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New Third Order Low-Storage SSP Explicit Runge–Kutta Methods

  • I. Higueras
  • T. RoldánEmail author
Article
  • 30 Downloads

Abstract

When a high dimension system of ordinary differential equations is solved numerically, the computer memory capacity may be exhausted. Thus, for such systems, it is important to incorporate low memory usage to some other properties of the scheme. In the context of strong stability preserving (SSP) schemes, some low-storage methods have been considered in the literature. In this paper we study 5-stage third order \(2N^*\) low-storage SSP explicit Runge–Kutta schemes. These are SSP schemes that can be implemented with 2N memory registers, where N is the dimension of the problem, and retain the previous time step approximation. This last property is crucial for a variable step size implementation of the scheme. In this paper, first we show that the optimal SSP methods cannot be implemented with \(2N^*\) memory registers. Next, two non-optimal SSP \(2N^*\) low-storage methods are constructed; although their SSP coefficients are not optimal, they achieve some other interesting properties. Finally, we show some numerical experiments.

Keywords

Runge–Kutta SSP Low-storage 2N implementations Stability 

Mathematics Subject Classification

65L06 65L05 65L04 

Notes

Acknowledgements

Supported by Ministerio de Economía y Competividad, Project MTM2016-77735-C3-2-P.

References

  1. 1.
    Calvo, M., Franco, J.M., Montijano, J.I., Rández, L.: On some new low storage implementations of time advancing Runge–Kutta methods. J. Comput. Appl. Math. 236(15), 3665–3675 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Calvo, M., Franco, J.M., Rández, L.: Minimum storage Runge–Kutta schemes for computational acoustics. Comput. Math. Appl. 45(1), 535–545 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cavaglieri, D., Bewley, T.: Low-storage implicit/explicit Runge–Kutta schemes for the simulation of stiff high-dimensional ode systems. J. Comput. Phys 286, 172–193 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ferracina, L., Spijker, M.N.: Stepsize restrictions for the total-variation-diminishing property in general Runge–Kutta methods. SIAM J. Numer. Anal. 42(3), 1073–1093 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ferracina, L., Spijker, M.N.: Strong stability of singly-diagonally-implicit Runge–Kutta methods. Appl. Numer. Math. 58(11), 1675–1686 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gottlieb, S., Ketcheson, D.I., Shu, C.W.: Strong Stability Preserving Runge–Kutta and Multistep Time Discretizations. World Scientific, Singapore (2011)CrossRefzbMATHGoogle Scholar
  7. 7.
    Gottlieb, S., Ketcheson, D.I., Shu, C.W.: High order strong stability preserving time discretizations. J. Sci. Comput. 38(3), 251–289 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gottlieb, S., Shu, C.W.: Total variation diminishing Runge–Kutta schemes. Math. Comput. 67(221), 73–85 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gottlieb, S., Shu, C.W., Tadmor, E.: Strong stability preserving high-order time discretization methods. SIAM Rev. 43(1), 89–112 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Happenhofer, N., Koch, O., Kupka, F.: IMEX methods for the ANTARES code. ASC report, 27 (2011)Google Scholar
  11. 11.
    Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I, Nonstiff Problems, 2 revised edn. Springer, Berlin (1993)zbMATHGoogle Scholar
  12. 12.
    Higueras, I.: Representations of Runge–Kutta methods and strong stability preserving methods. SIAM J. Numer. Anal. 43(3), 924–948 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Higueras, I.: Strong stability for additive Runge–Kutta methods. SIAM J. Numer. Anal. 44(4), 1735–1758 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kennedy, C.A., Carpenter, M.H., Lewis, R.M.: Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations. Appl. Numer. Math. 35(3), 177–219 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ketcheson, D.I.: Highly efficient strong stability preserving Runge–Kutta methods with low-storage implementations. SIAM J. Sci. Comput. 30(4), 2113–2136 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ketcheson, D.I.: Runge–Kutta methods with minimum storage implementations. J. Comput. Phys. 229(5), 1763–1773 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ketcheson, D.I., Macdonald, C.B., Gottlieb, S.: Optimal implicit strong stability preserving Runge–Kutta methods. Appl. Numer. Math. 59(2), 373–392 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ketcheson, D.I., Parsani, M., Ahmadia, A.J.: Rk-opt: software for the design of Runge–Kutta methods, version 0.2 (2013)Google Scholar
  19. 19.
    Kraaijevanger, J.F.B.M.: Absolute monotonicity of polynomials occurring in the numerical solution of initial value problems. Numer. Math. 48(3), 303–322 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kraaijevanger, J.F.B.M.: Contractivity of Runge–Kutta methods. BIT 31(3), 482–528 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002)CrossRefzbMATHGoogle Scholar
  22. 22.
    Ruuth, S.J.: Global optimization of explicit strong-stability-preserving Runge–Kutta methods. Math. Comput. 75(253), 183–208 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Shu, C.W.: Total-variation-diminishing time discretizations. SIAM J. Sci. Comput. 9(6), 1073–1084 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Shu, C.W., Osher, S.: Efficient implementation of essentially nonoscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439–471 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Spijker, M.N.: Contractivity in the numerical solution of initial value problems. Numer. Math. 42(3), 271–290 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Spijker, M.N.: Stepsize restrictions for stability of one-step methods in the numerical solution of initial value problems. Math. Comput. 45(172), 377–392 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Spijker, M.N.: Stepsize conditions for general monotonicity in numerical initial value problems. SIAM J. Numer. Anal. 45(3), 1226–1245 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Spiteri, R.J., Ruuth, S.J.: A new class of optimal high-order strong stability preserving time discretization methods. SIAM J. Numer. Anal. 40(2), 469–491 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Spiteri, R.J., Ruuth, S.J.: Non-linear evolution using optimal fourth-order strong-stability-preserving Runge–Kutta methods. Math. Comput. Simulat. 62(1–2), 125–135 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Van Der Houwen, P.J.: Construction of integration formulas for initial value problems. North Holland (1977)Google Scholar
  31. 31.
    Williamson, J.H.: Low-storage Runge-Kutta schemes. J. Comput. Phys. 35(1), 48–56 (1980)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Departamento de Estadística, Informática y MatemáticasUniversidad Pública de NavarraPamplonaSpain

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