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

New one-leg multistep time discretizations of nonlinear evolution equations are investigated. The main features of the scheme are the preservation of the nonneg- ativity 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 assump- tions 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.

Derrida–Lebowitz–Speer–Spohn equation ; diffusion equations ; entropy dissipation ; existence of solutions ; linear multistep methods ; population dynamics ; quantum drift-diffusion equation

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