plementing the characteristic equations with appropriate jump relations, i.e., by solving the corresponding local Rie-
The present work is concerned with an application of the theory of characteristics to conservation laws with source terms in one mann problem.
space dimension, such as the Euler equations for reacting flows.
Research in detonating flows was pioneered by Von
Space-time paths are introduced on which the flow/chemistry equa-Neumann [55], Zeldovich [57], and Doering [14] 50 years tions decouple to a characteristic set of ODE's for the corresponding ago and, subsequently, by others. Numerical integration homogeneous laws, thus allowing the introduction of functions of the governing equations, in high-resolution meshes, was analogous to the Riemann invariants in classical theory. The geometry of these paths depends on the spatial gradients of the solution. initiated by Fickett and Wood [18]. In the past, progress This particular decomposition can be used in the design of efficient was achieved in the study of the stability (see, e.g., Erpenunsplit algorithms for the numerical integration of the equations. beck [16], Lee and Stuart [29]) and the high-and low-As a first step, these ideas are implemented for the case of a scalar frequency asymptotic nature of detonations (see, e.g., conservation law with a nonlinear source term. The resulting algorithm belongs to the class of MUSCL-type, shock-capturing DiPerna and Majda [13], Majda and Rosales [36, 37], Choi schemes. Its accuracy and robustness are checked through a series and Majda [8], Majda and Roytburd [38], Kapila et al. [23]).
of tests. The stiffness of the source term is also studied. Then, Accurate algorithms for gas dynamics were first emthe algorithm is generalized for a system of hyperbolic equations, ployed in detonation problems in the late 1980s, using namely the Euler equations for reacting flows. A numerical study splitting techniques (see, e.g., Colella et al. [9], and Yee of unstable detonations is performed.