Related papers: Stability estimates for non-local scalar conservat…
In our previous papers we proposed a continuum model for the dynamics of the systems of self-propelling particles with conservative kinematic constraints on the velocities. We have determined a class of stationary solutions of this…
A recent study by Gassner et al. [J. Sci. Comput. 90:79 (2022)] demonstrates that local energy stability--that is, ensuring the asymptotic numerical growth rate does not exceed the continuous growth rate--is crucial for achieving accurate…
Consistency and stability are two essential ingredients in the design of numerical algorithms for partial differential equations. Robust algorithms can be developed by incorporating nonlinear physical stability principles in their design,…
We present a Godunov type numerical scheme for a class of scalar conservation laws with non-local flux arising for example in traffic flow models. The proposed scheme delivers more accurate solutions than the widely used Lax-Friedrichs type…
In this article, we explore some of the main mathematical problems connected to multidimensional fractional conservation laws driven by L\'evy processes. Making use of an adapted entropy formulation, a result of existence and uniqueness of…
We study the finite volume approximation of strong solutions to nonlinear systems of conservation laws. We focus on time-explicit schemes on unstructured meshes, with entropy satisfying numerical fluxes. The numerical entropy dissipation is…
Entropy stabilization of the compressible Euler system is achieved by adapting the averages that are applied to the density and internal energy variables. The approach achieves non-linear robustness despite the use of simplified symmetric…
High order entropy stable schemes provide improved robustness for computational simulations of fluid flows. However, additional stabilization and positivity preserving limiting can still be required for variable-density flows with…
We study one-dimensional conservation law. We develop a simple numerical method for computing the unique entropy admissible weak solution to the initial problem. The method basis on the equal-area principle and gives the solution for given…
Nonlinear scalar conservation laws are traditionally viewed as transport equations. We take instead the viewpoint of these PDEs as continuity equations with an implicitly defined velocity field. We show that a weak solution is the entropy…
The input/output stability of an interconnected system composed of an ordinary differential equation and a damped string equation is studied. Issued from the literature on time-delay systems, an exact stability result is firstly derived…
The paper is concerned with the boundary controllability of entropy weak solutions to hyperbolic systems of conservation laws. We prove a general result on the asymptotic stabilization of a system near a constant state. On the other hand,…
In this article, we consider inverse problems of determining a source term and a coefficient of a first-order partial differential equation and prove conditional stability estimates with minimum boundary observation data and relaxed…
We prove existence and uniqueness of solutions to a transport equation modelling vehicular traffic in which the velocity field depends non-locally on the downstream traffic density via a discontinuous anisotropic kernel. The result is…
We study a scalar integro-differential conservation law. The equation was first derived in [2] as the slow erosion limit of granular flow. Considering a set of more general erosion functions, we study the initial boundary value problem for…
This paper is concerned with the stability of stationary solutions of the conservation law $\partial_t u + \mathrm{div}_y A(y,u) -\Delta_y u=0$, where the flux $A$ is periodic with respect to its first variable. Essentially two kinds of…
In this paper, we consider scalar conservation laws with smoothly varying spatially heterogeneous flux that is convex in the conserved variable. We show that under certain assumptions, a shock wave connecting two constant states emerges in…
Global entropy solutions in $BV$ for a scalar nonlocal conservation law with fading memory are constructed as limits of vanishing viscosity approximate solutions. The uniqueness and stability of entropy solutions in $BV$ are established,…
In this paper we consider scalar conservation laws with a convex flux. Given a stationnary shock, we provide a feedback law acting at one boundary point such that this solution is now asymptotically stable in L 1-norm in the class of…
Many important initial value problems have the property that energy is non-increasing in time. Energy stable methods, also referred to as strongly stable methods, guarantee the same property discretely. We investigate requirements for…