New Hybrid Runge–Kutta Methods for Unsteady Reactive Flow Simulation

In the numerical simulation of transient reacting flow, standard explicit calculation is prohibitively expensive because of the small time steps needed to address the stiffness of a governing differential system. To circumvent this, new hybrid implicit–explicit methods proposed treat the stiffness, whereas the underlying time-step control is governed by the Courant stability criterion. Because the coefficients of both explicit and implicit operations are entirely determined by solving the necessary conditions of accuracy and L stability without any extra assumptions, the methods are more generalized than other similar methods in the literature. Two families of semi-implicit Runge– Kutta schemes are developed for split differential equations in the form of u′ = f (t, u) + g(t, u), where f is treated explicitly and g is simultaneously treated implicitly. Like the rest of all the developed schemes, a low-storage family of semi-implicit schemes is also derived to be globally high-order accurate and L stable for implicit calculations. In a companion paper (Yoh, J. J., and Zhong, X., “New Hybrid Runge–Kutta Methods for Unsteady Reactive Flow Simulation: Applications,” AIAA Journal, Vol. 42, No. 8, 2004, pp. 1601–1611) the new schemes are tested to solve a wide range of applications in high-speed flow physics involving combustion.