Section: New Results

Structure preserving numerical methods

In [7], Clément Cancès and Cindy Guichard proposed in the case of a simple degenerate parabolic equation a nonlinear Control Volume Finite Element (CFVE) scheme that was able to preserve at the discrete level some important features of the continuous problem, namely the positivity of the solution, the decay of the physical energy. The scheme is based on a suitable upwinding procedure and inherits key properties from the Two-Point Flux Approximation (TPFA) finite volume scheme even though the method is not monotone. The convergence of the scheme towards the solution of the continuous problem was also established. In [22], Clément Cancès, Moustafa Ibrahim, and Mazen Saad extend the approach of [7] to the case of the Keller-Segel system with volume filling effect. In [11], Ahmed Ait Hammou Oulhaj, Clément Cancès, and Claire Chainais-Hillairet extend this approach to the Richards equation modeling unsaturated flow in porous media.

In presence of strong anisotropy, the methodology described above may lack robustness: the method is first order accurate, but the error constant may become large in some particularly unfavorable situations. This motivated the development of a new family of schemes with locally positive metric tensor (this denomination was chosen in reference Otto's contribution [90]). The methodology is first developed by Clément Cancès and Cindy Guichard for the so-called Vertex Approximate Gradient (VAG) scheme [79] in [21]. The newly developed method is second order accurate in space and much more robust with respect to the anisotropy than the one of [7] based on upwinding. Then Clément Cancès, Claire Chainais-Hillairet, and Stella Krell extend the methodology to Discrete Duality Finite Volume (DDFV) schemes in [32] and [19].

In [11] (see also the short version [31]), Ahmed Ait Hammou Oulhaj propose an upstream mobility TPFA finite volume scheme for solving a degenerate cross-diffusion problem modeling the flow of two fluids in a porous medium. The scheme has the remarkable property to preserve at the discrete level the local conservation of mass, the positivity of the solution, the decay of the energy. Moreover, the scheme provides a control on the entropy dissipation rate. Thanks to these properties, the convergence of the scheme is established. Numerical simulation show the great robustness of the scheme.

In [37], Clément Cancès and Flore Nabet propose an upstream mobility TPFA finite volume scheme for solving the degenerate Cahn-Hilliard problem. The scheme is designed in order to maintain the positivity of the phase volume fractions, the local conservation of mass and the decay of the energy.

Many applications involve partial differential equations which admits nontrivial steady state solutions. The design of schemes which are able to describe correctly these equilibrium states may be challenging for numerical methods, in particular for high order ones. In [29], inspired by micro-macro decomposition methods for kinetic equations, Lorenzo Pareschi and Thomas Rey present a class of schemes which are capable to preserve the steady state solution and achieve high order accuracy for a class of time dependent partial differential equations including nonlinear diffusion equations and kinetic equations. Extension to systems of conservation laws with source terms are also discussed.