# American Institute of Mathematical Sciences

October  2011, 29(4): 1367-1391. doi: 10.3934/dcds.2011.29.1367

## A biharmonic-modified forward time stepping method for fourth order nonlinear diffusion equations

 1 Department of Mathematics, UCLA, Los Angeles, CA, 90095 2 Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences, Stillwater, OK 74078 3 Department of Mechanical Engineering, University of California, Los Angeles, CA, 90095-1555, United States

Received  December 2009 Revised  July 2010 Published  December 2010

We consider a class of splitting schemes for fourth order nonlinear diffusion equations. Standard backward-time differencing requires the solution of a higher order elliptic problem, which can be both computationally expensive and work-intensive to code, in higher space dimensions. Recent papers in the literature provide computational evidence that a biharmonic-modified, forward time-stepping method, can provide good results for these problems. We provide a theoretical explanation of the results. For a basic nonlinear 'thin film' type equation we prove $H^1$ stability of the method given very simple boundedness constraints of the numerical solution. For a more general class of long-wave unstable problems, we prove stability and convergence, using only constraints on the smooth solution. Computational examples include both the model of 'thin film' type problems and a quantitative model for electrowetting in a Hele-Shaw cell (Lu et al J. Fluid Mech. 2007). The methods considered here are related to 'convexity splitting' methods for gradient flows with nonconvex energies.
Citation: Andrea L. Bertozzi, Ning Ju, Hsiang-Wei Lu. A biharmonic-modified forward time stepping method for fourth order nonlinear diffusion equations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1367-1391. doi: 10.3934/dcds.2011.29.1367
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