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CNA Seminar/Colloquium/Joint Pitt-CNA Colloquium
Francesco Patacchini
Carnegie Mellon University
Title: A blob method for diffusion

Abstract: We derive a new deterministic particle method for linear and nonlinear diffusion. Inspired by classical vortex blob methods, we introduce a nonlocal regularization of our velocity field which ensures that initial Dirac masses remain Dirac masses as time evolves. We apply this to develop a numerical blob method for a range of diffusive partial differential equations of Wasserstein gradient flow type, including the heat, porous medium, Fokker--Planck, and Keller--Segel equations. Our choice of regularization is guided by this Wasserstein gradient flow structure, and the resulting regularized internal energy has a novel form combining aspects of the well-known interaction and internal energies. When restricting to diffusion of porous medium type with at least quadratic exponent, we show that, under sufficient regularity assumptions, the gradient flow for the regularized energy $\Gamma$-converges to the solution of the porous medium equation. We consider a range of numerical examples that demonstrate our method's rate of convergence to exact solutions and illustrate key qualitative properties, including asymptotic behavior of the Fokker-Planck equation and critical mass of the two-dimensional Keller--Segel equation. This is joint work with J. A. Carrillo and K. Craig.

Recording: http://mm.math.cmu.edu/recordings/cna/FrancescoPatacchini_small.mp4
Date: Tuesday, October 17, 2017
Time: 1:30 pm
Location: Wean Hall 7218
Submitted by:  Ian Tice