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Numerische Mathematik

, Volume 138, Issue 3, pp 557–579 | Cite as

Error analysis of implicit Runge–Kutta methods for quasilinear hyperbolic evolution equations

  • Marlis HochbruckEmail author
  • Tomislav Pažur
  • Roland Schnaubelt
Article
  • 321 Downloads

Abstract

We establish error bounds of implicit Runge–Kutta methods for a class of quasilinear hyperbolic evolution equations including certain Maxwell and wave equations on full space or with Dirichlet boundary conditions. Our assumptions cover algebraically stable and coercive schemes such as Gauß and Radau collocation methods. We work in a refinement of the analytical setting of Kato’s well-posedness theory.

Mathematics Subject Classification

Primary: 65M12 65J15 Secondary: 35Q61 35L90 

References

  1. 1.
    Akrivis, G., Lubich, C.: Fully implicit, linearly implicit and implicit–explicit backward difference formulae for quasi-linear parabolic equations. Numer. Math. 131(4), 713–735 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Benzoni-Gavage, S., Serre, D.: Multidimensional Hyperbolic Partial Differential Equations. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford (2007)zbMATHGoogle Scholar
  3. 3.
    Burrage, K., Butcher, J.C.: Stability criteria for implicit Runge–Kutta methods. SIAM J. Numer. Anal. 16(1), 46–57 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Busch, K., von Freymann, G., Linden, S., Mingaleev, S.F., Tkeshelashvili, L., Wegener, M.: Periodic nanostructures for photonics. Phys. Rep. 444(3–6), 101–202 (2007)CrossRefGoogle Scholar
  5. 5.
    Crandall, M.G., Souganidis, P.E.: Convergence of difference approximations of quasilinear evolution equations. Nonlinear Anal. 10(5), 425–445 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Crouzeix, M.: Sur la \(B\)-stabilité des méthodes de Runge–Kutta. Numer. Math. 32(1), 75–82 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations, Volume 194 of Graduate Texts in Mathematics. Springer, New York (2000)Google Scholar
  8. 8.
    Guès, O.: Problème mixte hyperbolique quasi-linéaire caractéristique. Commun. Partial Differ. Equ. 15(5), 595–645 (1990)CrossRefzbMATHGoogle Scholar
  9. 9.
    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, 2nd edn. Springer, Berlin (1996)zbMATHGoogle Scholar
  10. 10.
    Hochbruck, M., Pažur, T.: Implicit Runge–Kutta methods and discontinuous Galerkin discretizations for linear Maxwell’s equations. SIAM J. Numer. Anal. 53(1), 485–507 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hochbruck, M., Pažur, T.: Error analysis of implicit Euler methods for quasilinear hyperbolic evolution equations. Numer. Math. 135(2), 547–569 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kanda, S.: Convergence of difference approximations and nonlinear semigroups. Proc. Am. Math. Soc. 108(3), 741–748 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kato, T.: The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Ration. Mech. Anal. 58(3), 181–205 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kato, T.: Quasi-linear equations of evolution, with applications to partial differential equations. In: Spectral Theory and Differential Equations (Proc. Sympos., Dundee, 1974; dedicated to Konrad Jörgens). Lecture Notes in Math., vol. 448, pp. 25–70. Springer, Berlin (1975)Google Scholar
  15. 15.
    Kato, T.: Quasilinear equations of evolution in nonreflexive Banach spaces. In: Nonlinear Partial Differential Equations in Applied Science (Tokyo, 1982), Volume 81 of North-Holland Math. Stud., pp. 61–76. North-Holland, Amsterdam (1983)Google Scholar
  16. 16.
    Kobayashi, Y.: Difference approximation of Cauchy problems for quasi-dissipative operators and generation of nonlinear semigroups. J. Math. Soc. Jpn. 27(4), 640–665 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kovács, B., Lubich, C.: Stability and convergence of time discretizations of quasi-linear evolution equations of Kato type. Numer. Math., online first (2017). doi: 10.1007/s00211-017-0909-3
  18. 18.
    Lubich, C., Ostermann, A.: Runge–Kutta approximation of quasi-linear parabolic equations. Math. Comput. 64(210), 601–628 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mansour, D.: Gauss–Runge–Kutta time discretization of wave equations on evolving surfaces. Numer. Math. 129(1), 21–53 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Müller, D.: Well-posedness for a general class of quasilinear evolution equations with applications to Maxwell’s equations. Ph.D. thesis, Karlsruhe Institute of Technology. https://publikationen.bibliothek.kit.edu/1000042147 (2014)
  21. 21.
    Takahashi, T.: Convergence of difference approximation of nonlinear evolution equations and generation of semigroups. J. Math. Soc. Jpn. 28(1), 96–113 (1976)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  • Marlis Hochbruck
    • 1
    Email author
  • Tomislav Pažur
    • 2
  • Roland Schnaubelt
    • 1
  1. 1.Department of MathematicsKarlsruhe Institute of TechnologyKarlsruheGermany
  2. 2.AVL AST d.o.o. CroatiaZagrebCroatia

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