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 - On the spreading of characteristics for non-convex conservation laws
JF - Proc. Roy. Soc. Edinburgh Sect. A 131 (2001) 909-925
Y1 - 2001
A1 - Helge Kristian Jenssen
A1 - Carlo Sinestrari
AB - We study the spreading of characteristics for a class of one-dimensional scalar conservation laws for which the flux function has one point of inflection. It is well known that in the convex case the characteristic speed satisfies a one-sided Lipschitz estimate. Using Dafermos\\\' theory of generalized characteristics, we show that the characteristic speed in the non-convex case satisfies an HÃ¶lder estimate. In addition, we give a one-sided Lipschitz estimate with an error term given by the decrease of the total variation of the solution.
PB - Cambridge University Press
UR - http://hdl.handle.net/1963/3265
U1 - 1436
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - On the convergence of Godunov scheme for nonlinear hyperbolic systems
JF - Chinese Ann. Math. B, 2000, 21, 269
Y1 - 2000
A1 - Alberto Bressan
A1 - Helge Kristian Jenssen
PB - SISSA Library
UR - http://hdl.handle.net/1963/1473
U1 - 2690
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - Blowup asymptotics for scalar conservation laws with a source
JF - Comm. in Partial Differential Equations 24 (1999) 2237-2261
Y1 - 1999
A1 - Helge Kristian Jenssen
A1 - Carlo Sinestrari
PB - Taylor and Francis
UR - http://hdl.handle.net/1963/3482
U1 - 782
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -