Entropy dissipative one-leg multistep time approximations of nonlinear diffusive equations

TL;DR

This paper develops new one-leg multistep time discretization schemes for nonlinear diffusive equations that preserve nonnegativity and entropy dissipation, ensuring stability and convergence, with applications to population dynamics and quantum diffusion.

Contribution

It introduces a novel nonlinear variable transformation combined with G-stability to ensure entropy dissipation in discretizations of nonlinear diffusive equations.

Findings

Schemes preserve nonnegativity and entropy dissipation.

Proves existence of semi-discrete weak solutions for models.

Numerical experiments confirm theoretical stability and convergence.

Abstract

New one-leg multistep time discretizations of nonlinear evolution equations are investigated. The main features of the scheme are the preservation of the nonnegativity and the entropy-dissipation structure of the diffusive equations. The key ideas are to combine Dahlquist's G-stability theory with entropy-dissipation methods and to introduce a nonlinear transformation of variables which provides a quadratic structure in the equations. It is shown that G-stability of the one-leg scheme is sufficient to derive discrete entropy dissipation estimates. The general result is applied to a cross-diffusion system from population dynamics and a nonlinear fourth-order quantum diffusion model, for which the existence of semi-discrete weak solutions is proved. Under some assumptions on the operator of the evolution equation, the second-order convergence of solutions is shown. Moreover, some numerical experiments for the population model are presented, which underline the theoretical results.