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A. Jüngel, J. Milisic:
"Entropy dissipative one-leg multistep time approximations of nonlinear diffusive equations";
in: "ASC Report 42/2013", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2013, ISBN: 978-3-902627-06-3, 1 - 29.

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.

Linear multistep methods, entropy dissipation, diffusion equations, population dynamics, quantum drift-diffusion equation, Derrida-Lebowitz-Speer-Spohn equation, existence of solutions

http://www.asc.tuwien.ac.at/preprint/2013/asc42x2013.pdf