Positivity for Convective Semi-discretizations


We propose a technique for investigating stability properties like positivity and forward invariance of an interval for method-of-lines discretizations, and apply the technique to study positivity preservation for a class of TVD semi-discretizations of 1D scalar hyperbolic conservation laws. This technique is a generalization of the approach suggested in Khalsaraei (J Comput Appl Math 235(1): 137–143, 2010). We give more relaxed conditions on the time-step for positivity preservation for slope-limited semi-discretizations integrated in time with explicit Runge–Kutta methods. We show that the step-size restrictions derived are sharp in a certain sense, and that many higher-order explicit Runge–Kutta methods, including the classical 4th-order method and all non-confluent methods with a negative Butcher coefficient, cannot generally maintain positivity for these semi-discretizations under any positive step size. We also apply the proposed technique to centered finite difference discretizations of scalar hyperbolic and parabolic problems.

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Fig. 1
Fig. 2


  1. 1.

    Please note that the derivation of this formula in [12, Section 2] (denoted by \(\gamma (\kappa )\) there) contains some inconsistencies.

  2. 2.

    We thank Zoltán Horváth (Széchenyi István University, Hungary) for pointing this out.

  3. 3.

    Our Proposition 8 seems to directly contradict Theorem 1 in [13]. To explain the discrepancy, note that the polynomial \(P_3\) in our proof becomes negative along a 9-dimensional hyperface of the hypercube \([0,\varepsilon ]^{10}\) for any \(\varepsilon >0\); in [13] it seems that the non-negativity of the corresponding (but slightly different) polynomial was checked only at the vertices of the hypercube \([0,1]^{10}\).


  1. 1.

    Bolley, C., Crouzeix, M.: Conservation de la positivité lors de la discrétisation des problémes d’évolution paraboliques. RAIRO Anal. Numér. 12(3), 237–245 (1978)

  2. 2.

    Butcher, J.C.: Numerical Methods for Ordinary Differential Equations, 2nd edn. Wiley, Chichester (2008)

  3. 3.

    Dahlquist, G., Jeltsch, R.: Reducibility and contractivity of Runge–Kutta methods revisited. Bit Numer. Math. 46, 567–587 (2006)

  4. 4.

    Dormand, J.R., Prince, P.J.: A family of embedded Runge–Kutta formulae. J. Comput. Appl. Math. 6(1), 19–26 (1980)

  5. 5.

    Fehlberg, E.: Klassische Runge–Kutta-formeln fünfter und siebenter ordnung mit schrittweiten-kontrolle. Computing 4(2), 93–106 (1969)

  6. 6.

    Gottlieb, S., Ketcheson, D., Shu, C.-W.: Strong stability preserving Runge–Kutta and multistep time discretizations. World Scientific Publishing, Hackensack (2011)

  7. 7.

    Harten, A.: High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49, 357–393 (1983)

  8. 8.

    Higueras, I.: Strong stability for Runge-Kutta schemes on a class of nonlinear problems. J. Sci. Comput. 57(3), 518–535 (2013)

  9. 9.

    Hundsdorfer, W., Koren, B., van Loon, M., Verwer, J.G.: A positive finite-difference advection scheme. J. Comput. Phys. 117(1), 35–46 (1995)

  10. 10.

    Hundsdorfer, W., Verwer, J.: Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. Springer Series in Computational Mathematics, vol. 33. Springer, Berlin (2003)

  11. 11.

    Ketcheson, D.I., Robinson, A.C.: On the practical importance of the SSP property for Runge–Kutta time integrators for some common Godunov-type schemes. Int. J. Numer. Methods Fluids 48(3), 271–303 (2005)

  12. 12.

    Khalsaraei, M.M.: An improvement on the positivity results for 2-stage explicit Runge–Kutta methods. J. Comput. Appl. Math. 235(1), 137–143 (2010)

  13. 13.

    Khalsaraei, M.M.: Positivity of an explicit Runge–Kutta method. Ain Shams Eng. J. 6(4), 1217–1223 (2015)

  14. 14.

    Koren, B.: A robust upwind discretization method for advection, diffusion and source terms. In: Numerical methods for advection-diffusion problems, volume 45 of Notes Numer. Fluid Mech., pp. 117–138. Vieweg, Braunschweig (1993)

  15. 15.

    Kraaijevanger, J.F.B.M.: Contractivity of Runge–Kutta methods. BIT Numer. Math. 31(3), 482–528 (1991)

  16. 16.

    Ralston, A.: Runge–Kutta methods with minimum error bounds. Math. Comput. 16, 431–437 (1962)

  17. 17.

    Roe, P.L.: Characteristic-based schemes for the Euler equations. In: Annual Review of Fluid Mechanics, vol. 18, pp. 337–365. Annual Reviews, Palo Alto, CA (1986)

  18. 18.

    Ruuth, S. J., Spiteri, R. J.: Two barriers on strong-stability-preserving time discretization methods. In: Proceedings of the 5th International Conference on Spectral and High Order Methods (ICOSAHOM-01) (Uppsala), vol. 17, pp. 211–220 (2002)

  19. 19.

    van Leer, B.: Towards the ultimate conservative difference scheme. III: upstream-centered finite-difference schemes for ideal compressible flow. IV: a new approach to numerical convection. J. Comput. Phys. 23, 263–299 (1977)

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We are indebted to the referees of the manuscript for their suggestions that helped us improving the presentation of the material.

Author information

Correspondence to David I. Ketcheson.

Additional information

This work was supported by the King Abdullah University of Science and Technology (KAUST), 4700 Thuwal, 23955-6900, Saudi Arabia. The first author was also supported by the Tempus Public Foundation. The third author was also supported by the Department of Numerical Analysis, Eötvös Loránd University, and the Department of Differential Equations, Budapest University of Technology and Economics, Hungary.

Appendix: Some Mathematica Code

Appendix: Some Mathematica Code

Generating the Multivariable Polynomials

The first cell below contains the definition of a Mathematica function ERKpolynomials for generating the multivariable polynomials in (14). The two arguments A and b correspond to the Butcher tableau of the ERK method, and the output is a list of the \(m+1\) polynomials \(P_0\), ..., \(P_m\) in the variables \(\xi _\ell ^j\). Note that the superscripts in \(\xi _\ell ^j\) are not exponents; the symbols \(\xi _\ell ^j\) with different sub- or superscripts denote different variables.

The second cell illustrates how to obtain the \(4+1=5\) polynomials in \(\frac{4\cdot (4+1)}{2}=10\) variables corresponding to the classical ERK(4,4) method.


Non-negativity of Polynomials at the Vertices of a Hypercube

Here we provide a simple Mathematica code to test the non-negativity of a multivariable polynomial by evaluating it at each vertex of a hypercube. In this particular example, the polynomial \(P_{-1}(x,y,z,u)\) from Sect. 6.2 is evaluated at the \(2^4\) vertices of the hypercube \([0,\varepsilon ]^4\) with some \(\varepsilon >0\), and the resulting system of 16 inequalities \(P_{-1}\ge 0\) is solved.


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Fekete, I., Ketcheson, D.I. & Lóczi, L. Positivity for Convective Semi-discretizations. J Sci Comput 74, 244–266 (2018) doi:10.1007/s10915-017-0432-9

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  • Positivity
  • Runge–Kutta
  • Total variation diminishing
  • Strong stability preserving