A gradient flow scheme for nonlinear fourth order equations

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

A gradient flow scheme for nonlinear fourth order equations

Bertram Düring ^1, , Daniel Matthes ^1, and Josipa Pina Milišić ^2,

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

Abstract
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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.

Keywords: Wasserstein gradient flow, numerical solution., Higher-order diffusion equation.

Mathematics Subject Classification: Primary: 65M99; Secondary: 35K35, 35Q4.

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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