00820nas a2200121 4500008004300000245004900043210004800092520045600140100001700596700002100613700002800634856003600662 2007 en_Ud 00aBV instability for the Lax-Friedrichs scheme0 aBV instability for the LaxFriedrichs scheme3 aIt is proved that discrete shock profiles (DSPs) for the Lax-Friedrichs scheme for a system of conservation laws do not necessarily depend continuously in BV on their speed. We construct examples of $2 \\\\times 2$-systems for which there are sequences of DSPs with speeds converging to a rational number. Due to a resonance phenomenon, the difference between the limiting DSP and any DSP in the sequence will contain an order-one amount of variation.1 aBaiti, Paolo1 aBressan, Alberto1 aJenssen, Helge Kristian uhttp://hdl.handle.net/1963/233500891nas a2200121 4500008004300000245004100043210003800084520054500122100002100667700002800688700001700716856003600733 2006 en_Ud 00aAn instability of the Godunov scheme0 ainstability of the Godunov scheme3 aWe construct a solution to a $2\\\\times 2$ strictly hyperbolic system of conservation laws, showing that the Godunov scheme \\\\cite{Godunov59} can produce an arbitrarily large amount of oscillations. This happens when the speed of a shock is close to rational, inducing a resonance with the grid. Differently from the Glimm scheme or the vanishing viscosity method, for systems of conservation laws our counterexample indicates that no a priori BV bounds or $L^1$ stability estimates can in general be valid for finite difference schemes.1 aBressan, Alberto1 aJenssen, Helge Kristian1 aBaiti, Paolo uhttp://hdl.handle.net/1963/218300436nas a2200121 4500008004300000245008800043210006900131260001300200100001700213700002500230700002300255856003600278 2001 en_Ud 00aUniqueness of classical and nonclassical solutions for nonlinear hyperbolic systems0 aUniqueness of classical and nonclassical solutions for nonlinear bElsevier1 aBaiti, Paolo1 aLeFloch, Philippe G.1 aPiccoli, Benedetto uhttp://hdl.handle.net/1963/311301145nas a2200145 4500008004300000245007900043210006900122260001300191520067400204100002000878700001700898700002500915700002300940856003600963 1999 en_Ud 00aNonclassical Shocks and the Cauchy Problem for Nonconvex Conservation Laws0 aNonclassical Shocks and the Cauchy Problem for Nonconvex Conserv bElsevier3 aThe Riemann problem for a conservation law with a nonconvex (cubic) flux can be solved in a class of admissible nonclassical solutions that may violate the Oleinik entropy condition but satisfy a single entropy inequality and a kinetic relation. We use such a nonclassical Riemann solver in a front tracking algorithm, and prove that the approximate solutions remain bounded in the total variation norm. The nonclassical shocks induce an increase of the total variation and, therefore, the classical measure of total variation must be modified accordingly. We prove that the front tracking scheme converges strongly to a weak solution satisfying the entropy inequality.1 aAmadori, Debora1 aBaiti, Paolo1 aLeFloch, Philippe G.1 aPiccoli, Benedetto uhttp://hdl.handle.net/1963/331200381nas a2200109 4500008004100000245007600041210006900117260001000186653002200196100001700218856003600235 1997 en d00aOn Existence and Continuous Dependence for Systems of Conservation Laws0 aExistence and Continuous Dependence for Systems of Conservation bSISSA10aConservation laws1 aBaiti, Paolo uhttp://hdl.handle.net/1963/558800841nas a2200121 4500008004100000245006900041210006500110260001800175520045200193100001700645700002100662856003600683 1997 en d00aThe semigroup generated by a temple class system with large data0 asemigroup generated by a temple class system with large data bSISSA Library3 aWe consider the Cauchy problem $$u_t + [F(u)]_x=0, u(0,x)=\\\\bar u(x) (*)$$ for a nonlinear $n\\\\times n$ system of conservation laws with coinciding shock and rarefaction curves. Assuming the existence of a coordinates system made of Riemann invariants, we prove the existence of a weak solution of (*) that depends in a lipschitz continuous way on the initial data, in the class of functions with arbitrarily large but bounded total variation.1 aBaiti, Paolo1 aBressan, Alberto uhttp://hdl.handle.net/1963/1023