  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.
