Numerical Algorithms

, Volume 77, Issue 4, pp 1093–1116 | Cite as

Exact optimal values of step-size coefficients for boundedness of linear multistep methods

Original Paper


Linear multistep methods (LMMs) applied to approximate the solution of initial value problems—typically arising from method-of-lines semidiscretizations of partial differential equations—are often required to have certain monotonicity or boundedness properties (e.g., strong-stability-preserving, total-variation-diminishing or total-variation-boundedness properties). These properties can be guaranteed by imposing step-size restrictions on the methods. To qualitatively describe the step-size restrictions, one introduces the concept of step-size coefficient for monotonicity (SCM, also referred to as the strong-stability-preserving (SSP) coefficient) or its generalization, the step-size coefficient for boundedness (SCB). An LMM with larger SCM or SCB is more efficient, and the computation of the maximum SCM for a particular LMM is now straightforward. However, it is more challenging to decide whether a positive SCB exists, or determine if a given positive number is a SCB. Theorems involving sign conditions on certain linear recursions associated to the LMM have been proposed in the literature that allow us to answer the above questions: the difficulty with these theorems is that there are in general infinitely many sign conditions to be verified. In this work, we present methods to rigorously check the sign conditions. As an illustration, we confirm some recent numerical investigations concerning the existence of positive SCBs in the BDF and in the extrapolated BDF (EBDF) families. As a stronger result, we determine the optimal values of the SCBs as exact algebraic numbers in the BDF family (with 1 ≤ k ≤ 6 steps) and in the Adams–Bashforth family (with 1 ≤ k ≤ 3 steps).


Linear multistep methods Strong stability preservation Step-size coefficient for monotonicity Step-size coefficient for boundedness 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The author is indebted to the anonymous referees of the manuscript for their suggestions that helped improving the presentation of the material.


  1. 1.
    Beukers, F., Tijdeman, R.: One-sided power sum and cosine inequalities. Indag. Math. (N.S.) 24(2), 373–381 (2013). doi: 10.1016/j.indag.2012.11.009 MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    van de Griend, J.A., Kraaijevanger, J.F.B.M.: Absolute monotonicity of rational functions occurring in the numerical solution of initial value problems. Numer. Math. 49(4), 413–424 (1986). doi: 10.1007/BF01389539 MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Hairer, E., Nørsett, S.P., Wanner, G. Nonstiff problems: solving ordinary differential equations I. Springer, Berlin (2008)Google Scholar
  4. 4.
    Hairer, E., Wanner, G.: Stiff and differential-algebraic problems: solving ordinary differential equations II. Springer, Berlin (2002)Google Scholar
  5. 5.
    Hundsdorfer, W., Ruuth, S.J., Spiteri, R.J.: Monotonicity-preserving linear multistep methods. SIAM J. Numer. Anal. 41(2), 605–623 (2003). doi: 10.1137/S0036142902406326 MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Hundsdorfer, W., Ruuth, S.J.: On monotonicity and boundedness properties of linear multistep methods. Math. Comp. 75(254), 655–672 (2006). doi: 10.1090/S0025-5718-05-01794-1 MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Hundsdorfer, W., Mozartova, A., Spijker, M.N.: Special boundedness properties in numerical initial value problems. BIT 51(4), 909–936 (2011). doi: 10.1007/s10543-011-0349-x MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hundsdorfer, W., Spijker, M.N.: Boundedness and strong stability of Runge–Kutta methods. Math. Comp. 80(274), 863–886 (2011). doi: 10.1090/S0025-5718-2010-02422-6 MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Hundsdorfer, W., Mozartova, A., Spijker, M.N.: Stepsize restrictions for boundedness and monotonicity of multistep methods. J. Sci. Comput. 50(2), 265–286 (2012). doi: 10.1007/s10915-011-9487-1 MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Lóczi, L., Ketcheson, D.I.: Rational functions with maximal radius of absolute monotonicity. LMS J. Comput. Math. 17(1), 159–205 (2014). doi: 10.1112/S1461157013000326 MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Ouaknine, J., Worrell, J.: Decision problems for linear recurrence sequences. In: reachability problems, 6th International Workshop, RP 2012, Bordeaux, France, September 17–19 (2012). doi: 10.1007/978-3-642-33512-9_3 Google Scholar
  12. 12.
    Ouaknine, J., Worrell, J.: Positivity problems for low-order linear recurrence sequences. In: proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms (2014). doi: 10.1137/1.9781611973402.27 Google Scholar
  13. 13.
    Ouaknine, J., Worrell, J.: Automata, languages, and programming. Part II: on the positivity problem for simple linear recurrence Sequences, pp. 318–329. Springer, Heidelberg (2014)Google Scholar
  14. 14.
    Ouaknine, J., Worrell, J.: Ultimate positivity is decidable for simple linear recurrence sequences, pp. 330–341. Springer, Heidelberg (2014)MATHGoogle Scholar
  15. 15.
    Ruuth, S.J., Hundsdorfer, W.: High-order linear multistep methods with general monotonicity and boundedness properties. J. Comput. Phys. 209(1), 226–248 (2005). doi: 10.1016/ MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Spijker, M.N.: The existence of stepsize-coefficients for boundedness of linear multistep methods. Appl. Numer. Math. 63, 45–57 (2013). doi: 10.1016/j.apnum.2012.09.005 MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Numerical AnalysisEötvös Loránd University (ELTE)BudapestHungary

Personalised recommendations