Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 

 

An extension and analysis of the Shu-Osher representation of Runge-Kutta methods


Authors: L. Ferracina and M. N. Spijker
Journal: Math. Comp. 74 (2005), 201-219
MSC (2000): Primary 65M20; Secondary 65L05, 65L06
Published electronically: June 11, 2004
MathSciNet review: 2085408
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In the context of solving nonlinear partial differential equations, Shu and Osher introduced representations of explicit Runge-Kutta methods, which lead to stepsize conditions under which the numerical process is total-variation-diminishing (TVD). Much attention has been paid to these representations in the literature.

In general, a Shu-Osher representation of a given Runge-Kutta method is not unique. Therefore, of special importance are representations of a given method which are best possible with regard to the stepsize condition that can be derived from them.

Several basic questions are still open, notably regarding the following issues: (1) the formulation of a simple and general strategy for finding a best possible Shu-Osher representation for any given Runge-Kutta method; (2) the question of whether the TVD property of a given Runge-Kutta method can still be guaranteed when the stepsize condition, corresponding to a best possible Shu-Osher representation of the method, is violated; (3) the generalization of the Shu-Osher approach to general (possibly implicit) Runge-Kutta methods.

In this paper we give an extension and analysis of the original Shu-Osher representation, by means of which the above questions can be settled. Moreover, we clarify analogous questions regarding properties which are referred to, in the literature, by the terms monotonicity and strong-stability-preserving (SSP).


References [Enhancements On Off] (What's this?)

  • 1. K. BURRAGE (1982): Efficiently implementable algebraically stable Runge-Kutta methods, SIAM J. Numer. Anal. 19, 245-258. MR 83d:65235
  • 2. J.C. BUTCHER (1987): The numerical analysis of ordinary differential equations, John Wiley (Chichester). MR 88d:65002
  • 3. K. DEKKER AND J.G. VERWER (1984): Stability of Runge-Kutta methods for stiff nonlinear differential equations, North-Holland Publ. Comp. (Amsterdam). MR 86g:65003
  • 4. L. FERRACINA AND M.N. SPIJKER (2002): Stepsize restrictions for the total-variation-diminishing property in general Runge-Kutta methods, Report MI 2002-21 (2002), Mathematical Institute, University of Leiden. To appear in SIAM J. Numer. Anal.
  • 5. A. GERISH AND R. WEINER (2003): On the positivity of low order explicit Runge-Kutta schemes applied in splitting methods, Computers and Mathematics with Applications 45, 53-67.
  • 6. S. GOTTLIEB AND C.-W. SHU (1998): Total-variation-diminishing Runge-Kutta schemes, Math. Comp. 67, 73-85. MR 98c:65122
  • 7. S. GOTTLIEB, C.-W. SHU, AND E. TADMOR (2001): Strong-stability-preserving high-order time discretization methods, SIAM Review 43, 89-112. MR 2002f:65132
  • 8. E. HAIRER, S.P. NøRSETT, AND G. WANNER (1987): Solving ordinary differential equations I, Springer Verlag (Berlin). MR 87m:65005
  • 9. E. HAIRER AND G. WANNER (1996): Solving ordinary differential equations II. Stiff and differential-algebraic problems, Second Revised Edition, Springer (Berlin).MR 97m:65007
  • 10. A. HARTEN (1983): High resolution schemes for hyperbolic conservation laws, J. Comput. Phys. 49, 357-393. MR 84g:65115
  • 11. I. HIGUERAS (2002): On strong stability preserving time discretization methods, Report n.2 (2002), Departamento de Matemática e Informática, Universidad Pública de Navarra.
  • 12. W. HUNDSDORFER, S.J. RUUTH, AND R.J. SPITERI (2003): Monotonicity-preserving linear multistep methods, SIAM J. Numer. Anal. 41, 605-623.
  • 13. W. HUNDSDORFER AND J.G. VERWER (2003): Numerical solution of time-dependent advection-diffusion-reaction equations, Springer Ser. Comput. Math. 33, Springer-Verlag, Berlin.
  • 14. J.F.B.M. KRAAIJEVANGER (1991): Contractivity of Runge-Kutta methods, BIT 31, 482-528. MR 92i:65120
  • 15. D. KRÖNER (1997): Numerical schemes for conservation laws, Wiley, Teubner (Chichester, Stuttgart). MR 98b:65003
  • 16. C.B. LANEY (1998): Computational gas dynamics, Cambridge University Press (Cambridge). MR 2000e:76086
  • 17. R.J. LEVEQUE (2002): Finite volume methods for hyperbolic problems, Cambridge University Press (Cambridge). MR 2003h:65001
  • 18. K.W. MORTON (1980): Stability of difference approximations to a diffusion-convection equation, Int. J. Num. Meth. Eng. 15, 677-683. MR 82i:76080
  • 19. S.J. RUUTH AND R.J. SPITERI (2002): Two barriers on strong-stability-preserving time discretization methods, J. Sci. Comput. 17, 211-220.
  • 20. C.-W. SHU (1988): Total-variation-diminishing time discretizations, SIAM J. Sci. Stat. Comput. 9, 1073-1084. MR 90a:65196
  • 21. C.-W. SHU (2002): A survey of strong stability preserving high-order time discretizations, Collected Lectures on the Preservation of Stability under Discretization, D. Estep, S. Tavener Editors, SIAM (Philadelphia, PA), 51-65.
  • 22. C.-W. SHU AND S. OSHER (1988): Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys. 77, 439-471. MR 89g:65113
  • 23. M.N. SPIJKER (1983): Contractivity in the numerical solution of initial value problems, Numer. Math. 42, 271-290. MR 85b:65067
  • 24. R.J. SPITERI AND S.J. RUUTH (2002): A new class of optimal high-order strong-stability-preserving time discretization methods, SIAM J. Numer. Anal. 40, 469-491. MR 2003g:65083
  • 25. E.F. TORO (1999): Riemann Solvers and Numerical Methods for Fluid Dynamics, Second Edition, Springer-Verlag (Berlin). MR 2000f:76091

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65M20, 65L05, 65L06

Retrieve articles in all journals with MSC (2000): 65M20, 65L05, 65L06


Additional Information

L. Ferracina
Affiliation: Mathematical Institute, Leiden University, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands
Email: ferra@math.leidenuniv.nl

M. N. Spijker
Affiliation: Mathematical Institute, Leiden University, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands
Email: spijker@math.leidenuniv.nl

DOI: https://doi.org/10.1090/S0025-5718-04-01664-3
Keywords: Initial value problem, conservation law, method of lines (MOL), Runge-Kutta formula, Shu-Osher representation, total-variation-diminishing (TVD), strong-stability-preserving (SSP), monotonicity.
Received by editor(s): May 7, 2003
Received by editor(s) in revised form: August 3, 2003
Published electronically: June 11, 2004
Article copyright: © Copyright 2004 American Mathematical Society