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Continuity of solutions of a quasilinear hyperbolic equation with hysteresis

Petra Kordulová (2012)

Applications of Mathematics

This paper is devoted to the investigation of quasilinear hyperbolic equations of first order with convex and nonconvex hysteresis operator. It is shown that in the nonconvex case the equation, whose nonlinearity is caused by the hysteresis term, has properties analogous to the quasilinear hyperbolic equation of first order. Hysteresis is represented by a functional describing adsorption and desorption on the particles of the substance. An existence result is achieved by using an approximation of...

Discontinuous Galerkin method with Godunov-like numerical fluxes for traffic flows on networks. Part II: Maximum principle

Lukáš Vacek, Chi-Wang Shu, Václav Kučera (2025)

Applications of Mathematics

We prove the maximum principle for a discontinuous Galerkin (DG) method applied to the numerical solution of traffic flow problems on networks described by the Lighthill-Whitham-Richards equations. The paper is a followup of the preceding paper, Part I, where L 2 stability of the scheme is analyzed. At traffic junctions, we consider numerical fluxes based on Godunov’s flux derived in our previous work. We also construct a new Godunov-like numerical flux taking into account right of way at the junction...

Discontinuous Galerkin method with Godunov-like numerical fluxes for traffic flows on networks. Part I: L 2 stability

Lukáš Vacek, Chi-Wang Shu, Václav Kučera (2025)

Applications of Mathematics

We study the stability of a discontinuous Galerkin (DG) method applied to the numerical solution of traffic flow problems on networks. We discretize the Lighthill-Whitham-Richards equations on each road by DG. At traffic junctions, we consider two types of numerical fluxes that are based on Godunov’s numerical flux derived in a previous work of ours. These fluxes are easily constructible for any number of incoming and outgoing roads, respecting the drivers’ preferences. The analysis is split into...

Inverse problem for a physiologically structured population model with variable-effort harvesting

Ruslan V. Andrusyak (2017)

Open Mathematics

We consider the inverse problem of determining how the physiological structure of a harvested population evolves in time, and of finding the time-dependent effort to be expended in harvesting, so that the weighted integral of the density, which may be, for example, the total number of individuals or the total biomass, has prescribed dynamics. We give conditions for the existence of a unique, global, weak solution to the problem. Our investigation is carried out using the method of characteristics...

On the relation of delay equations to first-order hyperbolic partial differential equations

Iasson Karafyllis, Miroslav Krstic (2014)

ESAIM: Control, Optimisation and Calculus of Variations

This paper establishes the equivalence between systems described by a single first-order hyperbolic partial differential equation and systems described by integral delay equations. System-theoretic results are provided for both classes of systems (among them converse Lyapunov results). The proposed framework can allow the study of discontinuous solutions for nonlinear systems described by a single first-order hyperbolic partial differential equation under the effect of measurable inputs acting on...

Vertex centred Discretization of Two-Phase Darcy flows on General Meshes

Robert Eymard, Cindy Guichard, Raphaèle Herbin, Roland Masson (2012)

ESAIM: Proceedings

This paper concerns the discretization of multiphase Darcy flows, in the case of heterogeneous anisotropic porous media and general 3D meshes used in practice to represent reservoir and basin geometries. An unconditionally coercive and symmetric vertex centred approach is introduced in this paper. This scheme extends the Vertex Approximate Gradient scheme (VAG), already introduced for single phase diffusive problems in [9], to multiphase Darcy flows....

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