A GRADIENT FLOW SCHEME FOR NONLINEAR FOURTH ORDER EQUATIONS

作者：Duering Bertram; Matthes Daniel; Milisic Josipa Pina

来源：Discrete and Continuous Dynamical Systems-Series B, 2010, 14(3): 935-959.

DOI：10.3934/dcdsb.2010.14.935

• Wasserstein gradient flow
• Higher-order diffusion equation
• numerical solution

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.

• 出版日期2010-10