Skip to main content
Springer Nature Link
Log in

A High Order Compact Time/Space Finite Difference Scheme for the Wave Equation with Variable Speed of Sound

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

We consider fourth order accurate compact schemes, in both space and time, for the second order wave equation with a variable speed of sound. We demonstrate that usually this is much more efficient than lower order schemes despite being implicit and only conditionally stable. Fast time marching of the implicit scheme is accomplished by iterative methods such as conjugate gradient and multigrid. For conjugate gradient, an upper bound on the convergence rate of the iterations is obtained by eigenvalue analysis of the scheme. The implicit discretization technique is such that the spatial and temporal convergence orders can be adjusted independently of each other. In special cases, the spatial error dominates the problem, and then an unconditionally stable second order accurate scheme in time with fourth order accuracy in space is more efficient. Computations confirm the design convergence rate for the inhomogeneous, variable wave speed equation and also confirm the pollution effect for these time dependent problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. For the (4,4) scheme, \(\theta = \frac{1}{12}\) and thus \(Kh_x^2 = -\frac{1}{\theta CFL^2}<-12\) whenever \(CFL<1\), which is already guaranteed by the stability condition (see Sect. 3).

References

  1. Abdulkadir, Y.A.: Comparison of finite difference schemes for the wave equation based on dispersion. J. Appl. Math. Phys. 3, 1544–1562 (2015)

    Article  Google Scholar 

  2. Agut, C., Diaz, J., Ezziani, A.: High-order discretizations for the wave equation based on the modified equation technique. In: 10ème Congrès Français d’Acoustique, Lyon, France (2010)

  3. Alford, R.M., Kelley, K.R., Boore, D.M.: Accuracy of finite-difference modeling of the acoustic wave equation. Geophysics 39(6), 834–842 (1974)

    Article  Google Scholar 

  4. Babushka, I.M., Sauter, S.A.: Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? SIAM J. Numer. Anal. 34(6), 2392–2423 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Barucq, H., Calandra, H., Diaz, J., Ventimiglia, F.: High-order time discretization of the wave equation by Nabla-P scheme. ESAIM Proc. EDP Sci. 45, 67–74 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bayliss, A., Goldstein, C.I., Turkel, E.: On accuracy conditions for the numerical computation of waves. J. Comput. Phys. 59, 396–404 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  7. Britt, S., Tsynkov, S.V., Turkel, E.: A compact fourth order scheme for the Helmholtz equation in polar coordinates. J. Sci. Comput. 45, 26–47 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Britt, S., Tsynkov, S.V., Turkel, E.: Numerical simulation of time-harmonic waves in inhomogeneous media using compact high order schemes. Commun. Comput. Phys. 9, 520–541 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Britt, S., Tsynkov, S.V., Turkel, E.: A high order numerical method for the Helmholtz equation with non-standard boundary conditions. SIAM J. Sci. Comput. 35, A2255–A2292 (2013)

    Article  MATH  Google Scholar 

  10. Britt, S., Tsynkov, S.V., Turkel, E.: Numerical solution of the wave equation with variable wave speed on nonconforming domains by high-order difference potentials. J. Comput. Phys. 354, 26–42 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chabassier, J., Imperiale, S.: Introduction and study of fourth order theta schemes for linear wave equations. J. Comput. Appl. Math. 245, 194–212 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chabassier, J., Imperiale, S.: Fourth-order energy-preserving locally implicit time discretization for linear wave equations. Int. J. Numer. Methods Eng. 106, 593–622 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ciment, M., Leventhal, S.H.: Higher order compact implicit schemes for the wave equation. Math. Comput. 29, 985–994 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cohen, G.C.: Higher Order Numerical Methods for Transient Wave Equations. Springer, New York (2002)

    Book  MATH  Google Scholar 

  15. Cohen, G.C., Joly, P.: Construction analysis of fourth-order finite difference schemes for the acoustic wave equation in nonhomogeneous media. SIAM J. Numer. Anal. 33(4), 1266–1302 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dablain, M.A.: The application of high order differencing to the scalar wave equation. Geophysics 51(1), 54–66 (1986)

    Article  Google Scholar 

  17. Das, S., Liaob, W., Guptac, A.: An efficient fourth-order low dispersive finite difference scheme for a 2-D acoustic wave equation. J. Comput. Appl. Math. 258(1), 151–167 (2014)

    Article  MathSciNet  Google Scholar 

  18. Deraemaeker, A., Babushka, I., Bouillard, P.: Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimensions. Int. J. Numer. Methods Eng. 46(4), 471–499 (1999)

    Article  MATH  Google Scholar 

  19. Fernández, D.C.D.R., Hicken, J.E., Zingg, D.W.: Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations. Comput. Fluids 95, 171–196 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gilbert, J., Joly, P.: Higher order time stepping for second order hyperbolic problems and optimal CFL conditions. Partial Differ. Equ. 16, 67–93 (2008)

    MathSciNet  MATH  Google Scholar 

  21. Gottlieb, D., Turkel, E.: Dissipative two-four methods for time dependent problems. Math. Comput. 30, 703–723 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  22. Greenbaum, A.: Iterative Methods for Solving Linear systems. SIAM, Philadelphia (1997)

    Book  MATH  Google Scholar 

  23. Gustafsson, B., Mossberg, E.: Time compact high order difference methods for wave propagation. SIAM J. Sci. Comput. 26, 259–271 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hamilton, B., Bilbao, S.: Fourth order and optimized finite difference scheme for the 2-D wave equation. In: Proceedings of 16th International Conference on Digital Audio Effects (DAFx-13), Maynooth, Ireland, September 2–6 (2013)

  25. Henshaw, W.: A high-order accurate parallel solver for Maxwell’s equations on overlapping grids. SIAM J. Sci. Comput. 28(5), 1730–1765 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Joly, P., Rogriguez, J.: Optimized higher order time discretization of second order hyperbolic problems: construction and numerical study. J. Comput. Appl. Math. 234(6), 1953–1961 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kozdon, J.E., Wilcox, L.C.: Stable coupling of non-conforming high-order finite difference methods. SIAM J. Sci. Comput. 38(2), A923–A952 (2016)

    Article  MATH  Google Scholar 

  28. Kreiss, H.-O., Oliger, J.: Comparison of accurate methods for the integration of hyperbolic equations. Tellus 24(3), 199–215 (1972)

    Article  MathSciNet  Google Scholar 

  29. Lambert, J.D.: Computational Methods in Ordinary Differential Equations. Wiley, New York (1973)

    MATH  Google Scholar 

  30. Li, Z.: http://tinyurl.com/z5x7log or http://www.math.pku.edu.cn

  31. Liang, H., Liu, M.Z., Lv, W.: Stability of theta-schemes in the numerical solution of a partial differential equation with piecewise continuous arguments. Appl. Math. Lett. 23(2), 198–206 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Liao, W.Y.: On the dispersion, stability and accuracy of a compact higher-order finite difference scheme for 3D acoustic wave equation. J. Comput. Appl. Math. 270, 571–583 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  33. Liao, W., Yong, P., Dastour, H., Huang, J.: Efficient and accurate numerical simulation of acoustic wave propagation in a 2D heterogeneous media. Appl. Math. Comput. 321, 385–400 (2018)

    MathSciNet  Google Scholar 

  34. Mattsson, K., Ham, F., Iaccarino, G.: Stable boundary treatment for the wave equation on second-order form. J. Sci. Comput. 41(3), 366–383 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  35. Medvinsky, M., Tsynkov, S., Turkel, E.: The method of difference potentials for the Helmholtz equation using compact high order schemes. J. Sci. Comput. 53(1), 150–193 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  36. Nordström, J., Lundquist, T.: Summation-by-parts in time. J. Comput. Phys. 251, 487–499 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  37. Shubin, G.R., Bell, J.B.: The stability of numerical boundary treatments for compact high-order finite-difference schemes. J. Comput. Phys. 108, 272–295 (1993)

    Article  MathSciNet  Google Scholar 

  38. Singer, I., Turkel, E.: High-order finite difference methods for the Helmholtz equation. Comput. Methods Appl. Mech. Eng. 163(1–4), 343–358 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  39. Svärd, M., Nordström, J.: Review of summation-by-parts schemes for initial-boundary-value problems. J. Comput. Phys. 268, 17–38 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  40. Turkel, E., Gordon, D., Gordon, R., Tsynkov, S.: Compact 2D and 3D sixth order schemes for the Helmholtz equation with variable wave number. J. Comput. Phys. 232(1), 272–287 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  41. Virta, K., Mattsson, K.: Acoustic wave propagation in complicated geometries and heterogenous media. J. Sci. Comput. 61, 90–118 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  42. Wang, S., Kreiss, G.: Convergence of summation-by-parts finite difference methods for the wave equation. J. Sci. Comput. 71(1), 219–245 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  43. Zeumi, A.: Fourth order symmetric finite difference schemes for the acoustic wave equation. BIT Numer. Math. 45(3), 627–651 (2005)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, 69978, Tel Aviv, Israel

    Steven Britt & Eli Turkel

  2. Department of Mathematics, North Carolina State University, Box 8205, Raleigh, NC, 27695, USA

    Semyon Tsynkov

  3. Moscow Institute of Physics and Technology, Dolgoprudny, Russia, 141700

    Semyon Tsynkov

Authors
  1. Steven Britt
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Eli Turkel
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Semyon Tsynkov
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Steven Britt.

Additional information

Work supported by the US-Israel Binational Science Foundation (BSF) under Grant # 2014048 and by US Army Research Office (ARO) under Grant # W911NF-16-1-0115. S. Britt was supported by the Raymond and Beverly Sackler Post-Doctoral Scholarship at Tel Aviv University and a Fulbright Postdoctoral Scholarship funded by the US-Israel Educational Foundation.

Numerical Study of Iterative Methods for the Modified Helmholtz Equation

Numerical Study of Iterative Methods for the Modified Helmholtz Equation

In the course of our study of iterative methods for the wave equation, some observations were made regarding the use of MG and CG for solving the modified Helmholtz equation.

Fig. 3
figure 3

For the 2nd order central difference scheme (left), Jacobi without damping leads to divergence for the Poisson equation, which is well known. The 4th order compact scheme (right) exhibits convergence within a small number of V-cycles even without damping for the Poisson equation

Full size image

It is well known that the number of V-cycles needed for the Poisson equation when using the second-order central difference stencil grows with the grid size when using a Jacobi smoother without damping (i.e., \(\omega = 1\)) and it is therefore advantageous to seek a damping parameter for the high frequencies. In Fig. 3, we confirm this classical result for the 2nd order central difference scheme with \(\omega = 1\); however the compact scheme converges with \(\omega = 1\) in a small number of V-cycles. The test solution is \(u = \sin 5x\sin 3y\) on a square domain of side length \(s = \pi \) centered at the origin and Dirichlet boundary conditions on all edges. Each V-cycle uses \(\nu _1 = \nu _2 = 4\) pre- and post-sweeps of the Jacobi smoother. For the second order central difference scheme, a Jacobi smoother with \(\omega = 1\) was also divergent for the modified Helmholtz equation with \(K<0\) but converged rapidly with \(\omega = 4/5\), the classical optimal value for the Poisson equation in 2D. By contrast, use of the optimal damping parameter \(\omega ^*\) (see (45), Sect. 4.2) conferred no advantage for the 4th order compact scheme either for the Poisson or modified Helmholtz equation with \(K<0\).

In Sect. 4.1, our analysis showed that the error bound (38) for conjugate gradient was only well behaved when \(Kh_x^2\) does not tend to zero as the grid is refined. The wave equation resulted in favorable cases in which the modified Helmholtz equation satisfied \(Kh_x^2 = \frac{1}{\theta CFL^2}\), and this quantity is constant for the (4,4) scheme and actually increasing with the grid size for the (2,4) scheme with \(CFL = h_x\). In the following example, we solve the modified Helmholtz equation with a fixed parameter \(K = -50\) using the test solution \(u = \sin {15x}\sin {13y}\) on a square of side length 2 centered at the origin and Dirichlet BCs. The residual tolerance for terminating CG iterations is \(10^{-10}\) in Table 14, while the number of MG V-cycles is the point at which the residual converges.

Table 14 The number of CG iterations increases for the modified Helmholtz equation with a fixed parameter \(K = -50\) while the number of MG V-cycles remains constant
Full size table

Table 14 shows that the number of CG iterations doubles as the grid is refined by a factor of 2 for the case when K is fixed while the number of MG V-cycles remains small. This indicates that MG will in general be more efficient than CG when solving the modified Helmholtz equation. We ran a set of computations for various constant values of \(K=k^2\) and found that the error was a function of kh, indicating that there is no pollution effect for the modified Helmholtz equation.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Britt, S., Turkel, E. & Tsynkov, S. A High Order Compact Time/Space Finite Difference Scheme for the Wave Equation with Variable Speed of Sound. J Sci Comput 76, 777–811 (2018). https://doi.org/10.1007/s10915-017-0639-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-017-0639-9

Keywords