BV instability for the Lax-Friedrichs scheme
Paolo Baiti, Alberto Bressan, Helge Kristian Jenssen
TL;DR
The paper studies how discrete shock profiles (DSPs) for the Lax-Friedrichs scheme depend on the shock speed in the $BV$ norm. It proves that DSPs do not necessarily depend continuously in $BV$ on their speed for systems of conservation laws. By constructing explicit examples for $2×2$ systems, it shows sequences of DSPs with speeds converging to a rational number, where a resonance phenomenon makes the limit DSP differ from any member of the sequence by an order-one amount in $BV$. This reveals an intrinsic non-smooth dependence of discrete shock profiles on speed, with implications for the accuracy and stability of numerical schemes for conservation laws.
Abstract
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.