  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.
