841 ISSN 0965-5425, Computational Mathematics and Mathematical Physics, 2016, Vol. 56, No. 5, pp. 841–853. © Pleiades Publishing, Ltd., 2016. Original Russian Text © A.I. Lopato, P.S. Utkin, 2016, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2016, Vol. 56, No. 5, pp. 856–868. Detailed Simulation of the Pulsating Detonation Wave in the Shock-Attached Frame A. I. Lopato a,b and P. S. Utkin c a Institute for Computer Aided Design, Russian Academy of Sciences, Vtoraya Brestskaya ul. 19/18, Moscow, 123056 Russia b Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudnyi, Moscow oblast, 141700 Russia c Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia e-mail: lopato 2008@rambler.ru; pavel_utk@mail.ru Received May 29, 2015; in final form, September 8, 2015 Abstract—The paper is devoted to the numerical investigation of the stability of propagation of pulsat- ing gas detonation waves. For various values of the mixture activation energy, detailed propagation pat- terns of the stable, weakly unstable, irregular, and strongly unstable detonation are obtained. The mathematical model is based on the Euler system of equations and the one-stage model of chemical reaction kinetics. The distinctive feature of the paper is the use of a specially developed computational algorithm of the second approximation order for simulating detonation wave in the shock-attached frame. In distinction from shock capturing schemes, the statement used in the paper is free of compu- tational artifacts caused by the numerical smearing of the leading wave front. The key point of the computational algorithm is the solution of the equation for the evolution of the leading wave velocity using the second-order grid-characteristic method. The regimes of the pulsating detonation wave propagation thus obtained qualitatively match the computational data obtained in other studies and their numerical quality is superior when compared with known analytical solutions due to the use of a highly accurate computational algorithm. Keywords: pulsating detonation wave, mathematical modeling, activation energy, ENO scheme, grid- characteristic method, Euler equations. DOI: 10.1134/S0965542516050134 1. INTRODUCTION Detonation is a hydrodynamic wave process of the supersonic propagation of an exothermic reaction through a substance. The detonation wave is a self-sustained shock-wave discontinuity behind the front of which a chemical reaction is continuously initiated due to heating caused by adiabatic compression. In other words, the detonation wave (DW) is a complex consisting of the leading wave (LW) and attached chemical reaction zone. It is known from experimental (see [1]) and numerical (see [2]) studies that the propagation of the DW in space is characterized by a complicated nonlinear oscillatory process that involves: • pulsation of parameters behind the DW front in one-dimensional computations; • transverse compression waves that interact with the DW in two-dimensional computations and experiments on the propagation of DWs in narrow gaps; • transverse wave propagating on a spiral—spin in the three-dimensional case. The instability of the DW front is due to the strong interrelation between the rate of chemical reactions and the LW intensity. Indeed, the rate of chemical reactions ω depends on the temperature of the gas immediately behind the LW front—the von Neumann temperature T vN —by the Arrhenius law The dimensionless activation energy of the mixture E/(RT vN ) is the key parameter in theoretical studies of the stability of the DW propagation.   ω -     v ~ exp . N E RT