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Optimal Monotonicity-Preserving Perturbations of a Given Runge–Kutta Method

  • Inmaculada Higueras
  • David I. Ketcheson
  • Tihamér A. Kocsis
Article
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Abstract

Perturbed Runge–Kutta methods (also referred to as downwind Runge–Kutta methods) can guarantee monotonicity preservation under larger step sizes relative to their traditional Runge–Kutta counterparts. In this paper we study the question of how to optimally perturb a given method in order to increase the radius of absolute monotonicity (a.m.). We prove that for methods with zero radius of a.m., it is always possible to give a perturbation with positive radius. We first study methods for linear problems and then methods for nonlinear problems. In each case, we prove upper bounds on the radius of a.m., and provide algorithms to compute optimal perturbations. We also provide optimal perturbations for many known methods.

Keywords

Strong stability preserving Monotonicity Runge–Kutta methods Time discretization 

Mathematics Subject Classification

65L06 65L20 65M20 

References

  1. 1.
    Bogacki, P., Shampine, L.F.: An efficient Runge–Kutta (4, 5) pair. Comput. Math. Appl. 32(6), 15–28 (1996)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Calvo, M., Montijano, J.I., Rández, L.: A new embedded pair of Runge–Kutta formulas of orders 5 and 6. Comput. Math. Appl. 20(1), 15–24 (1990)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Conde, S., Gottlieb, S., Grant, Z.J., Shadid, J.N.: Implicit and implicit–explicit strong stability preserving Runge–Kutta methods with high linear order. J. Sci. Comput. 73(2), 667–690 (2017)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Donat, R., Higueras, I., Martínez-Gavara, A.: On stability issues for IMEX schemes applied to hyperbolic equations with stiff reaction terms. Math. Comput. 80, 2097–2126 (2011)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Dormand, J.R., Prince, P.J.: A family of embedded Runge–Kutta formulae. J. Comput. Appl. Math. 6(1), 19–26 (1980)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Fehlberg, E.: Klassische Runge–Kutta–Formeln fünfter und siebenter Ordnung mit Schrittweiten–Kontrolle. Computing 4(2), 93–106 (1969)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Ferracina, L., Spijker, M.N.: Stepsize restrictions for the total-variation-diminishing property in general Runge–Kutta methods. SIAM J. Numer. Anal. 42, 1073–1093 (2004)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Gottlieb, S., Ketcheson, D.I., Shu, C.W.: Strong Stability Preserving Runge–Kutta and Multistep Time Discretizations. World Scientific Publishing Company, Singapore (2011)CrossRefMATHGoogle Scholar
  9. 9.
    Gottlieb, S., Ruuth, S.J.: Optimal strong-stability-preserving time-stepping schemes with fast downwind spatial discretizations. J. Sci. Comput. 27, 289–303 (2006)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Gottlieb, S., Shu, C.W.: Total variation diminishing Runge–Kutta schemes. Math. Comput. 67(221), 73–85 (1998)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Springer, Berlin (1991)CrossRefMATHGoogle Scholar
  12. 12.
    Heun, K.: Neue methoden zur approximativen integration der differentialgleichungen einer unabhängigen veränderlichen. Z. Math. Phys. 45, 23–38 (1900)MATHGoogle Scholar
  13. 13.
    Higueras, I.: Representations of Runge–Kutta methods and strong stability preserving methods. SIAM J. Numer. Anal. 43, 924–948 (2005)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Higueras, I.: Strong stability for additive Runge–Kutta methods. SIAM J. Numer. Anal. 44(4), 1735–1758 (2006)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Higueras, I.: Positivity properties for the classical fourth order Runge–Kutta methods. Monografías de la Real Academia de Ciencias de Zaragoza 33, 125–139 (2010)MathSciNetGoogle Scholar
  16. 16.
    Higueras, I., Ketcheson, D.I.: Reproducibility repository for computations of optimal perturbations to Runge-Kutta methods, version 0.2 (2016).  https://doi.org/10.5281/zenodo.1146916
  17. 17.
    Horváth, Z.: On the positivity step size threshold of Runge–Kutta methods. Appl. Numer. Math. 53, 341–356 (2005)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Hundsdorfer, W., Koren, B., van Loon, M., Verwer, J.C.: A positive finite-difference advection scheme. J. Comput. Phys. 117(1), 35–46 (1995)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Ketcheson, D.I.: Highly efficient strong stability preserving Runge–Kutta Methods with low-storage implementations. SIAM J. Sci. Comput. 30, 2113–2136 (2008)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Ketcheson, D.I.: Computation of optimal monotonicity preserving general linear methods. Math. Comput. 78, 1497–1513 (2009)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Ketcheson, D.I.: High Order Strong Stability Preserving Time Integrators and Numerical Wave Propagation Methods for Hyperbolic PDEs. Doctoral thesis, University of Washington (2009)Google Scholar
  22. 22.
    Ketcheson, D.I.: Step sizes for strong stability preservation with downwind-biased operators. SIAM J. Numer. Anal. 49(4), 1649–1660 (2011)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Ketcheson, D.I.: Nodepy software version 0.6.1 (2015). Available from http://github.com/ketch/nodepy
  24. 24.
    Kraaijevanger, J.F.B.M.: Contractivity of Runge–Kutta methods. BIT 31, 482–528 (1991)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    LeVeque, R.J., Yee, H.C.: A study of numerical methods for hyperbolic conservation laws with stiff source terms. J. Comput. Phys. 210, 187–210 (1990)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Merson, R.H.: An operational method for the study of integration processes. In: Proceedings of the Symposium on Data Processing, pp. 1–25 (1957)Google Scholar
  27. 27.
    Prince, P.J., Dormand, J.R.: High order embedded Runge–Kutta formulae. J. Comput. Appl. Math. 7(1), 67–75 (1981)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Ruuth, S.J.: Global optimization of explicit strong-stability-preserving Runge–Kutta Methods. Math. Comput. 75, 183–207 (2006)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Ruuth, S.J., Spiteri, R.J.: High-order strong-stability-preserving Runge–Kutta methods with downwind-biased spatial discretizations. SIAM J. Numer. Anal. 42, 974–996 (2004)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439–471 (1988)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Zhang, X., Shu, C.W.: On maximum-principle-satisfying high order schemes for scalar conservation laws. J. Comput. Phys. 229(9), 3091–3120 (2010)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Public University of NavarrePamplonaSpain
  2. 2.King Abdullah University of Science and Technology (KAUST)ThuwalSaudi Arabia
  3. 3.Széchenyi István UniversityGyőrHungary

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