Semi-implicit Runge-kutta Schemes for Non-autonomous Diierential Equations in Reactive Flow Computations

This paper is concerned with time-stepping numerical methods for computing stii semi-discrete systems of ordinary diierential equations for transient hyper-sonic ows with thermo-chemical nonequilibrium. The stiiness of the equations is mainly caused by the vis-cous ux terms across the boundary layers and by the source terms modeling nite-rate thermo-chemical processes. Implicit methods are needed to treat the stii terms while more eecient explicit methods can still be used for the nonstii terms in the equations. For additively split autonomous diierential equations in the form of u 0 = f(u) + g(u), three diierent semi-implicit Runge-Kutta methods have been derived and tested in previous papers, where f is treated by explicit Runge-Kutta methods and g is simultaneously treated by three implicit Runge-Kutta methods. The coeecients 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 non-autonomous systems of u 0 = f(t; u) + g(t; u) because of the coupling between the f and g terms in the split Runge-Kutta methods. In this paper, we derive and test three diierent semi-implicit Runge-Kutta schemes of up to third-order accuracy for the non-autonomous diierential equations using the A-stability and accuracy conditions with four stages. The new schemes have been tested in computations of unsteady reactive ows with explicit time-dependent terms.