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 Hölder 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 -