# American Institute of Mathematical Sciences

2010, 14(3): 935-959. doi: 10.3934/dcdsb.2010.14.935

## A gradient flow scheme for nonlinear fourth order equations

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

Received  August 2009 Revised  March 2010 Published  July 2010

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.
Citation: Bertram Düring, Daniel Matthes, Josipa Pina Milišić. A gradient flow scheme for nonlinear fourth order equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 935-959. doi: 10.3934/dcdsb.2010.14.935
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