An energy and momentum conserving collisional bracket for the guiding-centre Vlasov–Maxwell–Landau model

This paper proposes a metric bracket for representing Coulomb collisions in the so-called guiding-centre Vlasov–Maxwell–Landau model. The bracket is manufactured to preserve the same energy and momentum functionals as does the Vlasov–Maxwell part and to simultaneously satisfy a revised version of the H-theorem, where the equilibrium distributions with respect to collisional dynamics are identified as Maxwellians. This is achieved by exploiting the special projective nature of the Landau collision operator and the simple form of the system's momentum functional. A discussion regarding a possible extension of the results to electromagnetic drift-kinetic and gyrokinetic systems is included. We anticipate that energy conservation and entropy dissipation can always be manufactured whereas guaranteeing momentum conservation is a delicate matter yet to be resolved.

About the Use of Entropy Production for the Landau–Fermi–Dirac Equation

In this paper, we present new estimates for the entropy dissipation of the Landau–Fermi–Dirac equation (with hard or moderately soft potentials) in terms of a weighted relative Fisher information adapted to this equation. Such estimates are used for studying the large time behaviour of the equation, as well as for providing new a priori estimates (in the soft potential case). An important feature of such estimates is that they are uniform with respect to the quantum parameter. Consequently, the same estimations are recovered for the classical limit, that is the Landau equation.

Finite Volume approximation of a two-phase two fluxes degenerate Cahn-Hilliard model

We study a time implicit Finite Volume scheme for degenerate Cahn-Hilliard model proposed in [W. E and P. Palffy-Muhoray. Phys. Rev. E , 55:R3844–R3846, 1997] and studied mathematically by the authors in [C. Canc\`es, D. Matthes, and F. Nabet. Arch. Ration. Mech. Anal. , 233(2):837-866, 2019]. The scheme is shown to preserve the key properties of the continuous model, namely mass conservation, positivity of the concentrations, the decay of the energy and the control of the entropy dissipation rate. This allows to establish the existence of a solution to the nonlinear algebraic system corresponding to the scheme. Further, we show thanks to compactness arguments that the approximate solution converges towards a weak solution of the continuous problems as the discretization parameters tend to 0. Numerical results illustrate the behavior of the numerical model.

Lagrangian Representations for Linear and Nonlinear Transport

In this note we present a unifying approach for two classes of first order partial differential equations: we introduce the notion of Lagrangian representation in the settings of continuity equation and scalar conservation laws. This yields, on the one hand, the uniqueness of weak solutions to transport equation driven by a two dimensional BV nearly incompressible vector field. On the other hand, it is proved that the entropy dissipation measure for scalar conservation laws in one space dimension is concentrated on countably many Lipschitz curves.

Study of Flux Limiters to Minimize the Numerical Dissipation Based on Entropy-Consistent Scheme

The use of limiters may impact both accuracy and resolution of a solution. And the accuracy and resolution are highly dependent on the amount of numerical dissipation in a scheme, so the ability of limiters to control numerical dissipation should be improved. In this view, based on the examination of several classical limiters to control dissipation, a class of general piecewise-linear flux limiters (termed GPL limiters) are presented in this paper for Multi-step time-space-separated unsteady schemes. The GPL limiters can satisfy the second-order TVD criterion and contain some existing limiters such as Superbee and Minmod. Using the decrement of discrete total entropy to represent the amount of numerical dissipation, an entropy dissipation function of GPL limiters is defined with three parameter variables. By proving the monotonicity of this function, a new GPL type limiter (named MDF individually), which can minimize the numerical dissipation and improve the calculation accuracy, is proposed. A high resolution entropy-consistent scheme is obtained by MDF limiter, which will be proved to satisfy entropy stability and entropy consistency. Computational results of this scheme for several 1-D and 2-D Euler test cases are presented, demonstrating the accuracy, monotonicity and robustness of MDF limiter.