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W. Auzinger, R. Frank, E. Weinmüller:
"On the efficient numerical integration of conservation laws with stiff source terms";
Report for Technical Report No.94/92, Institut für Angewandte und Numerische Mathematik; 1992.

We consider scalar 1D-conservation laws with stiff source terms, u_t + (f(u))_x = \psi(u). Such problems arise e.g. as simple models in the study of chemically reacting gas dynamics in combustion processes. While for pure conservation laws a variety of reliable numerical methods are known, the presence of a stiff source term causes severe additional difficulties. Several standard approaches (e.g. splitting into pure conservation laws and solution of stiff ODEs) fail unless extremely fine meshes are used. We propose an alternative method which successfully and efficiently solves certain model problems. The generalization of these ideas to more general problem classes remains to be studied.

http://www.math.tuwien.ac.at/~winfried/papers/ber94.pdf