Numerische Mathematik

, Volume 129, Issue 1, pp 21–53 | Cite as

Gauss–Runge–Kutta time discretization of wave equations on evolving surfaces

  • Dhia MansourEmail author


This paper is concerned with an error analysis for a full discretization of the linear wave equation on a moving surface. The equation is discretized in space by the evolving surface finite element method. Discretization in time is done by Gauß–Runge–Kutta (GRK) methods, aiming for higher-order accuracy in time and unconditional stability of the fully discrete scheme. The latter is established in the natural time-dependent norms by using the algebraic stability and the coercivity property of the GRK methods together with the properties of the spatial semi-discretization. Under sufficient regularity conditions, optimal-order error estimates for this class of fully discrete methods are shown. Numerical experiments are presented to confirm some of the theoretical results.

Mathematics Subject Classification (2000)

65M12 65M15 65M60 35L99 35R01 35R37 



The author would like to thank Christian Lubich for introducing him to the topic of this paper, for the fruitful discussions and insightful suggestions.


  1. 1.
    Bastian, P., Blatt, M., Dedner, A., Engwer, C., Fahlke, J., Gräser, C., Klöfkorn, R., Nolte, M., Ohlberger, M., Sander, O. (2013)
  2. 2.
    Burrage, K., Hundsdorfer, W.H., Verwer, J.G.: A study of B-convergence of Runge–Kutta methods. Computing 36(1), 17–34 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Deckelnick, K., Elliott, C., Styles, V.: Numerical diffusion-induced grain boundary motion. Interfaces Free Bound. 3, 393–414 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Dekker, K., Kraaijevanger, J.F.B.M., Spijker, M.N.: The order of B-convergence of the Gaussian Runge–Kutta method. Computing 36(1), 35–41 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Dedner, A., Klöfkorn, R., Nolte, M., Ohlberger, M.: A generic interface for parallel and adaptive scientific computing: abstraction principles and the DUNE-FEM module. Computing 90, 165–196 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Dedner, A., Klöfkorn, R., Nolte, M., Ohlberger, M. (2013)
  7. 7.
    Dziuk, G., Elliott, C.M.: Finite elements on evolving surfaces. IMA J. Numer. Anal. 27, 262–292 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Dziuk, G., Elliott, C.M.: \(L^2\)-estimates for the evolving surface finite element method. Math. Comp. 82, 1–24 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Dziuk, G., Elliott, C.M.: A fully discrete evolving surface finite element method., vol. 50, No. 5, pp. 2677–2694. ISSN 1095–7170 (2012)Google Scholar
  10. 10.
    Dziuk, G., Elliott, C.M.: SIAM J. Numer. Anal. 50(5), 2677–2694 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Dziuk, G., Lubich, C., Mansour, D.: Runge–Kutta time discretization of parabolic differential equations on evolving surfaces. IMA J. Numer. Anal. 32, 394–416 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin (1983)CrossRefzbMATHGoogle Scholar
  13. 13.
    Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration. In: Structure-Preserving Algorithms for Ordinary Differential Equations. 2nd ed. Springer, Berlin (2006)Google Scholar
  14. 14.
    Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations. I. Nonstiff Problems, 2nd edn. Springer, Berlin (1993)zbMATHGoogle Scholar
  15. 15.
    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations. II. Stiff and differential-algebraic problems, 2nd edn. Springer, Berlin (1996)CrossRefzbMATHGoogle Scholar
  16. 16.
    Halpern, D., Jenson, O.E., Grotberg, J.B.: A theoretical study of surfactant and liquid delivery into the lung. J. Appl. Physiol. 85, 333–352 (1998)Google Scholar
  17. 17.
    James, A., Lowengrub, J.: A surfactant-conserving volume-of-fluid method for interfacial flows with insoluble surfactant. J. Comp. Phys. 201, 685–722 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Kraaijevanger, J.F.B.M.: B-convergence of the implicit midpoint rule and the trapezoidal rule. BIT Numer. Math. 25(4), 652–666 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Lenz, M., Nemadjieu, S.F., Rumpf, M.: A convergent finite volume scheme for diffusion on evolving surfaces. SIAM J. Numer. Anal. 49(1), 15–37 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Lubich, C., Ostermann, A.: Runge–Kutta methods for parabolic equations and convolution quadrature. Math. Comp. 60, 105–131 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Lubich, C., Ostermann, A.: Interior estimates for time discretizations of parabolic equations. Appl. Numer. Math. 18, 241–251 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Lubich, C., Mansour, D.: Variational discretization of linear wave equations on evolving surfaces. Math. Comp. (to appear)Google Scholar
  23. 23.
    Lubich, C., Mansour, D., Venkataraman, Ch.: Backward difference time discretization of parabolic differential equations on evolving surfaces. IMA J. Numer. Anal., (2013). doi: 10.1093/imanum/drs044
  24. 24.
    Mansour, D.: Dissertation (PhD thesis), Univ. Tübingen (2013)Google Scholar
  25. 25.
    Marsden, J.E., West, M.: Discrete mechanics and variational integrators. Acta Numer. 10, 357–514 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Memoli, F., Sapiro, G., Thompson, P.: Implicit brain imaging. Human Brain Mapp. 23, 179–188 (2004)Google Scholar
  27. 27.
    Sanz-Serna, J.M., Verwer, J.G., Hundsdorfer, W.H.: Convergence and order reduction of Runge-Kutta schemes applied to evolutionary problems in partial differential equations. Numer. Math. 50, 405–418 (1987)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität TübingenTübingenGermany

Personalised recommendations