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Positivity for Convective Semi-discretizations

Abstract

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

Notes

  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 \([0,1]^{10}\).

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Acknowledgements

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

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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.

figurea

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.

figureb

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

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Keywords

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