  Hyperbolic conservation laws with stiff source terms appear in the study of a
variety of physical systems. Early work showed that the use of formally
second-order accurate semi-implicit methods could lead to a substantial loss of
accuracy, due to inconsistencies between the flux calculation without sources
and the limiting equilibrium behavior of the gas. In this paper we present an
efficient second order accurate scheme to treat stiff source terms within the
framework of higher order Godunov's methods. We employ Duhamel's formula to
devise a modified predictor step which accounts for the effects of stiff source
terms on the conservative fluxes and recovers the correct isothermal behavior
in the limit of an infinite cooling/reaction rate. Source term effects on the
conservative quantities are fully accounted for by means of a one-step, second
order accurate semi-implicit corrector scheme based on the deferred correction
method of Dutt et. al. We demostrate the accurate, stable and convergent
results of the proposed method through a set of benchmark problems for a
variety of stiffness conditions and source types.
