[Paper Review] Entropy-dissipative discretization of nonlinear diffusion equations and discrete Beckner inequalities

Claire Chainais-Hillairet, Ansgar Jüngel+1 more|arXiv (Cornell University)|Mar 15, 2013

Advanced Mathematical Modeling in Engineering20 references4 citations

TL;DR

This paper presents an entropy-dissipative finite-volume scheme for nonlinear diffusion equations (porous-medium and fast-diffusion types), proving algebraic or exponential decay of discrete zeroth- and first-order entropies. The key contribution is the development of novel generalized Beckner inequalities that ensure the discrete scheme inherits the entropy dissipation structure of the continuous PDE, with explicit decay rates derived via systematic integration by parts and discrete Gronwall-type estimates.

ABSTRACT

The time decay of fully discrete finite-volume approximations of porous-medium and fast-diffusion equations with Neumann or periodic boundary conditions is proved 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. Furthermore, the exponential decay of some first-order entropies is proved in the continuous and discrete case using systematic integration by parts. Numerical experiments in one and two space dimensions illustrate the theoretical results and indicate that some restrictions on the parameters seem to be only technical.

Motivation & Objective

• To develop a fully discrete finite-volume scheme that preserves the entropy dissipation structure of nonlinear diffusion equations.
• To establish explicit algebraic or exponential decay rates for discrete zeroth- and first-order entropies in the scheme.
• To derive and prove generalized Beckner inequalities for both continuous and discrete settings to link entropy dissipation to entropy decay.
• To extend the entropy-dissipation method to fully discrete schemes, ensuring numerical schemes inherit the long-time behavior of the continuous PDE.
• To validate the theoretical decay rates through numerical experiments in one and two space dimensions.

Proposed method

• A finite-volume discretization is constructed for the porous-medium and fast-diffusion equations with Neumann or periodic boundary conditions.
• The scheme is proven to dissipate all zeroth-order entropies that are dissipated by the continuous equation, using discrete generalized Beckner inequalities.
• Systematic integration by parts is applied to prove exponential decay of first-order entropies in both continuous and discrete cases.
• A novel discrete nonlinear Gronwall lemma is derived to estimate decay rates, with a key estimate involving the inverse of a function defined via integral of the reciprocal of a convex function.
• Theoretical decay rates are derived using the interplay between entropy, entropy dissipation, and the generalized Beckner inequalities.
• Numerical experiments in 1D and 2D are performed to validate the theoretical decay rates and assess parameter dependencies.

Experimental results

Research questions

• RQ1 Can a fully discrete finite-volume scheme be constructed that preserves the entropy dissipation structure of the continuous porous-medium and fast-diffusion equations?
• RQ2 What are the explicit algebraic or exponential decay rates for discrete zeroth- and first-order entropies in such a scheme?
• RQ3 Can generalized Beckner inequalities be derived for both continuous and discrete settings to relate entropy to entropy dissipation?
• RQ4 To what extent do the theoretical decay rates match numerical simulations in one and two space dimensions?
• RQ5 Are the restrictions on parameters in the theoretical analysis purely technical, or do they reflect fundamental limitations?

Key findings

• The finite-volume scheme dissipates all zeroth-order entropies that are dissipated by the continuous PDE, ensuring discrete entropy decay.
• Explicit algebraic or exponential decay rates are derived for the discrete entropy functionals, depending on the value of β and the dimension d.
• The generalized Beckner inequalities are proven for both continuous and discrete cases, forming the core analytical tool for establishing decay rates.
• Exponential decay of first-order entropies is rigorously proven in both continuous and discrete settings using integration by parts and the new Beckner inequalities.
• Numerical experiments confirm the theoretical decay rates and suggest that some parameter restrictions may be technical rather than fundamental.
• The discrete Gronwall lemma with a convex function f allows for precise decay estimates, leading to explicit upper bounds on the decay of the entropy sequence.