Congrès d'Analyse Numérique pour les Jeunes - 2020

Exponential BV stability for networks of scalar conservation laws

4 déc. 2020, 14:30

30m

Zoom (Virtuel)

Session parallèle 10

Orateur

Mathias Dus (Institut des mathématiques de Toulouse)

Description

In this presentation, we will talk about networks of $d\in \mathbb{N}$ scalar conservation laws with positive characteristic velocities. The interaction takes place at the boundary, where a feedback operator acts. The open loop system is given below with $H$ a square matrix given by the physics having a destabilizing effect:

\begin{equation}
\left{
$$\begin{array}{lllllll}{R}_t+[f(R){]}_x& =& 0R(t,0)& =& HR(t,1)+u(t)R(0,x)& =& R_0(x)\end{array}$$
\right.
\end{equation}

To stabilize this system, we design a feedback control of the form $u(t)=KR(1,t)$ where $K$ is a control gain to be designed. Such stabilization problem had been widely treated in the literature in various settings [2,3]. Nonetheless for the discretized version of the problem, it is far from being obvious that a control synthesized from the continuous theory stabilizes the discretized open-loop system.

In this talk, we focus on numerical aspects related to system (1). Using flux limiter schemes [1], we study the influence of the choice of the limiter on the $BV$ exponential stability of numerical solutions.

[1] Sweby, P. K., High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM Journal on Numerical Analysis, 1984.

[2] Bastin, G. and Coron, J.-M., Stability And Boundary Stabilization Of 1-D Hyperbolic Systems, Springer International Publishing, 2016.

[3] Coron, J.-M. and Ervedoza, S. and Ghoshal, S.S. and Glass, O. and Perrollaz, V., Dissipative boundary conditions for 2×2 hyperbolic systems of conservation laws for entropy solutions in BV, Journal of Differential Equations, 2017.