On stability issues for IMEX schemes applied to 1D scalar hyperbolic equations with stiff reaction terms

The application of a Method of Lines to a hyperbolic PDE with source terms gives rise to a system of ODEs containing terms that may have very different stiffness properties. In this case, Implicit-Explicit Runge-Kutta (IMEX-RK) schemes are particularly useful as high order time integrators because they allow an explicit handling of the convective terms, which can be discretized using the highly developed shock capturing technology, together with an implicit treatment of the source terms, necessary for stability reasons. Motivated by the structure of the source term in a model problem introduced by LeVeque and Yee in [J. Comput. Phys. 86 (1990)], in this paper we study the preservation of certain invariant regions as a weak stability constraint. For the class of source terms considered in this paper, the unit interval is an invariant region for the model balance law. In the first part of the paper, we consider first order time discretizations, which are the basic building blocks of higher order IMEX-RK schemes, and study the conditions that guarantee that [0, 1] is also an invariant region for the numerical scheme. In the second part of the paper, we study the conditions that ensure the preservation of this property for higher order IMEX schemes.