Additive Semi-Implicit Runge-Kutta Methods for Computing High-Speed Nonequilibrium Reactive Flows

This paper is concerned with time-stepping numerical methods for computing sti semi-discrete systems of ordinary di erential equations for transient hypersonic ows with thermo-chemical nonequilibrium. The sti ness of the equations is mainly caused by the viscous ux terms across the boundary layers and by the source terms modeling nite-rate thermo-chemical processes. Implicit methods are needed to treat the sti terms while more e cient explicit methods can still be used for the nonsti terms in the equations. This paper studies three di erent semi-implicit Runge-Kutta methods for additively split di erential equations in the form of u0 = 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 coe cients 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.