Section: New Results

Structure-preserving numerical methods

In [54], C. Cancès et al. propose a Finite Element scheme for the numerical approximation of degenerate parabolic problems in the form of a nonlinear anisotropic Fokker–Planck equation. The scheme is energy-stable, only involves physically motivated quantities in its definition, and is able to handle general unstructured grids. Its convergence is rigorously proven thanks to compactness arguments, under very general assumptions. Although the scheme is based on Lagrange Finite Elements of degree 1, it is locally conservative after a local post-processing giving rise to an equilibrated flux. This also allows to derive a guaranteed a posteriori error estimate for the approximate solution. Numerical experiments are presented in order to give evidence of a very good behavior of the proposed scheme in various situations involving strong anisotropy and drift terms.

In [55], C. Chainais-Hillairet and M. Herda apply an iterative energy method à la de Giorgi in order to establish $L^{\infty }$ bounds for numerical solutions of noncoercive convection-diffusion equations with mixed Dirichlet-Neumann boundary conditions.

In [23], C. Cancès, C. Chainais-Hillairet et al. study a finite volume scheme for a degenerate cross-diffusion system describing the ion transport through biological membranes. The strongly coupled equations for the ion concentrations include drift terms involving the electric potential, which is coupled to the concentrations through the Poisson equation. The finite volume scheme is based on two-point flux approximations with “double” upwind mobilities. The existence of solutions to the fully discrete scheme is proven. When the particles are not distinguishable and the dynamics is driven by cross-diffusion only, it is shown that the scheme preserves the structure of the equations like nonnegativity, upper bounds, and entropy dissipation.

In [51], C. Cancès and B. Gaudeul propose a two-point flux approximation finite volume scheme for the approximation of the solutions to an entropy dissipative cross-diffusion system. The scheme is shown to preserve several key properties of the continuous system, among which positivity and decay of the entropy. Numerical experiments illustrate the behavior of the scheme.

In [48], C. Cancès, C. Chainais-Hillairet, B. Gaudeul et al. consider an unipolar degenerate drift-diffusion system arising in the modeling of organic semiconductors. They design four different finite volume schemes based on four different formulations of the fluxes. They provide a stability analysis and existence results for the four schemes; the convergence is established for two of them.

In [24], C. Cancès et al. compare energy-stable finite volume schemes for multiphase flows in porous media with schemes based on the Wasserstein gradient flow structure of the equations, that has recently been highlighted in [3]. The model is approximated by means of the minimizing movement (or JKO) scheme, that C. Cancès et al. solve thanks to the ALG2-JKO scheme proposed in [76].

In [50], C. Cancès et al. propose a variational finite volume scheme for the computation of Wasserstein gradient flows. The discrete solution is the minimizer of a discrete action, keeping track at the discrete level of the optimal character of the gradient flow. The spatial discretization relies on upstream mobility fluxes, while an implicit linearization of the Wasserstein distance is used in order to reduce the computational cost by avoiding an inner time-stepping as in the related contributions of the literature.

In [61], T. Rey et al. present a new finite volume method for computing numerical approximations of a system of nonlocal transport equations modeling interacting species. In this work, the nonlocal continuity equations are treated as conservative transport equations with a nonlocal, nonlinear, rough velocity field. Some properties of the method are analyzed, and numerical simulations are performed.

In [15], I. Lacroix-Violet et al. are interested in the numerical integration in time of nonlinear Schrödinger equations using different methods preserving the energy or a discrete analog of it. In particular, they give a rigorous proof of the order of the relaxation method (presented in [78] for cubic nonlinearities) and they propose a generalized version that allows to deal with general power law nonlinearities. Numerical simulations for different physical models show the efficiency of these methods.