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

Abstract

New one-leg multistep time discretizations of nonlinear evolution equations are investigated. The main features of the scheme are the preservation of the non-negativity 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 semidiscrete 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.

Blow-up of solutions to semi-discrete parabolic-elliptic Keller-Segel models

“…Lemma 8 (Conservation of mass). Let n k ∈ L 1 (R 2 ) be a solution to (27) such that m i ∈ L 1 (R 2 ) and…”

“…Proof. Using an approximation of |x| 2 as a test function in (27) and passing to the deregularization limit (see step 5 of the proof of Theorem 2), we find that…”

Structure preserving approximation of dissipative evolution problems

“…Let us briefly mention some particular results: One-leg multistep methods and implicit Runge-Kutta methods have been investigated for the time discretization of dissipative evolution problems in [15,16]. Apart from the implicit Euler method, however, the assumptions required for the rigorous analysis of these schemes seem rather restrictive.…”

Two Structure-Preserving Time Discretizations for Gradient Flows

“…Moreover, as the result of [31] shows, the lack of smoothness may decrease the optimal convergence order. For quadratic potentials φ, the G-stability by Dahlquist allows for energy-dissipating schemes, but requires to redefine the energy as a function of (u i , u i−1 ) instead of u i alone [26]. General energy-dissipative Runge-Kutta schemes are studied in [27], but again, they do not replicate the exact energy dissipation dynamics of the problem.…”