Hyperbolic Systems with Relaxation: Characterization of Stiff Well-Posedness and Asymptotic Expansions

Abstract

The Cauchy problem for linear constant-coefficient hyperbolic systems ut+∑jA(j)uxj=(1/δ)Bu+Cu in d space dimensions is analyzed. Here (1/δ)Bu is a large relaxation term, and we are mostly interested in the critical case where B has a non-trivial null-space. A concept of stiff well-posedness is introduced that ensures solution estimates independent of 0<δ⪡1. Stiff well-posedness is characterized algebraically and—under mild assumptions on B—is shown to be equivalent to the existence of a limit of the L2-solution as δ→0. The evolution of the limit is governed by a reduced hyperbolic system, the so-called equilibrium system, which is related to the original system by a phase speed condition. We also show that stiff well-posedness—which is a weaker requirement than the existence of an entropy—leads to the validity of an asymptotic expansion. As an application, we consider a linearized version of a generic model of two-phase flow in a porous medium and show stiff well-posedness using a general result on strictly hyperbolic systems. To confirm the theory, the leading terms of the asymptotic expansion are computed and compared with a numerical solution of the full problem.