Section: New Results

large-time behavior of some numerical schemes

In [19], C. Chainais-Hillairet, A. Jüngel and S. Schuchnigg prove the time decay of fully discrete finite-volume approximations of porous-medium and fast-diffusion equations with Neumann or periodic boundary conditions in the entropy sense. The algebraic or exponential decay rates are computed explicitly. In particular, the numerical scheme dissipates all zeroth-order entropies which are dissipated by the continuous equation. The proofs are based on novel continuous and discrete generalized Beckner inequalities.

In [13], M. Bessemoulin-Chatard and C. Chainais-Hillairet study the large-time behavior of a numerical scheme discretizing drift-diffusion systems for semiconductors. The numerical method is based on a  generalization of the classical Scharfetter-Gummel scheme which allows to consider both linear or nonlinear pressure laws.They study the convergence of approximate solutions towards an approximation of the thermal equilibrium state as time tends to infinity, and obtain a decay rate by controlling the discrete relative entropy with the entropy production. This result is proved under assumptions of existence and uniform-in-time $L^{\infty }$ estimates for numerical solutions, which are then discussed.

The question of uniform-in-time $L^{\infty }$ estimates for the scheme proposed in [13] has then be tackled by M. Bessemoulin-Chatard, C. Chainais-Hillairet and A. Jüngel. The result is obtained via a Moser’s iteration technique adapted to the discrete setting. Related to this question, the existence of a positive lower bound for the numerical solution of a convection-diffusion equation has been studied by C. Chainais-Hillairet, B. Merlet and A. Vasseur. They apply a method due to De Giorgi in order to establish a positive lower bound for the numerical solution of a stationary convection-diffusion equation. These results are submitted for publication in the FVCA8 conference (to be held in June 2017).

In [11] B. Merlet et al. consider a second-order two-step time discretization of the Cahn-Hilliard equation with an analytic nonlinearity. They study the long time behavior of the discrete solution and show that if the time-step is chosen small enough, the sequence generated by the scheme converges to a steady state as time tends to infinity. Convergence rates are also provided. This parallels the behavior of the solutions of the non-discretized solutions and shows the reliability of the scheme for long time simulations. The method of proof is based on the Lojasiewicz-Simon inequality and on the study of the pseudo-energy associated with the discretization which is shown to be non-increasing.