Communications in Mathematical Sciences

Volume 18 (2020)

Number 5

Fully discrete positivity-preserving and energy-dissipating schemes for aggregation-diffusion equations with a gradient-flow structure

Pages: 1259 – 1303

DOI: https://dx.doi.org/10.4310/CMS.2020.v18.n5.a5

Authors

Rafael Bailo (Department of Mathematics, Imperial College London, United Kingdom)

José A. Carrillo (Department of Mathematics, Imperial College London, United Kingdom)

Jingwei Hu (Department of Mathematics, Purdue University, West Lafayette, Indiana, U.S.A.)

Abstract

We propose fully discrete, implicit-in-time finite-volume schemes for a general family of non-linear and non-local Fokker–Planck equations with a gradient-flow structure, usually known as aggregation-diffusion equations, in any dimension. The schemes enjoy the positivity-preservation and energy-dissipation properties, essential for their practical use. The first-order scheme verifies these properties unconditionally for general non-linear diffusions and interaction potentials, while the second-order scheme does so provided a CFL condition holds. Sweeping dimensional splitting permits the efficient construction of these schemes in higher dimensions while preserving their structural properties. Numerical experiments validate the schemes and show their ability to handle complicated phenomena typical in aggregation-diffusion equations, such as free boundaries, metastability, merging and phase transitions.

Keywords

gradient flows, finite-volume methods, fully discrete schemes, positivity preservation, energy dissipation, integro-differential equations

2010 Mathematics Subject Classification

35Q70, 35Q91, 45K05, 65M08

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J.A.C. was partially supported by the EPSRC grant number EP/P031587/1.

J.H. was supported by NSF grant DMS-1620250 and NSF CAREER grant DMS-1654152.

Received 13 December 2018

Accepted 13 February 2020

Published 23 September 2020