Semi-implicit Runge-Kutta schemes for stiff multi-dimensional reacting flows

Abstract

Reacting flow calculations are stiff for time-stepping. Recently, Zhong derived three sets of semi-implicit Runge-Kutta(SIRK) methods for split ordinary differential equations in the form of u' = /(«) 4g(u), where the nonstiff term / is treated explicitly while the stiff term g is simultaneously treated implicitly. The purpose of this work is twofold: first, to simulate multidimensional detonation waves using the high-order Rosenbrock semi-implicit Runge-Kutta time-stepping scheme(SIRK-3C); secondly to quantitatively compare the performance of SIRK with the second-order timesplitting schemes in dealing with the hyperbolic conservation laws with stiff relaxation terms. In the simulation of two-dimensional detonations, a full set of elementary hydrogen-oxygen kinetics with nine species are linear-implicitly treated while the explicit discretization of the basic advection equations are achieved by the third-order ENO schemes. Several model problems of hyperbolic systems with stiff relaxation are considered in the comparison of the Time-Splitting and the SIRK-3C scheme. A set of low-storage SIRK(LSSIRK) schemes is derived and tested to be high-order accurate and strongly A-stable. The results show that high-order SIRK time-stepping methods are suitable for stiff reactive flow simulations like the multi-dimensional detonations. INTRODUCTION In the reactive flow field solutions and the combustion related problems, the existence of several nonequilibrium states impose additional difficulties in the solutions of the reactive Euler equations with stiff relaxation terms. Since the smallest time scale is introduced from the chemical reaction kinetics, the second-order time-splitting method of Strang^ allows an independent calculation of the stiff ODE for the source term via implicit methods in one step, and an explicit highorder convective calculation in the following time steps. By the main source of stiffness from the basic hyperbolic conservation laws, it can accomplish a •Graduate Student, Member AIAA 'Assistant Professor, Mechanical and Aerospace Engineering Department, Member AIAA Copyright ©1997 by American Institute of Aeronautics and Astronautics, Inc. AH rights reserved. robust and easy-to-implement computation of reactive flow of current interests. Despite the second-order in time accuracy and the robustness of these methods, there are two main difficulties with the stiffness which have been minimally handled. The spurious solutions are one of the often observed phenomena, which are stable and free of oscillations, and yet may be completely incorrect.'-' Some type of resolution technique is needed to treat this known difficulties.^'!] The degrading of the second-order time accuracy to the first-order is another and quite recent observation when there exist solutions with many small-scaled structures of high-degree of complication. Jin® has modified the Strang splitting into the secondorder time-splitting methods which preserve its highorder accuracy when the considered solutions are complicated. New third-order semi-implicit Rosenbrock type Runge-Kutta scheme, based on the original version of Zhong'---', is computationally efficient with the linearized source term integration, thus requiring no iterations during the implicit treatment of the stiff source term. Third-order ENO schemes discretize the convective fluxes such that both implicit and explicit terms are treated at each of the three Runge-Kutta stages. This high-order Runge-Kutta scheme(SIRK-3C) still possesses the typical under-resolved characteristics of unphysical spurious numerical results of the TimeSplitting method. However the reduction to lower order if the small relaxation time is not temporally wellresolved, is successfully removed. Multi-dimensional Detonations Detonation waves are multi-dimensional and unstable phenomenon in nature as demonstrated by the early experiments of Urtiew and Oppenheinv-'. Existence of the triple points'^, consisting of an incident shock, a reflecting shock and a Mach stem, is the main characteristic of the reacting region behind the propagating shock front, and these detaching triple points from the leading front further contributes to the rolling up of vortices of opposite strength. One distinctive feature of the instability process observed in experiments is the formation of regular cell structures as triple points collide as they come together in the incident shock and move away from each other in the newly formed Mach stem. Figure[13] depicts this process of triple point col-