The equilibrium state method for hyperbolic conservation laws with stiff reaction terms

A new fractional-step method is proposed for numerical simulations of hyperbolic conservation laws with stiff source terms arising from chemically reactive flows. In stiff reaction problems, a well-known spurious numerical phenomenon, the incorrect propagation speed of discontinuities, may occur in general fractional-step algorithm due to the underresolved numerical solution in both space and time. The basic idea of the present proposed scheme is to replace the cell average representation with a two-equilibrium states reconstruction during the reaction step, which allows us to obtain the correct propagation of discontinuities for stiff reaction problems in an underresolved mesh. Because the definition of these two-equilibrium states for each transition cell is independent of its neighboring cells, the proposed method can be extended to multi-dimensional problems directly. In addition, this method is promising to deal with more complicated real-world problems after being extended to multi-species/multi-reactions system. Extensive numerical examples for one- and two-dimensional scalar and Euler system demonstrate the reliability and robustness of this novel method.