New Hybrid Runge-Kutta Methods for Unsteady Reactive Flow Simulation

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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 ,V ol. 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.