%0 Report
%D 2007
%T BV instability for the Lax-Friedrichs scheme
%A Paolo Baiti
%A Alberto Bressan
%A Helge Kristian Jenssen
%X It 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.
%G en_US
%U http://hdl.handle.net/1963/2335
%1 1681
%2 Mathematics
%3 Functional Analysis and Applications
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-11-05T09:55:21Z\\nNo. of bitstreams: 1\\n0502043v1.pdf: 257155 bytes, checksum: 5055aa260394ac1339fa775b656f3b15 (MD5)
%0 Journal Article
%J Comm. Pure Appl. Math. 59 (2006) 1604-1638
%D 2006
%T An instability of the Godunov scheme
%A Alberto Bressan
%A Helge Kristian Jenssen
%A Paolo Baiti
%X We 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.
%B Comm. Pure Appl. Math. 59 (2006) 1604-1638
%G en_US
%U http://hdl.handle.net/1963/2183
%1 2061
%2 Mathematics
%3 Functional Analysis and Applications
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-05T07:52:50Z\\nNo. of bitstreams: 1\\n0502125v1.pdf: 337578 bytes, checksum: 41193faac363c00e9f80d6c05c0b098c (MD5)
%R 10.1002/cpa.20141
%0 Journal Article
%J J. Differential Equations 172 (2001) 59-82
%D 2001
%T Uniqueness of classical and nonclassical solutions for nonlinear hyperbolic systems
%A Paolo Baiti
%A Philippe G. LeFloch
%A Benedetto Piccoli
%B J. Differential Equations 172 (2001) 59-82
%I Elsevier
%G en_US
%U http://hdl.handle.net/1963/3113
%1 1220
%2 Mathematics
%3 Functional Analysis and Applications
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-10-15T10:27:54Z\\nNo. of bitstreams: 1\\nuniqueness.pdf: 278381 bytes, checksum: d70f89c690a2695e7b44e73737a6aff8 (MD5)
%R 10.1006/jdeq.2000.3869
%0 Journal Article
%J J. Differential Equations 151 (1999) 345-372
%D 1999
%T Nonclassical Shocks and the Cauchy Problem for Nonconvex Conservation Laws
%A Debora Amadori
%A Paolo Baiti
%A Philippe G. LeFloch
%A Benedetto Piccoli
%X The 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.
%B J. Differential Equations 151 (1999) 345-372
%I Elsevier
%G en_US
%U http://hdl.handle.net/1963/3312
%1 1018
%2 Mathematics
%3 Functional Analysis and Applications
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-11-20T11:49:26Z\\nNo. of bitstreams: 1\\nNonclassical_shocks.pdf: 261875 bytes, checksum: bd41bb6490895996b965941b1eeb6797 (MD5)
%R 10.1006/jdeq.1998.3513
%0 Thesis
%D 1997
%T On Existence and Continuous Dependence for Systems of Conservation Laws
%A Paolo Baiti
%K Conservation laws
%I SISSA
%G en
%U http://hdl.handle.net/1963/5588
%1 5418
%2 Mathematics
%3 Functional Analysis and Applications
%4 -1
%$ Submitted by Gerardina Cargnelutti (gerry@sissa.it) on 2012-03-08T13:48:00Z\\nNo. of bitstreams: 1\\nPhD_Baiti_Paolo.pdf: 5888073 bytes, checksum: fadab5be3940cd6a300d30600eb61194 (MD5)
%0 Journal Article
%J Differential Integral Equations 10 (1997), no. 3, 401-418
%D 1997
%T The semigroup generated by a temple class system with large data
%A Paolo Baiti
%A Alberto Bressan
%X We 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.
%B Differential Integral Equations 10 (1997), no. 3, 401-418
%I SISSA Library
%G en
%U http://hdl.handle.net/1963/1023
%1 2833
%2 Mathematics
%3 Functional Analysis and Applications
%$ Made available in DSpace on 2004-09-01T12:41:49Z (GMT). No. of bitstreams: 0\\n Previous issue date: 1995