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 ref. 12. 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.
Cite
@article{arxiv.1610.00228,
title = {Positivity for convective semi-discretizations},
author = {Imre Fekete and David I. Ketcheson and Lajos Lóczi},
journal= {arXiv preprint arXiv:1610.00228},
year = {2017}
}