Positivity for Convective Semi-discretizations

Imre Fekete, David I. Ketcheson, Lajos Loczi

Research output: Contribution to journalArticlepeer-review

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.
Original languageEnglish (US)
Pages (from-to)244-266
Number of pages23
JournalJournal of Scientific Computing
Volume74
Issue number1
DOIs
StatePublished - Apr 19 2017

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