TY - RPRT
T1 - BV instability for the Lax-Friedrichs scheme
Y1 - 2007
A1 - Paolo Baiti
A1 - Alberto Bressan
A1 - Helge Kristian Jenssen
AB - 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.
UR - http://hdl.handle.net/1963/2335
U1 - 1681
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - An instability of the Godunov scheme
JF - Comm. Pure Appl. Math. 59 (2006) 1604-1638
Y1 - 2006
A1 - Alberto Bressan
A1 - Helge Kristian Jenssen
A1 - Paolo Baiti
AB - 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.
UR - http://hdl.handle.net/1963/2183
U1 - 2061
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - Uniqueness of classical and nonclassical solutions for nonlinear hyperbolic systems
JF - J. Differential Equations 172 (2001) 59-82
Y1 - 2001
A1 - Paolo Baiti
A1 - Philippe G. LeFloch
A1 - Benedetto Piccoli
PB - Elsevier
UR - http://hdl.handle.net/1963/3113
U1 - 1220
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - Nonclassical Shocks and the Cauchy Problem for Nonconvex Conservation Laws
JF - J. Differential Equations 151 (1999) 345-372
Y1 - 1999
A1 - Debora Amadori
A1 - Paolo Baiti
A1 - Philippe G. LeFloch
A1 - Benedetto Piccoli
AB - 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.
PB - Elsevier
UR - http://hdl.handle.net/1963/3312
U1 - 1018
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - THES
T1 - On Existence and Continuous Dependence for Systems of Conservation Laws
Y1 - 1997
A1 - Paolo Baiti
KW - Conservation laws
PB - SISSA
UR - http://hdl.handle.net/1963/5588
U1 - 5418
U2 - Mathematics
U3 - Functional Analysis and Applications
U4 - -1
ER -
TY - JOUR
T1 - The semigroup generated by a temple class system with large data
JF - Differential Integral Equations 10 (1997), no. 3, 401-418
Y1 - 1997
A1 - Paolo Baiti
A1 - Alberto Bressan
AB - 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.
PB - SISSA Library
UR - http://hdl.handle.net/1963/1023
U1 - 2833
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -