  We investigate the late-time asymptotic behavior of solutions to nonlinear
hyperbolic systems of conservation laws containing stiff relaxation terms.
First, we introduce a Chapman-Enskog-type asymptotic expansion and derive an
effective system of equations describing the late-time/stiff relaxation
singular limit. The structure of this new system is discussed and the role of a
mathematical entropy is emphasized. Second, we propose a new finite volume
discretization which, in late-time asymptotics, allows us to recover a discrete
version of the same effective asymptotic system. This is achieved provided we
suitably discretize the relaxation term in a way that depends on a
matrix-valued free-parameter, chosen so that the desired asymptotic behavior is
obtained. Our results are illustrated with several models of interest in
continuum physics, and numerical experiments demonstrate the relevance of the
proposed theory and numerical strategy.
