A blob method for diffusion

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.