGoogle Scholar
  4. 4.
    Carpenter, M.H., 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(2), 220–236 (1994) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cohen, G., Joly, P.: Construction and analysis of fourth-order finite difference schemes for the acoustic wave equation in nonhomogeneous media. SIAM J. Numer. Anal. 33(4), 1266–1302 (1996) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Diener, P., Dorband, E.N., Schnetter, E., Tiglio, M.: Optimized high-order derivative and dissipation operators satisfying summation by parts, and applications in three-dimensional multi-block evolutions. J. Sci. Comput. 32(1), 109–145 (2007) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Grote, M., Schneebeli, A., Schötzau, D.: Discontinuous Galerkin finite element method for the wave equation. SIAM J. Numer. Analysis 44, 2408–2431 (2006) MATHCrossRefGoogle Scholar
  8. 8.
    Grote, M., Schneebeli, A., Schötzau, D.: Interior penalty discontinuous Galerkin method for maxwell’s equations: Energy norm error estimates. J. Comput. Appl. Math. 204, 375–386 (2007) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Gustafsson, B., Kreiss, H.O., Sundström, A.: Stability theory of difference approximations for mixed initial boundary value problems. Math. Comput. 26(119), 649–686 (1972) MATHCrossRefGoogle Scholar
  10. 10.
    Hagstrom, T.: Radiation boundary conditions for the numerical simulation of waves. Acta Numer. 8, 47–106 (1999) CrossRefMathSciNetGoogle Scholar
  11. 11.
    Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer, New York (2008) MATHGoogle Scholar
  12. 12.
    Kelly, K.R., Ward, R.W., Treitel, S., Alford, R.M.: Synthetic seismograms: A finite difference approach. Geophysics 41, 2–27 (1976) CrossRefGoogle Scholar
  13. 13.
    Kreiss, H.-O., Petersson, N.A.: An embedded boundary method for the wave equation with discontinuous coefficients. SIAM J. Sci. Comput. 28, 2054–2074 (2006) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    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, 1141–1167 (2006) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kreiss, H.-O., Petersson, N.A., Yström, J.: Difference approximations for the second order wave equation. SIAM J. Numer. Anal. 40, 1940–1967 (2002) CrossRefGoogle Scholar
  16. 16.
    Kreiss, H.-O., Petersson, N.A., Yström, J.: Difference approximations of the Neumann problem for the second order wave equation. SIAM J. Numer. Anal. 42, 1292–1323 (2004) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Kreiss, H.-O., Scherer, G.: Finite element and finite difference methods for hyperbolic partial differential equations. In: Mathematical Aspects of Finite Elements in Partial Differential Equations. Academic Press, San Diego (1974) Google Scholar
  18. 18.
    Kreiss, H.-O., Oliger, J.: Comparison of accurate methods for the integration of hyperbolic equations. Tellus XXIV, 3 (1972) MathSciNetGoogle Scholar
  19. 19.
    Lehner, L., Neilsen, D., Reula, O., Tiglio, M.: The discrete energy method in numerical relativity: towards long-term stability. Class. Quantum Gravity 21, 5819–5848 (2004) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Lehner, L., Reula, O., Tiglio, M.: Multi-block simulations in general relativity: high-order discretizations, numerical stability and applications. Class. Quantum Gravity 22, 5283–5321 (2005) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Lele, S.K.: Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103, 16–42 (1992) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Mattsson, K.: Boundary procedures for summation-by-parts operators. J. Sci. Comput. 18, 133–153 (2003) MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Mattsson, K., Ham, F., Iaccarino, G.: Stable and accurate wave propagation in discontinuous media. J. Comput. Phys. 227, 8753–8767 (2008) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Mattsson, K., Nordström, J.: Summation by parts operators for finite difference approximations of second derivatives. J. Comput. Phys. 199(2), 503–540 (2004) MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Mattsson, K., Nordström, J.: High order finite difference methods for wave propagation in discontinuous media. J. Comput. Phys. 220, 249–269 (2006) MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Mattsson, K., Svärd, M., Carpenter, M.H., Nordström, J.: High-order accurate computations for unsteady aerodynamics. Comput. Fluids 36, 636–649 (2006) CrossRefGoogle Scholar
  27. 27.
    Mattsson, K., Svärd, M., Nordström, J.: Stable and accurate artificial dissipation. J. Sci. Comput. 21(1), 57–79 (2004) MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Mattsson, K., Svärd, M., Shoeybi, M.: Stable and accurate schemes for the compressible Navier-Stokes equations. J. Comput. Phys. 227(4), 2293–2316 (2008) MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Nordström, J., Mattsson, K., Swanson, R.C.: Boundary conditions for a divergence free velocity-pressure formulation of the incompressible Navier-Stokes equations. J. Comput. Phys. 225, 874–890 (2007) MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Nycander, J.: Tidal generation of internal waves from a periodic array of steep ridges. J. Fluid Mech. 567, 415–432 (2006) MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Olsson, P.: Summation by parts, projections, and stability I. Math. Comput. 64, 1035 (1995) MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Olsson, P.: Summation by parts, projections, and stability II. Math. Comput. 64, 1473 (1995) MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Shubin, G.R., Bell, J.B.: A modified equation approach to constructing fourth order methods for acoustic wave propagation. SIAM J. Sci. Stat. Comput. 8(2), 135–151 (1987) MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Strand, B.: Summation by parts for finite difference approximations for d/dx. J. Comput. Phys. 110, 47–67 (1994) MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Svärd, M., Gong, J., Nordström, J.: An accuracy evaluation of unstructured node-centred finite volume methods. Appl. Numer. Math. 58(8), 1142–1158 (2008) MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Svärd, M., Nordström, J.: On the order of accuracy for difference approximations of initial-boundary value problems. J. Comput. Phys. 218, 333–352 (2006) MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Szilagyl, B., Kreiss, H.-O., Winicour, J.W.: Modeling the black hole excision problem. Phys. Rev. D 71, 104035 (2005) MathSciNetGoogle Scholar
  38. 38.
    Tsynkov, S.V.: Numerical solution of problems on unbounded domains: a review. Appl. Numer. Math. 27, 465–532 (1998) MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Virieux, J.: Sh-wave propagation in heterogeneous media: Velocity-stress finite-difference method. Geophysics 49, 1933–1957 (1984) CrossRefGoogle Scholar
  40. 40.
    Virieux, J.: P-sv wave propagation in heterogeneous media: Velocity-stress finite-difference method. Geophysics 51, 889–901 (1986) CrossRefGoogle Scholar
  41. 41.
    Yee, K.S.: Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media. IEEE Trans. Antennas Propag. 14, 302–307 (1966) MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of Information TechnologyUppsala UniversityUppsalaSweden
  2. 2.Mechanical Engineering—Center for Integrated Turbulence SimulationsStanford UniversityStanfordUSA
  3. 3.Mechanical Engineering—Flow Physics and ComputationStanford UniversityStanfordUSA

Personalised recommendations