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Publication 18-CNA-003

A Blob Method For Diffusion

Josť Antonio Carrillo
Department of Mathematics
Imperial College London

Katy Craig
Department of Mathematics
University of California
Santa Barbara, CA 93117, USA

Francesco S. Patacchini
Deparment of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15203, USA

Abstract: As a counterpoint to classical stochastic particle methods for diffusion, we develop a deterministic particle method for linear and nonlinear diffusion. At first glance, deterministic particle methods are incompatible with diffusive partial differential equations since initial data given by 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 velocity field that ensures particles do remain particles and apply this to develop a numerical blob method for a range of diffusive partial differential equations of Wasserstein gradient ow type, including the heat equation, the porous medium equation, the Fokker-Planck equation, and the Keller-Segel equation and its variants. Our choice of regularization is guided by the Wasserstein gradient ow structure, and the corresponding energy has a novel form, combining aspects of the well-known interaction and potential energies. In the presence of a confining drift or interaction potential, we prove that minimizers of the regularized energy exists and, as the regularization is removed, converge to the minimizers of the unregularized energy. We then restrict our attention to nonlinear diffusion of porous medium type with at least quadratic exponent. Under suffcient regularity assumptions, we prove that gradient ows of the regularized energies converge to solutions of the porous medium equation. As a corollary, we obtain convergence of our numerical blob method, again under sufficient regularity assumptions. We conclude by considering a range of numerical examples to demonstrate our method's rate of convergence to exact solutions and to illustrate key qualitative properties preserved by the method, including asymptotic behavior of the Fokker-Planck equation and critical mass of the two-dimensional Keller-Segel equation.

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