Zhang et al. propose a new fractional-step method for the numerical solution of advection equations with stiff source terms. In general, it is too difficult to obtain satisfactory numerical approximate solutions for stiff reaction problems, because most of the classical numerical schemes produce non-physical oscillatory solutions.

With this method, the convection and reaction steps are solved separately. In order to solve the convection step, the authors use a second-order upwind scheme, whereas for the reaction step, an implicit ordinary differential equation (ODE) solver is used. Instead of using the cell averages, the authors use a two-equilibrium states reconstruction during the reaction step. This helps to capture the correct location of the reaction front even with the coarse meshes.

The proposed method preserves the order of accuracy of a corresponding standard method for smooth solutions. In addition, it captures the correct discontinuity speed, and one can extend this method to higher-dimensional problems. Several numerical examples are solved by this new method to show efficiency and accuracy.