A Modified Higher Order Godunov's Scheme for Stiff Source Conservative   Hydrodynamics
  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. 
Francesco Miniati (ETH, Zuerich), Phillip Colella (LBL, Berkeley)