Wave dynamics of linear hyperbolic relaxation systems

Abstract

We consider linear hyperbolic systems with a stable rank 1 relaxation term and establish that the characteristic polynomial for the individual Fourier components of the solution can be written as a convex combination of the characteristic polynomials for the formal stiff and non-stiff limits. This allows us to provide a direct and elementary proof of the equivalence between linear stability and the subcharacteristic condition. In a similar vein, a maximum principle follows: The velocity of each individual Fourier component is bounded by the minimum and maximum eigenvalues of the non-stiff limit system.