Positivity for convective semi-discretizations

TL;DR

This paper introduces a technique to analyze the stability and positivity preservation of semi-discretizations for hyperbolic conservation laws, revealing limitations of high-order Runge-Kutta methods in maintaining positivity.

Contribution

It generalizes existing methods to assess positivity in semi-discretizations and establishes sharp step-size conditions, showing many high-order methods cannot preserve positivity under any positive step size.

Findings

Many high-order explicit Runge-Kutta methods cannot maintain positivity.

The proposed technique provides relaxed and sharp step-size conditions.

Application to centered finite difference discretizations extends the analysis.

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.