Semi-Implicit Runge-Kutta Schemes for non-autonomous differential equations in reactive flow computations

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 nonequilibrium. 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. For additively split autonomous differential equations in the form of u' = f ( u ) + g(u), three different semiimplicit Runge-Kutta methods have been derived and tested in previous papers, where / is treated by explicit Runge-Kutta methods and g is simultaneously treated by three implicit Runge-Kutta methods. The coefficients of up to third-order accuracy have been derived such that the methods are both high-order accurate and strongly A-stable for the implicit terms. However, these semi-implicit Runge-Kutta methods for the autonomous systems cannot be extended to nonautonomous systems of u' = f ( t , u) + g(t, u) because of the coupling between the / and g terms in the split Runge-Kutta methods. In this paper, we derive and test three different semi-implicit Runge-Kutta schemes of up to third-order accuracy for the non-autonomous differential equations using the A-stability and accuracy conditions with four stages. The new schemes have been tested in computations of unsteady reactive flows with explicit time-dependent terms.