Summary: We present an efficient second-order accurate scheme to treat stiff source terms within the framework of higher-order Godunov methods. We employ Duhamel 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 A. Dutt
et al. [BIT 40, No. 2, 241–266 (2000; Zbl 0959.65084
)]. We demonstrate 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.