Journal of Scientific Computing

, Volume 74, Issue 1, pp 244–266 | Cite as

Positivity for Convective Semi-discretizations

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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.

Keywords

Positivity Runge–Kutta Total variation diminishing Strong stability preserving 
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Notes

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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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Imre Fekete
    • 1
  • David I. Ketcheson
    • 1
  • Lajos Lóczi
    • 1
  1. 1.Division of Computer, Electrical, and Mathematical Sciences & EngineeringKing Abdullah University of Science and Technology (KAUST)ThuwalSaudi Arabia

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