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2010, Volume 14, Issue 3: 935-959. Doi: 10.3934/dcdsb.2010.14.935

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A gradient flow scheme for nonlinear fourth order equations

1.
Institut für Analysis und Scientific Computing, Technische Universität Wien, 1040 Wien
2.
Faculty of Electrical Engineering and Computing, University of Zagreb, 10000 Zagreb

Received: August 2009

Revised: March 2010

Published: October 2010

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Abstract

We propose a method for numerical integration of Wasserstein gradient flows based on the classical minimizing movement scheme. In each time step, the discrete approximation is obtained as the solution of a constrained quadratic minimization problem on a finite-dimensional function space. Our method is applied to the nonlinear fourth-order Derrida-Lebowitz-Speer-Spohn equation, which arises in quantum semiconductor theory. We prove well-posedness of the scheme and derive a priori estimates on the discrete solution. Furthermore, we present numerical results which indicate second-order convergence and unconditional stability of our scheme. Finally, we compare these results to those obtained from different semi- and fully implicit finite difference discretizations.

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Mathematics Subject Classification: Primary: 65M99; Secondary: 35K35, 35Q40.

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