A biharmonic-modified forward time steppingmethod for fourth order nonlinear diffusion equations

Andrea L. Bertozzi; Ning Ju; Hsiang-Wei Lu

doi:10.3934/dcds.2011.29.1367

Article Contents

2011, Volume 29, Issue 4: 1367-1391. Doi: 10.3934/dcds.2011.29.1367 open access

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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

Received: November 30, 2009

Revised: June 30, 2010

Published: September 2011

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Abstract

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.

Keywords:

Fourth order equations,
convex splitting.

Mathematics Subject Classification: 65M06, 65M12, 65M70.

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