A blob method for diffusion

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Title: A blob method for diffusion
Authors: Carrillo de la Plata, JA
Craig, K
Patacchini, F
Item Type: Journal Article
Abstract: As a counterpoint to classical stochastic particle methods for diffusion, we developa deterministic particle method for linear and nonlinear diffusion. At first glance, deterministicparticle methods are incompatible with diffusive partial differential equations since initial data givenby sums of Dirac masses would be smoothed instantaneously: particles do not remain particles.Inspired by classical vortex blob methods, we introduce a nonlocal regularization of our velocityfield that ensures particles do remain particles and apply this to develop a numerical blob methodfor a range of diffusive partial differential equations of Wasserstein gradient flow type, includingthe heat equation, the porous medium equation, the Fokker–Planck equation, and the Keller–Segelequation and its variants. Our choice of regularization is guided by the Wasserstein gradient flowstructure, and the corresponding energy has a novel form, combining aspects of the well-knowninteraction and potential energies. In the presence of a confining drift or interaction potential,we prove that minimizers of the regularized energy exist and, as the regularization is removed,converge to the minimizers of the unregularized energy. We then restrict our attention to nonlineardiffusion of porous medium type with at least quadratic exponent. Under sufficient regularityassumptions, we prove that gradient flows of the regularized porous medium energies converge tosolutions of the porous medium equation. As a corollary, we obtain convergence of our numericalblob method. We conclude by considering a range of numerical examples to demonstrate ourmethod’s rate of convergence to exact solutions and to illustrate key qualitative properties preservedby the method, including asymptotic behavior of the Fokker–Planck equation and critical mass ofthe two-dimensional Keller–Segel equation.
Issue Date: 1-Apr-2019
Date of Acceptance: 4-Jan-2019
URI: http://hdl.handle.net/10044/1/66347
DOI: 10.1007/s00526-019-1486-3
ISSN: 1432-0835
Publisher: Springer Verlag
Journal / Book Title: Calculus of Variations and Partial Differential Equations
Volume: 58
Copyright Statement: © The Author(s) 2019. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Sponsor/Funder: Engineering & Physical Science Research Council (EPSRC)
Funder's Grant Number: EP/P031587/1
Keywords: Science & Technology
Physical Sciences
Mathematics, Applied
General Mathematics
0101 Pure Mathematics
0102 Applied Mathematics
Publication Status: Published
Article Number: ARTN 53
Online Publication Date: 2019-02-16
Appears in Collections:Mathematics
Applied Mathematics and Mathematical Physics
Faculty of Natural Sciences

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