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We develop a version of Haar and Holmgren methods which applies to discontinuous solutions of nonlinear hyperbolic systems and allows us to control the L1 distance between two entropy solutions. The main difficulty is to cope with linear hyperbolic systems with discontinuous coefficients. Our main observation is that, while entropy solutions contain compressive shocks only, the averaged matrix associated with two such solutions has compressive or undercompressive shocks, but no rarefaction-shocks -- which are recognized as a source for non-uniqueness and instability. Our Haar-Holmgren-type method rests on the geometry associated with the averaged matrix and takes into account adjoint problems and wave cancellations along generalized characteristics. It extends the method proposed earlier by LeFloch et al. for genuinely nonlinear systems. In the present paper, we cover solutions with small total variation and a class of systems with general flux that need not be genuinely nonlinear and includes for instance fluid dynamics equations. We prove that solutions generated by Glimm or front tracking schemes depend continuously in the L1 norm upon their initial data, by exhibiting an L1 functional controling the distance between two solutions.
The ubiquitous Lanczos method can approximate $f(A)x$ for any symmetric $n times n$ matrix $A$, vector $x$, and function $f$. In exact arithmetic, the methods error after $k$ iterations is bounded by the error of the best degree-$k$ polynomial unifor
In this article, we investigate the determination of the spatial component in the time-dependent second order coefficient of a hyperbolic equation from both theoretical and numerical aspects. By the Carleman estimates for general hyperbolic operators
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