Long-time behavior of a finite volume discretization for a fourth order diffusion equation

Abstract

Abstract. We consider a non-standard finite-volume discretization of a strongly non-linear fourth order diffusion equation on the d-dimensional cube, for arbitrary d ≥ 1. The scheme preserves two important structural properties of the equation: the first is the interpretation as a gradient flow in a mass transportation metric, and the second is an intimate relation to a linear Fokker-Planck equation. Thanks to these structural properties, the scheme possesses two discrete Lyapunov functionals. These functionals approximate the entropy and the Fisher information, respectively, and their dissipation rates converge to the optimal ones in the discrete-to-continuous limit. Using the dissipation, we derive estimates on the long-time asymptotics of the discrete solutions. Finally, we present results from numerical experiments which indicate that our discretization is able to capture significant features of the complex original dynamics, even with a rather coarse spatial resolution.

Entropy dissipation of Fokker-Planck equations on graphs

“…For example, Mass and Erbar studied the discrete heat flow, and further gave the Ricci curvature lower bound in the discrete space [10]. More generalizations are followed in [11,13,17]. Mielke proposed the discrete reaction diffusion equation [20].…”

Discretization of flux-limited gradient flows: $\Gamma $-convergence and numerical schemes

“…Further, recalling the lower bound (16) on c k and estimate (21), we obtain for all sufficiently large k thaẗ…”

“…Hence, it suffices to consider a sequence (γ k ) such that E τ k (γ k ) converges to a finite value. From (21), one directly concludes k-uniform boundedness of˜c k dγ k . Thanks to the bound (16) on c k , it follows for every t > 0 that γ k 's mass in |x − y| ≥ τ + t goes to zero as…”

“…This is extremely efficient in one space dimension [7,22,23], but becomes significantly more cumbersome -and difficult to analyze -in multiple dimensions [5,12,13,18]. Various alternatives to the Lagrangian approach are available, including finite volume methods [21], blob methods [11] etc.…”

“…To connect this auxiliary result to the claim of the Lemma, recall that η Proof. We use the estimates from Lemma 2: thanks to (21), the E τ k are bounded below for all sufficiently small k. And thanks to (20), the γ's in the sublevels of E τ k have uniformly bounded second moment, hence are relatively compact with respect to narrow convergence. Moreover, it is easily seen that E τ k is the sum of three convex (in the sense of convex combinations of measures) functionals, and thus is lower semi-continuous with respect to narrow convergence.…”