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A Finite-Volume Method for Nonlinear Nonlocal Equations with a Gradient Flow Structure

Published online by Cambridge University Press:  28 November 2014

José A. Carrillo ,
Alina Chertock  and
Yanghong Huang
José A. Carrillo
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
Alina Chertock
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA
Yanghong Huang*
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
*
*Email addresses:carrillo@imperial.ac.uk(J. A. Carrillo), chertock@math.ncsu.edu(A. Cherock), yanghong.huang@imperial.ac.uk(Y. Huang)
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Abstract

We propose a positivity preserving entropy decreasing finite volume scheme for nonlinear nonlocal equations with a gradient flow structure. These properties allow for accurate computations of stationary states and long-time asymptotics demonstrated by suitably chosen test cases in which these features of the scheme are essential. The proposed scheme is able to cope with non-smooth stationary states, different time scales including metastability, as well as concentrations and self-similar behavior induced by singular nonlocal kernels. We use the scheme to explore properties of these equations beyond their present theoretical knowledge.

Keywords

65M0835R0934B10Finite-volume methodgradient flownonlinear nonlocal equations

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Type
Research Article
Information
Communications in Computational Physics , Volume 17 , Issue 1 , January 2015 , pp. 233 - 258
Copyright
Copyright © Global Science Press Limited 2015 

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