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PAPALEXANDRIS, M. V., LEONARD, A., & DIMOTAKIS, P. E. 1996
"Unsplit Schemes for Hyperbolic Conservation Laws with Source Terms in
One Space Dimension,"
*J. Comp. Phys.* **134**, 31-61.

*Abstract*

The present work is concerned with an application of the theory of
characteristics to conservation laws with source terms in one space
dimension, such as the Euler equations for reacting flows. Space-time
paths are introduced on which the flow/chemistry equations decouple to
a characteristic set of O.D.E's for the corresponding homogeneous laws,
thus allowing the introduction of functions analogous to the Riemann
Invariants in the classical theory.
The geometry of these paths depends on the spatial gradients of the
solution.
This particular decomposition can be used in the design of efficient
unsplit algorithms for the numerical integration of the equations. As a
first step, these ideas are implemented for the case of a scalar
conservation law with a nonlinear source term. The resulting algorithm
belongs to the class of MUSCL-type, shock-capturing schemes. Its
accuracy and robustness are checked through a series of tests. The
stiffness of the source term is also studied.
Then, the algorithm is generalized for a system of hyperbolic
equations, namely the Euler equations for reacting flows. A numerical
study of unstable detonations is performed.