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

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Abstract

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.

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References

  1. Bastian, P., Blatt, M., Dedner, A., Engwer, C., Fahlke, J., Gräser, C., Klöfkorn, R., Nolte, M., Ohlberger, M., Sander, O. (2013) http://www.dune-project.org

  2. Burrage, K., Hundsdorfer, W.H., Verwer, J.G.: A study of B-convergence of Runge–Kutta methods. Computing 36(1), 17–34 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  3. Deckelnick, K., Elliott, C., Styles, V.: Numerical diffusion-induced grain boundary motion. Interfaces Free Bound. 3, 393–414 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  6. Dedner, A., Klöfkorn, R., Nolte, M., Ohlberger, M. (2013) http://dune.mathematik.uni-freiburg.de

  7. Dziuk, G., Elliott, C.M.: Finite elements on evolving surfaces. IMA J. Numer. Anal. 27, 262–292 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dziuk, G., Elliott, C.M.: \(L^2\)-estimates for the evolving surface finite element method. Math. Comp. 82, 1–24 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  10. Dziuk, G., Elliott, C.M.: SIAM J. Numer. Anal. 50(5), 2677–2694 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  12. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin (1983)

    Book  MATH  Google Scholar 

  13. Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration. In: Structure-Preserving Algorithms for Ordinary Differential Equations. 2nd ed. Springer, Berlin (2006)

  14. Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations. I. Nonstiff Problems, 2nd edn. Springer, Berlin (1993)

    MATH  Google Scholar 

  15. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations. II. Stiff and differential-algebraic problems, 2nd edn. Springer, Berlin (1996)

    Book  MATH  Google Scholar 

  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. James, A., Lowengrub, J.: A surfactant-conserving volume-of-fluid method for interfacial flows with insoluble surfactant. J. Comp. Phys. 201, 685–722 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  21. Lubich, C., Ostermann, A.: Interior estimates for time discretizations of parabolic equations. Appl. Numer. Math. 18, 241–251 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  22. Lubich, C., Mansour, D.: Variational discretization of linear wave equations on evolving surfaces. Math. Comp. (to appear)

  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. Mansour, D.: Dissertation (PhD thesis), Univ. Tübingen (2013)

  25. Marsden, J.E., West, M.: Discrete mechanics and variational integrators. Acta Numer. 10, 357–514 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  26. Memoli, F., Sapiro, G., Thompson, P.: Implicit brain imaging. Human Brain Mapp. 23, 179–188 (2004)

    Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

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

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Authors and Affiliations

  1. Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 , Tübingen, Germany

    Dhia Mansour

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  1. Dhia Mansour
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Correspondence to Dhia Mansour.

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This work was supported by DFG, SFB/TR 71 “Geometric Partial Differential Equations”.

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Mansour, D. Gauss–Runge–Kutta time discretization of wave equations on evolving surfaces. Numer. Math. 129, 21–53 (2015). https://doi.org/10.1007/s00211-014-0632-2

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