Additive semi-implicit Runge-Kutta methods for computing high-speed nonequilibrium reactive flows

This paper is concerned with time-stepping numerical methods for computing stiff semi-discrete systems of ordinary differential equations for transient hypersonic flows with thermo-chemical non-equilibrium. The stiffness of the equations is mainly caused by the viscous flux terms across the boundary layers and by the source terms modeling finite-rate thermo-chemical processes. Implicit methods are needed to treat the stiff terms while more efficient explicit methods can still be used for the nonstiff terms in the equations. This paper studies three different semi-implicit Runge-Kutta methods for additively split differential equations in the form of u' = f(u) + g(u), where f is treated by explicit Runge-Kutta methods and g is simultaneously treated by three implicit Runge-Kutta methods: a diagonally implicit Runge-Kutta method and two linearized implicit Runge-Kutta methods. The coefficients of up to third-order accurate additive semi-implicit Runge-Kutta methods have been derived such that the methods are both high-order accurate and strongly A-stable for the implicit terms. The results of two numerical tests on the stability and accuracy properties of these methods are also presented in the paper. 26 refs., 10 figs., 2 tabs