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Publication 18-CNA-003
José Antonio Carrillo Katy Craig Francesco S. Patacchini 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.Get the paper in its entirety as 18-CNA-003.pdf |