Advertisement

Optimized High Order Explicit Runge-Kutta-Nyström Schemes

  • Marc DurufléEmail author
  • Mamadou N’diaye
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 119)

Abstract

Runge-Kutta-Nyström (RKN) schemes have been developed to solve a non-linear ordinary differential equation (ODE) of the type y″ = f(t, y). In Chawla and Sharma (Computing, 26:247–256, 1981), the stability condition (the Courant-Friedrichs-Lewy or CFL) associated with these schemes have been studied for order 3, 4 and 5. In this paper, we extend this study for higher orders and we propose a new algorithm to compute numerically the CFL. By using this algorithm, we compute optimal coefficients for RKN schemes of orders 6, 7, 8 and 10 which maximize the CFL. Herein, the obtained schemes are used to solve non-linear Maxwell’s equations in 1-D.

References

  1. 1.
    M. Chawla, S. Sharma, Families of fifth-order Nyström methods for y″ = f(x, y) and intervals of periodicity. Computing 26, 247–256 (1981)Google Scholar
  2. 2.
    M. Chawla, S. Sharma, Interval of periodicity and absolute stability of explicit Nyström methods for y″ = f(x, y). BIT 21, 455–464 (1981)Google Scholar
  3. 3.
    E. Hairer, Méthodes de Nyström pour l’équation différentielle y″ = f(x, y). Numer. Math. 27, 283–300 (1977)Google Scholar
  4. 4.
    E. Hairer, A one-step method of order 10 for y″(t) = f(t, y). IMA J. Numer. Anal. 2, 83–94 (1982)Google Scholar
  5. 5.
    E. Hairer, S.P. Norsett, G. Wanner, Solving Ordinary Differential Equations I - Nonstiff Problems (Springer, Berlin, 2008)zbMATHGoogle Scholar
  6. 6.
    P. Joly, J.-C. Gilbert, Higher order time stepping for second order hyperbolic problems and optimal CFL conditions. Comput. Methods Appl. Sci. 16, 67–93 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    J. Neigemann, R. Diehl, K. Bush, Efficient low-storage Runge-Kutta schemes with optimized stability regions. J. Comput. Phys. 231, 364–372 (2012)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Magique-3DInria Centre de Recherche Bordeaux Sud-OuestTalenceFrance
  2. 2.Laboratoire de Mathématiques et leurs ApplicationsUniversity of Pau and Pays de L’AdourPauFrance

Personalised recommendations