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\title{Error estimators for the position of discontinuities 
in hyperbolic conservation laws with source terms which are 
solved using operator splitting}
\author{Rolf Jeltsch \\
Petra Klingenstein, Seminar for Applied Mathematics, 
\\ETH Zurich, CH 8092 Zurich, Switzerland}

\begin{document}
\date{}
\maketitle
It is well known that solving nonlinear hyperbolic conservation
laws with stiff source terms using operator splitting may create 
wrong shock speeds if the timestep is to large. One first introduces 
the numerical position of the discontinuity. Then an error 
estimator for this position is derived. The main idea is 
developed for the Riemann problem of a scalar hyperbolic 
conservation law with a source term. In that situation the 
solution has a Taylor expansion on both sides of the 
discontinuity if the source and the flux funtions are 
smooth enough. Hence the position of the discontinuity is 
a smooth function and one can compare the Taylor expansions
of the exact and the numerical solution. The estimator is then
applied to Burgers equation with a stiff source term,
the combustion model of Majda and the reacting Euler equations 
in one and two space dimension. In two space dimensions the 
estimator is only used for discontinuities which are in space 
smooth curves. One simply applies the one dimensional estimator
in direction normal to this curve. The results are the contents 
of the Ph. D. thesis of P. Klingenstein
\end{document}

Rolf Jeltsch
Seminar for Applied Mathematics
CH-8092 Zuerich
Switzerland

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